Nonparametric estimation for linear SPDEs from local measurements

TL;DR

This paper develops rate-optimal, robust nonparametric estimators for the leading coefficient of linear SPDEs using localized continuous observations, with theoretical analysis and numerical validation.

Contribution

It introduces new nonparametric estimation methods for linear SPDEs from local measurements, achieving parametric rates and robustness to perturbations.

Findings

Establishes asymptotic properties of estimators in fixed time and shrinking spatial resolution regimes.

Provides scaling limits of the PDE and SPDE on expanding domains.

Demonstrates estimator robustness to lower order operator perturbations.

Abstract

The coefficient function of the leading differential operator is estimated from observations of a linear stochastic partial differential equation (SPDE). The estimation is based on continuous time observations which are localised in space. For the asymptotic regime with fixed time horizon and with the spatial resolution of the observations tending to zero, we provide rate-optimal estimators and establish scaling limits of the deterministic PDE and of the SPDE on growing domains. The estimators are robust to lower order perturbations of the underlying differential operator and achieve the parametric rate even in the nonparametric setup with a spatially varying coefficient. A numerical example illustrates the main results.

Figures

Equations

$$dX(t)=A_{\vartheta }X(t)dt+BdW(t),t\in [0,T],$$

$$A_{\vartheta }={\Delta }_{\vartheta }+A_0,\mathcal{D}(A_{\vartheta })=H_0^1(\Lambda )\cap H^2(\Lambda ),$$

$$\{\begin{array}{l}dX\left(t\right)=A_{\vartheta }X\left(t\right)dt+BdW\left(t\right),0<t⩽T,\\ X\left(0\right)=X_0,\\ X(t){|}_{\partial \Lambda }=0,0<t⩽T,\end{array}$$

$$X(t)=S_{\vartheta }(t)X_0+{\int }_0^t{S}_{\vartheta }(t-s)BdW(s).$$

$$d⟨X(t),z⟩=⟨X(t),A_{\vartheta }^{\ast }z⟩dt+d⟨BW\left(t\right),z⟩.$$

$$⟨X(t),z⟩=⟨S_{\vartheta }\left(t\right)X_0,z⟩+{\int }_0^t⟨S_{\vartheta }^{\ast }\left(t-s\right)z,BdW\left(s\right)⟩.$$

$$\mathrm{Cov}(\ell (t,z),\ell (t^{\prime },z^{\prime }))={\int }_0^{t\wedge t^{\prime }}⟨B^{\ast }{S}_{\vartheta }^{\ast }\left(t-s\right)z,B^{\ast }{S}_{\vartheta }^{\ast }\left(t^{\prime }-s\right)z^{\prime }⟩ds.$$

$${\Lambda }_{\delta ,x_0}$$

$$z_{\delta ,x_0}(x)$$

$$X_{\delta ,x_0}(t)$$

$$X_{\delta ,x_0}^{\Delta }(t)$$

$$d{X}_{\delta ,x_0}\left(t\right)=⟨X\left(t\right),A_{\vartheta }^{\ast }{K}_{\delta ,x_0}⟩dt+\Vert B^{\ast }{K}_{\delta ,x_0}\Vert d\overline{W}\left(t\right)$$

$$A_{0,\delta ,x_0}^{\ast }z$$

$$\frac{d}{dt}w(t)={\delta }^{-2}(A_{\vartheta ,\delta ,x_0}^{\ast }{S}_{\vartheta ,\delta ,x_0}^{\ast }(t{\delta }^{-2})z{)}_{\delta ,x_0}=A_{\vartheta }^{\ast }w(t).$$

$$\frac{d}{dt}u(t)=A_{\vartheta }^{\ast }u(t),t⩾0,u(0)=z_{\delta ,x_0}.\text{qed}$$

$$\Psi (z,z^{\prime })$$

$$\Psi \left(\Delta K,\Delta K\right)⩽\Vert B_{0,x_0}^{\ast }{\Vert }_{L^2({\mathbb{R}}^d)}^2{\int }_0^{\infty }\Vert e^{s\Delta }\Delta K{\Vert }_{L^2({\mathbb{R}}^d)}^{2}ds⩽\frac{\Vert B_{0,x_0}^{\ast }{\Vert }_{L^2({\mathbb{R}}^d)}^2}{2}\Vert \nabla K{\Vert }_{L^2({\mathbb{R}}^d)}^2,$$

$$\dot{X}\left(t,x\right)=A_{\vartheta }X\left(t,x\right)+\sigma (x)\dot{W}\left(t,x\right),0<t⩽T,x\in \Lambda .$$

$$\Psi (\Delta z,z^{\prime })=-\frac{{\sigma }^2(x_0)}{2}⟨z,z^{\prime }{⟩}_{L^2({\mathbb{R}}^d)}$$

$$\text{Cov}({\delta }^{-1}{X}_{\delta ,x_0}(t{\delta }^2),{\delta }^{-1}{X}_{\delta ,x_0}(t^{\prime }{\delta }^2))$$

$$={\int }_0^{t\wedge t^{\prime }}{⟨B_{\delta ,x_0}^{\ast }{S}_{\vartheta ,\delta ,x_0}^{\ast }\left(t-s\right)K,B_{\delta ,x_0}^{\ast }{S}_{\vartheta ,\delta ,x_0}^{\ast }\left(t^{\prime }-s\right)K⟩}_{L^2({\Lambda }_{\delta ,x_0})}ds.$$

$$\frac{d}{dt}{u}^{(\delta )}(t)$$

$$\frac{d}{dt}u(t)=\vartheta (x_0)\Delta u(t),u(0)=z,$$

$$\{\begin{array}{l}dY\left(t\right)=\vartheta (x_0)\Delta Y(t)dt+B_{0,x_0}dW(t),t>0,\\ Y\left(0\right)=0.\end{array}$$

$${\int }_0^{t\wedge t^{\prime }}⟨B_{\delta ,x_0}^{\ast }{S}_{\vartheta ,\delta ,x_0}^{\ast }(t-s)(z{|}_{{\Lambda }_{\delta ,x_0}}),B_{\delta ,x_0}^{\ast }{S}_{\vartheta ,\delta ,x_0}^{\ast }(t^{\prime }-s)(z^{\prime }{|}_{{\Lambda }_{\delta ,x_0}}){⟩}_{L^2({\Lambda }_{\delta ,x_0})}ds.$$

$$\mathrm{Cov}(Z_0(t,z),Z_0(t^{\prime },z^{\prime }))={\int }_0^{t\wedge t^{\prime }}⟨B_{0,x_0}^{\ast }{e}^{\vartheta (x_0)(t-s)\Delta }z,B_{0,x_0}^{\ast }{e}^{\vartheta (x_0)(t^{\prime }-s)\Delta }{z}^{\prime }{⟩}_{L^2({\mathbb{R}}^d)}ds.$$

$${\int }_0^T⟨S_{\vartheta }(t)X_0,(\Delta z{)}_{\delta ,x_0}{⟩}^{2}dt=o({\ell }_{d,2}(\delta {)}^{-1}{\delta }^{\beta }),\delta \to 0,$$

$${\hat{\vartheta }}_{\delta }^{LS}=\frac{{\int }_0^T{X}_{\delta ,x_0}^{\Delta }(t)d{X}_{\delta ,x_0}(t)}{{\int }_0^T{X}_{\delta ,x_0}^{\Delta }(t{)}^{2}dt},$$

$$d{X}_{\delta ,x_0}(t)=\vartheta m_{\vartheta }(t)dt+\Vert K{\Vert }_{L^2({\mathbb{R}}^d)}d\tilde{W}(t),t\in [0,T],$$

$$m_{\vartheta }(t)$$

$$\mathcal{L}\left(\vartheta ,X_{\delta ,x_0}\right)$$

$${\hat{\vartheta }}_{\delta }^A=\frac{{\int }_0^T{X}_{\delta ,x_0}^{\Delta }(t)d{X}_{\delta ,x_0}\left(t\right)}{{\int }_0^T{X}_{\delta ,x_0}^{\Delta }(t{)}^{2}dt}.$$

Full text

MSC 2000 subject classification: Primary: 60H15, 60F05; Secondary: 62G05, 35J15

Keywords: Nonparametric estimation, fourth moment theorem, Gaussian process, Feynman-Kac, localised scaling limits.

1 Introduction

While there is a large amount of work on probabilistic, analytical and recently also computational aspects of stochastic partial differential equations (SPDEs), many natural statistical questions are open. With this work we want to enlarge the scope of statistical methodology in two major directions. First, we consider observations of a solution path that are local in space and we ask whether the underlying differential operator or rather its local characteristics can be estimated from this local information only. Second, we allow the coefficients in the differential operator to vary in space and we pursue nonparametric estimation of the coefficient functions, as opposed to parametric estimation approaches for finite-dimensional global parameters in the coefficients. Naturally, both directions are intimately connected.

As a concrete model we consider the parabolic SPDE

[TABLE]

with the second-order differential operator $A_{\vartheta}z:=\text{div}(\vartheta\nabla z)+\langle a,\nabla z\rangle+bz$ on some bounded domain $\Lambda\subseteq\mathbb{R}^{d}$ with Dirichlet boundary conditions, see Section 2 for formal details. The coefficient functions $\vartheta,a,b$ are unknown on $\Lambda$ and we aim at estimating $\vartheta:\Lambda\to\mathbb{R}^{+}$, which models the diffusivity in a stochastic heat equation. The functions $a,b$ as well as the operator $B$ in front of the driving space-time white noise process $dW$ form an unknown nuisance part. Linear SPDEs of this form appear in many applications, including neuroscience (Walsh, [38]), oceanography (Frankignoul, [14]), geostatistics (Sigrist et al., [36]), surface growth (Edwards & Wilkinson, [10]) and finance (Cont, [8]).

Measurements of a solution process $X$ necessarily have a minimal spatial resolution $\delta>0$ and we dispose of the observations $\langle X(t),K_{\delta,x_{0}}\rangle$, where the solution is integrated in the spatial domain against a kernel function $K_{\delta,x_{0}}$ with support of diameter $\delta$ around some $x_{0}\in\Lambda$. We keep the time span $T$ fixed and construct an estimator, called proxy MLE, which for the resolution asymptotics $\delta\to 0$ converges at rate $\delta$ to $\vartheta(x_{0})$ and satisfies a CLT, which we derive in the case of a local multiplication covariance operator $B$ in the SPDE. Another estimator, the so-called augmented MLE, will even converge under far more general conditions and exhibit a smaller asymptotic variance, but requires a second local observation process $\langle X(t),\Delta K_{\delta,x_{0}}\rangle$ in terms of the Laplace operator $\Delta$. Clearly, if we have access to these observations around all $x_{0}\in\Lambda$, then both estimators can be used to estimate the diffusivity function $\vartheta$ nonparametrically on all of $\Lambda$.

These results are statistically remarkable. First of all, even for the parametric case that $\vartheta$ is a constant, it is not immediately clear that $\vartheta$ is identified (i.e., exactly recovered) from local observations in a shrinking neighbourhood around some $x_{0}\in\Lambda$ only. Probabilistically, this means that the local observation laws are mutually singular for different values of $\vartheta$. What is more, the bias-variance trade-off paradigm in nonparametric statistics does not apply: asymptotic bias and standard deviation are both of order $\delta$ and the CLT provides us even with a simple pointwise confidence interval for $\vartheta$. The robustness of the estimators to lower order parts in the differential operator and unknown $B$ is very attractive for applications. The rate $\delta$ is shown to be the best achievable rate in a minimax sense even for constant $\vartheta$ without nuisance parts.

The fundamental probabilistic structure behind these results is a universal scaling limit of the observation process for $\delta\to 0$. At a highly localised level, the differential operator $A_{\vartheta}$ behaves like $\vartheta(x_{0})\Delta$, as expressed in Corollary 3.6 below, and the construction of the estimators shows a certain scaling invariance with respect to $B$. To study these scaling limits, we need to consider the deterministic PDE on growing domains via the stochastic Feynman-Kac approach and to deduce tight asymptotics for the action of the semigroup and the heat kernels. Further tools like the fourth moment theorem or the Feldman-Hajek Theorem rely on the underlying Gaussian structure, but extensions to semi-linear SPDEs seem possible.

Let us compare our localisation approach to the spectral approach, introduced by Huebner et al., [18] and then in Huebner & Rozovskii, [20] for parametric estimation. In the simplest case $A_{\vartheta}=\vartheta\Delta$ for some $\vartheta>0$ and $B$ commuting with $A_{\vartheta}$, the SPDE solution can be expressed in the eigenbasis of the Laplace operator $\Delta$. If the first $N$ coefficient processes (Fourier modes of $X$) are observed, then a maximum-likelihood estimator for $\vartheta$ is asymptotically efficient as $N\to\infty$. This approach has turned out to be very versatile, allowing also for estimating time-dependent $\vartheta(t)$ nonparametrically (Huebner & Lototsky, [19]) or to cover nonlinear SPDEs (Cialenco & Glatt-Holtz, [6], Pasemann & Stannat, [31]). In particular, it helps to understand that the coefficient in the leading order of the differential operator can be estimated with better rates than lower order coefficients. The methodology, however, is intrinsically bound to observations in the spectral domain and to operators $A_{\vartheta}$ whose eigenfunctions, at least in the leading order, are independent of $\vartheta$. In contrast, we work with local observations in space and the unknown spectrum of the operators $A_{\vartheta}$ does not harm us. More conceptually, we rely on the local action of the differential operator $A_{\vartheta}$, while the spectral approach also applies to an abstract operator in a Hilbert space setting.

Our case of spatially varying coefficients has been considered first by Aihara & Sunahara, [3] (with $a=b=0$) in a filtering problem. The corresponding nonparametric estimation problem is then addressed by Aihara & Bagchi, [2] with a sieve least squares estimator, but they achieve consistency only for global observations with a growing time horizon $T\to\infty$. In a stationary one-dimensional setting Bibinger & Trabs, [4] ask whether the parameter $\vartheta>0$ can be estimated when observing the solution only at $x_{0}$ over a fixed time interval $[0,T]$. Interestingly, in the case $B=\sigma^{2}I$ the parameter $\vartheta$ cannot be recovered if the level $\sigma$ of the space-time white noise is unknown (see also lower bounds of Hildebrandt & Trabs, [17]). For a recent and exhaustive survey on statistics for SPDEs we refer to Cialenco, [5].

In Section 2 the SPDE and the observation model are introduced and in Section 3 the scaling properties along with the resolution level $\delta$ are discussed. Section 4 derives our estimators via a least-squares and a likelihood approach and provides basic insight into their error analysis. The main convergence results as well as a minimax lower bound are presented in Section 5. The findings are illustrated by a numerical example in Section 6. While the main steps in the proofs are presented together with the results, all more technical arguments are delegated to the Appendix.

2 The model

2.1 Notation

Let $\Lambda$ be a bounded open set in $\mathbb{R}^{d}$ with $C^{2}$-boundary $\partial\Lambda$ and consider $L^{2}(\Lambda)$ with the usual $L^{2}$-norm $\lVert\cdot\rVert:=\lVert\cdot\rVert_{L^{2}(\Lambda)}$. For any open set $U$ in $\mathbb{R}^{d}$ and any linear operator $A:L^{2}(U)\rightarrow L^{2}(U)$ let $\lVert A\rVert_{L^{2}(U)}:=\lVert A\rVert_{L^{2}(U)\rightarrow L^{2}(U)}$ denote the operator norm, and let $H^{k}(U)$ for $k\in\mathbb{N}$ be the $L^{2}$-Sobolev spaces. Define $H_{0}^{1}(\Lambda)$ as the closure of $C_{c}^{\infty}(\overline{\Lambda})$ in $H^{1}(\Lambda)$. We write $\langle\cdot,\cdot\rangle_{\mathbb{R}^{d}}$ for the Euclidean inner product and $|\cdot|$ for the norm. Let us define a second order elliptic operator with Dirichlet boundary conditions

[TABLE]

where $\Delta_{\vartheta}z=\text{div}(\vartheta\nabla z)=\sum_{i=1}^{d}\partial_{i}(\vartheta\partial_{i}z)$ is the weighted Laplace operator with spatially varying diffusivity $\vartheta\in C^{1+\alpha}(\overline{\Lambda})$ for $\alpha>0$, $\min_{x\in\overline{\Lambda}}\vartheta(x)>0$, and where $A_{0}z=\langle a,\nabla z\rangle_{\mathbb{R}^{d}}+bz$ with functions $a\in C^{1+\alpha}(\overline{\Lambda};\mathbb{R}^{d})$, $b\in C^{\alpha}(\overline{\Lambda})$. The regularity conditions on $\vartheta,a,b$ are such that the deterministic PDE $\tfrac{d}{dt}u(t)=A_{\vartheta}^{*}u(t)$ with initial value $z\in L^{2}(\Lambda)$ has a sufficiently smooth solution (see proof of Proposition 3.5 below). Let $(S_{\vartheta}(t))_{t\geqslant 0}$ denote the analytic semigroup on $L^{2}(\Lambda)$ generated by $A_{\vartheta}$ (cf. Theorem 3.1.3 of Lunardi, [28]), while $(e^{t\Delta})_{t\geqslant 0}$ is the heat semigroup on $L^{2}(\mathbb{R}^{d})$ generated by $\Delta=\Delta_{1}$ with domain $H^{2}(\mathbb{R}^{d})$.

2.2 The SPDE model

Throughout this work $T<\infty$ is fixed. Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{0\leqslant t\leqslant T},\mathbb{P})$ be a filtered probability space with a cylindrical Brownian motion $W$ on $L^{2}(\Lambda)$ ($dW$ is also referred to as space-time white noise), and let $B:L^{2}(\Lambda)\rightarrow L^{2}(\Lambda)$ be a bounded linear operator, which is not assumed to be trace class. We study the linear stochastic partial differential equation

[TABLE]

with deterministic initial value $X_{0}\in L^{2}(\Lambda)$.

Our statistical analysis below relies on linear functionals of $X(t)$ rather than on $X(t)$ itself. We therefore use the weak solution concept of Da Prato & Zabczyk, [9]. If $\int_{0}^{T}\lVert S_{\vartheta}\left(t\right)B\rVert_{HS(L^{2}(\Lambda))}^{2}dt<\infty$ with Hilbert-Schmidt norm $\lVert\cdot\rVert_{HS(L^{2}(\Lambda))}$, then the unique weak solution $(X(t))_{0\leqslant t\leqslant T}$ of the SPDE (2.1) is given by the variation of constants formula, cf. Theorem 5.4 of Da Prato & Zabczyk, [9],

[TABLE]

It takes values in $L^{2}(\Lambda)$ and satisfies for $z\in H_{0}^{1}(\Lambda)\cap H^{2}(\Lambda)$

[TABLE]

Clearly, for $z\in L^{2}(\Lambda)$

[TABLE]

If $\int_{0}^{T}\lVert S_{\vartheta}\left(t\right)B\rVert_{HS(L^{2}(\Lambda))}^{2}dt=\infty$, then the stochastic integral in (2.2) is well-defined only in a space of distributions. For example, if $H^{-s}(\Lambda)$ is a fractional Sobolev space of negative order with $s>d/4$, then the natural embedding $\iota:L^{2}(\Lambda)\rightarrow H^{-s}(\Lambda)$ is a Hilbert-Schmidt operator such that $\int_{0}^{T}\lVert\iota S_{\vartheta}\left(t\right)B\rVert_{HS(L^{2}(\Lambda),H^{-s}(\Lambda))}^{2}dt<\infty$, and $X(t)$ takes values in $H^{-s}(\Lambda)$ (cf. Remark 5.6 of Hairer, [16]). Still, (2.3) and (2.4) remain valid, if $\langle X(t),z\rangle$ and $\langle X(t),A_{\vartheta}^{*}z\rangle$ are interpreted as dual pairings between $H^{-s}(\Lambda)$ and its dual space for $z\in C_{c}^{\infty}(\overline{\Lambda})$.

On the other hand, denote the right hand side of the equation in (2.4) by $\ell(t,z)$ and observe that it is always well-defined for any $z\in L^{2}(\Lambda)$, independent of the space in which (2.2) makes sense, cf. Lemma 2.4.2 of Liu & Röckner, [26]. The resulting process $\ell:=(\ell(t,z))_{0\leqslant t\leqslant T,z\in L^{2}(\Lambda)}$ thus extends the linear forms $z\mapsto\langle X(t),z\rangle$ from $C_{c}^{\infty}(\overline{\Lambda})$ to $L^{2}(\Lambda)$. It has the following properties.

2.1 Proposition**.**

$\ell$* is a Gaussian process with mean function $(t,z)\mapsto\left\langle S_{\vartheta}\left(t\right)X_{0},z\right\rangle$ and covariance function at $0\leqslant t,t^{\prime}\leqslant T$, $z,z^{\prime}\in L^{2}(\Lambda)$ given by*

[TABLE]

Moreover, $\ell$ satisfies (2.3) for $z\in H_{0}^{1}(\Lambda)\cap H^{2}(\Lambda)$, if $\langle X(t),z\rangle$ and $\langle X(t),A_{\vartheta}^{*}z\rangle$ are replaced by $\ell(t,z)$ and $\ell(t,A_{\vartheta}^{*}z)$.

Proof.

By (2.4), $\ell(t,z)$ for $z\in L^{2}(\Lambda)$ is Gaussian with mean $\left\langle S_{\vartheta}\left(t\right)X_{0},z\right\rangle$. Itô’s isometry (Proposition 4.28 of Da Prato & Zabczyk, [9]) proves (2.5). If $z\in C_{c}^{\infty}(\overline{\Lambda})$, then $\ell(t,z)=\langle X(t),z\rangle$ satisfies $d\,\ell(t,z)=\ell(t,A_{\vartheta}^{*}z)dt+d\left\langle BW\left(t\right),z\right\rangle$. This extends to $z\in H_{0}^{1}(\Lambda)\cap H^{2}(\Lambda)$ by approximation and continuity of $\ell:[0,T]\times L^{2}(\Lambda)\mapsto L^{2}(\mathbb{P})$ from (2.5). ∎

In the following, justified by this proposition, we write $\left\langle X(t),z\right\rangle$ for $0\leqslant t\leqslant T$ and $z\in L^{2}(\Lambda)$ instead of $\ell(t,z)$.

2.3 Local observations

Throughout this work let $x_{0}\in\Lambda$ be fixed. The following rescaling will be useful in the sequel: for $z\in L^{2}(\mathbb{R}^{d})$ and $\delta>0$ set

[TABLE]

Fix a function $K\in H^{2}(\mathbb{R}^{d})$, called kernel, with compact support in $\Lambda_{\delta,x_{0}}$. The compact support ensures that $K_{\delta,x_{0}}$ is localized around $x_{0}$ and $K_{\delta,x_{0}}\in H_{0}^{1}(\Lambda)\cap H^{2}(\Lambda)$, $\lVert K_{\delta,x_{0}}\rVert=\lVert K\rVert_{L^{2}(\mathbb{R}^{d})}$. Local measurements of $X$ at $x_{0}$ with resolution level $\delta$ until time $T$ are described by the real-valued processes $X_{\delta,x_{0}}=(X_{\delta,x_{0}}(t))_{0\leqslant t\leqslant T}$, $X_{\delta,x_{0}}^{\Delta}=(X_{\delta,x_{0}}^{\Delta}(t))_{0\leqslant t\leqslant T}$,

[TABLE]

Note that it is sufficient to observe $X_{\delta,x}(t)$ for $x$ in a neighbourhood of $x_{0}$ in order to provide us with $X_{\delta,x_{0}}^{\Delta}(t)=\Delta X_{\delta,\cdot}(t)|_{x=x_{0}}$. Examples for $K$ can be found in Section 6.

The process $X_{\delta,x_{0}}$ satisfies $X_{\delta,x_{0}}(0)=\langle X_{0},K_{\delta,x_{0}}\rangle$ and

[TABLE]

with the scalar Brownian motion $\overline{W}(t)=\langle BW(t),K_{\delta,x_{0}}\rangle/\|B^{\ast}K_{\delta,x_{0}}\|$, whenever $\lVert B^{*}K_{\delta,x_{0}}\rVert>0$.

3 Scaling assumptions

3.1 Rescaled operators and semigroups

Let us study how $A_{\vartheta}^{*}$ and $S_{\vartheta}^{*}(t)$ act on localized functions $z_{\delta,x_{0}}$. For this note first that $A_{\vartheta}^{*}=\Delta_{\vartheta}+A_{0}^{*}$ with $A_{0}^{*}z=-\text{div}(az)+bz$ has domain $\mathcal{D}(A_{\vartheta}^{*})=H_{0}^{1}(\Lambda)\cap H^{2}(\Lambda)$. For $\delta>0$ define similarly the operator $A_{\vartheta,\delta,x_{0}}^{*}=\Delta_{\vartheta(x_{0}+\delta\cdot)}+A_{0,\delta,x_{0}}^{*}$ with domain $\mathcal{D}(A_{\vartheta,\delta,x_{0}}^{*})=H_{0}^{1}(\Lambda_{\delta,x_{0}})\cap H^{2}(\Lambda_{\delta,x_{0}})$, where for $z\in C_{c}^{\infty}({\Lambda}_{\delta,x_{0}})$

[TABLE]

The operator $A_{\vartheta,\delta,x_{0}}^{*}$ generates again an analytic semigroup $(S_{\vartheta,\delta,x_{0}}^{*}(t))_{t\geqslant 0}$ on $L^{2}(\Lambda_{\delta,x_{0}})$ (Lemma 7.3.4 of Pazy, [32]). The following scaling properties are fundamental for our analysis:

3.1 Lemma**.**

For $\delta>0$:

1. (i)

If $z\in H_{0}^{1}(\Lambda_{\delta,x_{0}})\cap H^{2}(\Lambda_{\delta,x_{0}})$, then $A_{\vartheta}^{*}z_{\delta,x_{0}}=\delta^{-2}(A_{\vartheta,\delta,x_{0}}^{*}z)_{\delta,x_{0}}$. 2. (ii)

If $z\in L^{2}(\Lambda_{\delta,x_{0}})$, then $S_{\vartheta}^{*}\left(t\right)z_{\delta,x_{0}}=(S_{\vartheta,\delta,x_{0}}^{*}(t\delta^{-2})z)_{\delta,x_{0}}$, $t\geqslant 0$.

Proof.

It suffices to prove the result for $z\in C_{c}^{\infty}(\overline{\Lambda}_{\delta,x_{0}})$. In this case, (i) follows immediately, noting that $z_{\delta,x_{0}}\in C_{c}^{\infty}(\overline{\Lambda})$. For (ii) set $w(t)=(S_{\vartheta,\delta,x_{0}}^{*}(t\delta^{-2})z)_{\delta,x_{0}}\in L^{2}(\Lambda)$. As $(S_{\vartheta,\delta,x_{0}}^{*}(t))_{t\geqslant 0}$ is an analytic semigroup, we have $S_{\vartheta,\delta,x_{0}}^{*}(t)z\in\mathcal{D}(A_{\vartheta,\delta,x_{0}}^{*})=H_{0}^{1}(\Lambda_{\delta,x_{0}})\cap H^{2}(\Lambda_{\delta,x_{0}})$ and so by (i)

[TABLE]

Since $w(0)=z_{\delta,x_{0}}$, we conclude that $w(t)=S_{\vartheta}^{*}(t)z_{\delta,x_{0}}$ from $u(t)=S_{\vartheta}^{*}(t)z_{\delta,x_{0}}$ being the unique solution in $C([0,\infty);H_{0}^{1}(\Lambda)\cap H^{2}(\Lambda))\cap C^{1}([0,\infty);L^{2}(\Lambda))$ of

[TABLE]

Applying $S_{\vartheta}^{*}(t)$ to a localized function $z_{\delta,x_{0}}$ is therefore equivalent to applying a different semigroup, rescaled in time and space, to the fixed function $z$.

3.2 Scaling of $B$

Just as with $A_{\vartheta}^{*}$ we also need that $B^{*}$ behaves nicely when applied to localized functions. For this we shall assume a scaling limit for $B^{*}$, which does not degenerate in combination with $K$.

3.2 Assumption**.**

There are bounded linear operators $B_{\delta,x_{0}},B_{0,x_{0}}:L^{2}(\mathbb{R}^{d})\rightarrow L^{2}(\mathbb{R}^{d})$ such that $B^{*}(z_{\delta,x_{0}})=(B_{\delta,x_{0}}^{*}z)_{\delta,x_{0}}$ for $z\in L^{2}(\mathbb{R}^{d})$ with support in $\Lambda_{\delta,x_{0}}$ and $B_{\delta,x_{0}}^{*}z\rightarrow B_{0,x_{0}}^{*}z$ for $z\in L^{2}(\mathbb{R}^{d})$ and $\delta\rightarrow 0$. Introducing

[TABLE]

assume the non-degeneracy conditions $\lVert B_{0,x_{0}}^{*}K\rVert_{L^{2}(\mathbb{R}^{d})}>0$, $\Psi(\Delta K,\Delta K)>0$.

3.3 Remark**.**

We shall see that after an appropriate rescaling $\vartheta(x_{0})^{-1}\Psi(z,z^{\prime})$ becomes the limiting covariance in (2.5) (cf. Proposition A.8 below). $\Psi(\Delta K,\Delta K)$ is always nonnegative and finite because

[TABLE]

using $\lVert e^{s\Delta}\Delta K\rVert_{L^{2}(\mathbb{R}^{d})}^{2}=\langle e^{2s\Delta}\Delta K,\Delta K\rangle_{L^{2}(\mathbb{R}^{d})}$ and $\int_{0}^{\infty}e^{2s\Delta}\Delta K\,ds=-\frac{1}{2}K$.

3.4 Examples**.**

1. (a)

For a bounded continuous function $\sigma:\mathbb{R}^{d}\to(0,\infty)$ define the multiplication operator $M_{\sigma}:L^{2}(\Lambda)\rightarrow L^{2}(\Lambda)$, $M_{\sigma}z(x):=(\sigma z)(x)=\sigma(x)z(x)$. With $B=B^{*}=M_{\sigma}$ the SPDE in (2.1) can be written informally as

[TABLE]

Note that $B^{*}$ commutes with $A_{\vartheta}$ only if $\sigma$ is constant. For $z\in L^{2}(\Lambda_{\delta,x_{0}})$ we find that $B^{*}z_{\delta,x_{0}}=(M_{\sigma(\delta\cdot+x_{0})}z)_{\delta,x_{0}}$ and so $B_{\delta,x_{0}}=M_{\sigma(\delta\cdot+x_{0})}$. Then $\lVert B_{\delta,x_{0}}^{*}z-\sigma(x_{0})z\rVert_{L^{2}(\mathbb{R}^{d})}\rightarrow 0$ for $z\in L^{2}(\mathbb{R}^{d})$, $\delta\rightarrow 0$, and thus $B_{0,x_{0}}^{*}=M_{\sigma(x_{0})}$ is the multiplication operator on $L^{2}(\mathbb{R}^{d})$ with the constant $\sigma(x_{0})$. For $z\in H^{2}(\mathbb{R}^{d})$, $z^{\prime}\in L^{2}(\mathbb{R}^{d})$ we have (cf. Remark 3.3)

[TABLE]

and integration by parts shows $\Psi(\Delta K,\Delta K)=\frac{\sigma^{2}(x_{0})}{2}\lVert\nabla K\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$. The non-degeneracy conditions are clearly satisfied. 2. (b)

Let $\sigma$ be as in (a) and consider with bounded $\eta\in C^{2}(\mathbb{R}^{d})$, $\min_{x\in\mathbb{R}^{d}}\eta(x)>0$, the perturbed multiplication operator $B=B^{*}=M_{\sigma}+(-\Delta_{\eta})^{-\gamma}$, $\gamma>0$. By functional calculus $B^{*}z_{\delta,x_{0}}=(B_{\delta,x_{0}}^{*}z)_{\delta,x_{0}}$ for $z\in L^{2}(\Lambda_{\delta,x_{0}})$ with $B_{\delta,x_{0}}=M_{\sigma(\delta\cdot+x_{0})}+\delta^{2\gamma}(-\Delta_{\eta(\delta\cdot+x_{0})})^{-\gamma}$ and $\lVert B_{\delta,x_{0}}^{*}z-\sigma(x_{0})z\rVert_{L^{2}(\mathbb{R}^{d})}\rightarrow 0$ for $z\in L^{2}(\mathbb{R}^{d})$, $\delta\rightarrow 0$. $B_{0,x_{0}}$ and $\Psi(\Delta K,\Delta K)$ are as in (a). 3. (c)

Assumption 3.2 excludes $B=(-\Delta)^{-\gamma}$, $\gamma>0$, a typical choice to obtain smooth solutions $X$, cf. Da Prato & Zabczyk, [9, Chapter 5.5]. Indeed, by (b) $B_{\delta,x_{0}}^{*}=\delta^{2\gamma}(-\Delta)^{-\gamma}$ and so $B_{0,x_{0}}^{*}=0$, violating the non-degeneracy conditions. This problem can be solved by modifying the test function $K_{\delta,x_{0}}$. For example, if $A_{\vartheta}=\vartheta\Delta$ for constant $\vartheta>0$ and $X_{0}\in\mathcal{D}((-\Delta)^{\gamma})$, then assume we have access to $\langle X(t),(-\Delta)^{\gamma}K_{\delta,x_{0}}\rangle$, $\langle X(t),(-\Delta)^{\gamma}\Delta K_{\delta,x_{0}}\rangle$ instead of (2.6), (2.7). Since $B$ and $A_{\vartheta}$ commute, $\langle X(\cdot),(-\Delta)^{\gamma}K_{\delta,x_{0}}\rangle$ has the same distribution as $\langle\tilde{X}(\cdot),K_{\delta,x_{0}}\rangle$, where $\tilde{X}$ corresponds to the SPDE (2.1) with $B=I$ and $\tilde{X}_{0}=(-\Delta)^{\gamma}X_{0}$, and so Assumption 3.2 is satisfied.

3.3 From bounded to unbounded domains

Lemma 3.1 and Assumption 3.2 allow us to rewrite the covariance function of $X_{\delta,x_{0}}$ for $t,t^{\prime}\geqslant 0$:

[TABLE]

In order to see how this behaves when $\delta\rightarrow 0$, note that the domain $\Lambda_{\delta,x_{0}}$ grows and we find from (3.1) that $A_{\vartheta,\delta,x_{0}}^{*}K\rightarrow\vartheta(x_{0})\Delta K$ in $L^{2}(\mathbb{R}^{d})$. This motivates the following result, proved in Appendix A.2.

3.5 Proposition**.**

For $t>0$:

1. (i)

If $\delta>0$ and $z\in C(\overline{\Lambda}_{\delta,x_{0}})$, then $|(S_{\vartheta,\delta,x_{0}}^{*}\left(t\right)z)(x)|\leqslant c_{3}e^{c_{1}\delta^{2}t}(e^{c_{2}t\Delta}\left|z\right|)(x)$ for all $x\in\Lambda_{\delta,x_{0}}$ with universal constants $c_{1},c_{2},c_{3}>0$. 2. (ii)

If $z\in L^{2}(\mathbb{R}^{d})$, then $S_{\vartheta,\delta,x_{0}}^{*}\left(t\right)(z|_{\Lambda_{\delta,x_{0}}})\rightarrow e^{\vartheta(x_{0})t\Delta}z$ in $L^{2}(\mathbb{R}^{d})$ for $\delta\rightarrow 0$.

This means that the solution of

[TABLE]

on $L^{2}(\Lambda_{\delta,x_{0}})$ with bounded domain $\Lambda_{\delta,x_{0}}$ converges to the solution of the heat equation

[TABLE]

on $L^{2}(\mathbb{R}^{d})$ with unbounded domain $\mathbb{R}^{d}$. This scaling limit, which seems natural but is nevertheless non-trivial, lies at the heart of the analysis for the covariance function. Yet, the convergence in Proposition 3.5(ii) does not hold uniformly in $z$, which complicates the approximations in the covariance analysis.

Applying the proposition to (3.4) also implies a scaling limit for the SPDE in (2.1), where for simplicity a zero initial condition is assumed:

3.6 Theorem**.**

Let $X_{0}=0$ and set $Z_{\delta}(t,z):=\delta^{-1}\langle X(t\delta^{2}),(z|_{\Lambda_{\delta,x_{0}}})_{\delta,x_{0}}\rangle$ for $t\geqslant 0$, $z\in L^{2}(\mathbb{R}^{d})$. Under Assumption 3.2 the finite dimensional distributions of $(Z_{\delta}(t,z))_{t\geqslant 0,z\in L^{2}(\mathbb{R}^{d})}$ converge to those of $(Z_{0}(t,z))_{t\geqslant 0,z\in L^{2}(\mathbb{R}^{d})}$, $Z_{0}(t,z)=\langle Y(t),z\rangle_{L^{2}(\mathbb{R}^{d})}$, solving the stochastic heat equation on $L^{2}(\mathbb{R}^{d})$ with space-time white noise $dW$ on $L^{2}(\mathbb{R}^{d})$:

[TABLE]

Proof.

According to (3.4) $Z_{\delta}$ is a centered Gaussian process with covariance function $\text{Cov}(Z_{\delta}(t,z),Z_{\delta}(t^{\prime},z^{\prime}))$ for $t,t^{\prime}\geqslant 0$, $z,z^{\prime}\in L^{2}(\mathbb{R}^{d})$ equal to

[TABLE]

It is enough to show that this converges to

[TABLE]

Approximating $z$ by continuous functions, the semigroup bound in Proposition 3.5(i) gives $\sup_{0<\delta\leqslant 1}\sup_{s\leqslant t}\lVert S_{\vartheta,\delta,x_{0}}^{*}(s)(z|_{\Lambda_{\delta,x_{0}}})\rVert_{L^{2}(\Lambda_{\delta,x_{0}})}<\infty$, while Assumption 3.2 implies $B_{\delta,x_{0}}^{*}u^{(\delta)}\to B_{0,x_{0}}^{*}u$ for any $u^{(\delta)}\to u$, invoking the uniform boundedness principle. By Proposition 3.5(ii) we have $S_{\vartheta,\delta,x_{0}}^{*}(s)(z|_{\Lambda_{\delta,x_{0}}})\rightarrow e^{\vartheta(x_{0})s\Delta}z$ in $L^{2}(\mathbb{R}^{d})$. Arguing in the same way with respect to $z^{\prime}$, the dominated convergence theorem shows the claim. ∎

This theorem demonstrates the strength of local measurements that at small scales the highest order differential operator dominates, together with the local coefficient $\vartheta(x_{0})$ and the local operator $B_{0,x_{0}}$ in the noise.

3.4 The initial condition

For $X_{0}$ we require the following scaling behaviour:

3.7($z$;$\beta$) Assumption****.

For $\beta>0$ and $z\in H^{2}(\mathbb{R}^{d})$ with compact support in $\Lambda_{\delta,x_{0}}$ for $\delta>0$, the initial condition $X_{0}$ satisfies

[TABLE]

where $\ell_{d,2}(\delta)=\log(\delta^{-1})$ for $d=2$ and $\ell_{d,2}(\delta)=1$ otherwise.

Under this assumption the initial condition becomes negligible in the estimation procedure. It is true under general conditions.

3.8 Lemma**.**

Assumption 3.7($z$;$\beta$) Assumption($z$;$\beta$) is satisfied for all $z\in H^{2}(\mathbb{R}^{d})$ with compact support in $\Lambda_{\delta,x_{0}}$ for $\delta>0$ and

1. (i)

$\beta=2$* if $X_{0}\in L^{p}(\Lambda)$ for some $p>2$, in particular if $X_{0}\in C(\overline{\Lambda})$,* 2. (ii)

$\beta=3$* if $X_{0}\in\mathcal{D}(A_{\vartheta})$.*

Proof.

This follows from Lemma A.7(ii,iii) below, noting $\gamma(d,p)>2$ in (ii) for $p>2$, $d\geqslant 1$. ∎

4 The two estimation methods

4.1 Motivation and construction

We give two motivations for deriving the estimators in the parametric case $A_{\vartheta}=\vartheta\Delta$ with constant $\vartheta>0$, $B=I$. As we shall see later, these estimators will then work quite universally for nonparametric specifications of $\vartheta$ and general $A_{\vartheta}$ and $B$.

Least squares approach.

In the deterministic situation of (2.8) without driving noise (i.e. $A_{\vartheta}=\vartheta\Delta$ and $B=0$) we recover $\vartheta$ via $\dot{X}_{\delta,x_{0}}(t)=\vartheta X_{\delta,x_{0}}^{\Delta}(t)$ for all $t\in[0,T]$. A standard least-squares ansatz in the noisy situation would therefore lead to an estimator $\hat{\vartheta}=\operatorname{argmin}_{\vartheta}\int_{0}^{T}(\dot{X}_{\delta,x_{0}}(t)-\vartheta X_{\delta,x_{0}}^{\Delta}(t))^{2}dt$. While this itself is certainly not well defined, the corresponding normal equations yield the feasible estimator

[TABLE]

compare with the approach by Maslowski & Tudor, [29] for fractional noise.

Likelihood approach.

Assume that only $X_{\delta,x_{0}}$ in (2.8) is observed with $A_{\vartheta}=\vartheta\Delta$, $B=I$. Denote by $\mathbb{P}_{\vartheta}^{\delta,x_{0}}$ and $\mathbb{P}_{0}$ the laws of $X_{\delta,x_{0}}$ and $\lVert K\rVert_{L^{2}(\mathbb{R}^{d})}\overline{W}$ on the canonical path space $(C([0,T]),\lVert\cdot\rVert_{\infty})$ equipped with its Borel sigma algebra. Typically, the likelihood of $\mathbb{P}_{\vartheta}^{\delta,x_{0}}$ with respect to $\mathbb{P}_{0}$ is determined via Girsanov’s theorem. This is not immediate from (2.8), because $X_{\delta,x_{0}}^{\Delta}$ cannot be obtained from $X_{\delta,x_{0}}$ for fixed $x_{0}$. Therefore we employ Liptser & Shiryaev, [25, Theorem 7.17] and write $X_{\delta,x_{0}}$ as the diffusion-type process

[TABLE]

with a different scalar Brownian motion $\tilde{W}=(\tilde{W}(t))_{0\leqslant t\leqslant T}$, adapted to the filtration generated by $X_{\delta,x_{0}}$, and

[TABLE]

Girsanov’s theorem in the form of Liptser & Shiryaev, [25, Theorem 7.18] applies and we find that $\mathbb{P}_{\vartheta}^{\delta,x_{0}}$ has with respect to $\mathbb{P}_{0}$ the likelihood

[TABLE]

Computing the conditional expectation $m_{\vartheta}(t)$ is a non-explicit filtering problem, even in the parametric case $A_{\vartheta}=\vartheta\Delta$. In particular, $m_{\vartheta}$ depends on $\vartheta$ in a highly nonlinear way. We pursue two different modifications:

Augmented MLE.

If we observe $X_{\delta,x_{0}}^{\Delta}$ additionally, then we can just replace the conditional expectation $m_{\vartheta}(t)$ in the likelihood by its argument $X_{\delta,x_{0}}^{\Delta}(t)$, which is in particular independent of $\vartheta$. Maximizing this modified likelihood leads to the augmented MLE

[TABLE]

We remark that $\hat{\vartheta}_{\delta}^{A}=\hat{\vartheta}_{\delta}^{LS}$.

Proxy MLE.

If we do not dispose of additional observations, we can approximate $m_{\vartheta}(t)$ by the conditional expectation $\mathbb{E}_{\vartheta}[X_{\delta,x_{0}}^{\Delta}(t)\,|\,X_{\delta,x_{0}}(t)]$. In our simplified setup with $A_{\vartheta}=\vartheta\Delta$ and $B=I$ the projected finite-dimensional process $(\langle X(t),z_{i}\rangle)_{1\leqslant i\leqslant m}$ for $z_{i}\in L^{2}(\Lambda)$ admits a stationary solution $\langle X(t),z_{i}\rangle=\int_{-\infty}^{t}\langle S_{\vartheta}(t-s){z_{i}},dW(s)\rangle$, $i=1,\ldots,m$, with a two-sided cylindrical Brownian motion $(W(t))_{t\in\mathbb{R}}$, provided the variances remain finite. Note that we need not require $X_{0}=\int_{-\infty}^{t}S_{\vartheta}^{*}(t-s)dW(s)$ to exist, but only that the finite-dimensional projection $(\langle X_{0},z_{i}\rangle)_{1\leqslant i\leqslant m}$ follows the right law, which is always feasible. If we choose $z_{1}=K_{\delta,x_{0}}$, $z_{2}=\Delta K_{\delta,x_{0}}$, then the process $(X_{\delta,x_{0}},X_{\delta,x_{0}}^{\Delta})$ is stationary with

[TABLE]

In general, $\langle(-\Delta)^{-1}K_{\delta,x_{0}},K_{\delta,x_{0}}\rangle$ may not exist, but if we assume the existence of $\tilde{K}\in H^{4}(\mathbb{R}^{d})$ with $\Delta\tilde{K}=K$ and compact support in $\Lambda_{\delta,x_{0}}$, then by the scaling in Lemma 3.1, $\operatorname{Var}(X_{\delta,x_{0}}(t))=\frac{\delta^{2}}{2\vartheta}\lVert\nabla\tilde{K}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}<\infty$ follows. In this situation we therefore find that $\mathbb{E}_{\vartheta}[X_{\delta,x_{0}}^{\Delta}(t)\,|\,X_{\delta,x_{0}}(t)]$ equals

[TABLE]

This expression is again independent of $\vartheta$. Using it as an approximation of $m_{\vartheta}(t)$ in the likelihood and neglecting the boundary terms in the identity $2\int_{0}^{T}X_{\delta,x_{0}}(t)dX_{\delta,x_{0}}\left(t\right)=(X_{\delta,x_{0}}^{2}(T)-X_{\delta,x_{0}}^{2}(0))-\langle X_{\delta,x_{0}}\rangle_{T}$ with quadratic variation $\langle X_{\delta,x_{0}}\rangle_{T}$, we obtain the proxy MLE

[TABLE]

Note that the quadratic variation $\langle X_{\delta,x_{0}}\rangle_{T}=T\|B^{*}K_{\delta,x_{0}}\|^{2}$ is known to us from observing $X_{\delta,x_{0}}$ continuously in time.

4.1 Remark**.**

A sufficient condition for the existence of $\tilde{K}$ is $\int_{\mathbb{R}^{d}}K(x)dx=0$, $\int_{\mathbb{R}^{d}}xK(x)dx=0$ by Lemma A.5(iii) below.

4.2 Basic error decomposition

Let us discuss the basic error analysis for the augmented MLE $\hat{\vartheta}_{\delta}^{A}$ and the proxy MLE $\hat{\vartheta}_{\delta}^{P}$ in the general nonparametric framework of Section 2. Since we only use local measurements around $x_{0}$, it might be expected that asymptotically we are lead to estimating $\vartheta(x_{0})$. Let us point out that this is indeed true, but a priori not clear because all values of $\vartheta(x)$ enter into the observations $X_{\delta,x_{0}}$ and it must be excluded that the resulting bias spoils the estimator.

Augmented MLE.

Consider $\hat{\vartheta}_{\delta}^{A}(x_{0})=\hat{\vartheta}_{\delta}^{A}$ from (4.1). Then insertion of Equation (2.8) for $dX_{\delta,x_{0}}(t)$ yields the decomposition

[TABLE]

with

[TABLE]

Let us note that $\mathcal{I}_{\delta}^{A}$ is not the observed Fisher information in a strict sense (due to the appearance of $m_{\vartheta}$ in the likelihood), but it plays the same role, compare the analysis of the MLE for Ornstein-Uhlenbeck processes in Kutoyants, [24]. In particular, it forms the quadratic variation of the martingale $M_{\delta}^{A}$. In the specific case $A_{\vartheta}=\vartheta\Delta$ for some parametric $\vartheta>0$ the term $R_{\delta}^{A}$ vanishes, otherwise it induces a bias due to the variations of $\vartheta$ around $\vartheta(x_{0})$ and due to first and zero order differential operators that may appear in $A_{\vartheta}$.

As the error structure suggests, the augmented MLE $\hat{\vartheta}_{\delta}^{A}(x_{0})$ is a consistent estimator for $\delta\to 0$ if the observed Fisher information satisfies $\mathcal{I}_{\delta}^{A}\to\infty$. In the simple stationary case of (4.2) we obtain $\mathbb{E}[\mathcal{I}_{\delta}^{A}]=\frac{T}{2\vartheta}\langle(-\Delta)K_{\delta,x_{0}},K_{\delta,x_{0}}\rangle$, which by the scaling properties is of order $\delta^{-2}$. Physically, this can be interpreted as an increase in energy in $X_{\delta,x_{0}}^{\Delta}$ under $\delta$-localisation due to the Laplacian in the drift, while the energy from the space-time white noise remains unchanged. This is in fact the same phenomenon as the increasing signal-to-noise ratio for high Fourier modes in the spectral approach by Huebner & Rozovskii, [20].

Proxy MLE.

Consider $\hat{\vartheta}_{\delta}^{P}(x_{0})=\hat{\vartheta}_{\delta}^{P}$ from (4.3). The only stochastic part is

[TABLE]

in the denominator. In the general model (2.8) we shall see that $\mathcal{I}_{\delta}^{P}$ converges to $\vartheta(x_{0})^{-1}T\Psi(K,K)$, compare also Remark 3.3 with $K=\Delta\tilde{K}$. Asking for consistency $\hat{\vartheta}_{\delta}^{P}(x_{0})\to\vartheta(x_{0})$ leads to requiring the identity $\lVert\nabla\tilde{K}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\lVert B_{0,x_{0}}^{*}K\rVert_{L^{2}(\mathbb{R}^{d})}^{2}=2\lVert K\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\Psi(K,K)$. This does not hold for any operator $B_{0,x_{0}}$. We therefore restrict to our main specification $B=M_{\sigma}$, for which the identity holds by (3.3). In contrast to the augmented MLE, the proxy MLE works with the observation of $X_{\delta,x_{0}}$ alone, but asks for new structural assumptions on $B$ and $K$. If they are not fulfilled, other likelihood approximations should be pursued. Compare also the suboptimal behaviour of $\hat{\vartheta}_{\delta}^{P}(x_{0})$ for the kernel $K^{(2)}$ in the simulations of Section 6 below.

5 Main results

5.1 Results for the augmented MLE

Recall the function $\Psi$ from (3.2) and the error decomposition (4.4). We show first that the observed Fisher information and the bias, after rescaling, converge to deterministic quantities. The propositions are proved in Appendix A.1.

5.1 Proposition**.**

Grant Assumptions 3.2 and 3.7($z$;$\beta$) Assumption($K$;$2$). Then for any $d\geqslant 1$ as $\delta\rightarrow 0$

[TABLE]

5.2 Proposition**.**

*** Grant Assumptions 3.2, 3.7($z$;$\beta$) Assumption($K$;$2$), and for $d=1$ assume $\vartheta\in C^{1+\alpha^{\prime}}(\overline{\Lambda})$ for $\alpha^{\prime}>1/2$ and $\int_{\mathbb{R}}K(x)dx=0$. Then for $\delta\rightarrow 0$*

[TABLE]

where $\beta(x)=\Delta(\langle\nabla\vartheta(x_{0}),x\rangle_{\mathbb{R}^{d}}K)(x)-\langle\nabla\vartheta(x_{0})-a(x_{0}),\nabla K(x)\rangle_{\mathbb{R}^{d}},$ $x\in\mathbb{R}^{d}$.

From this it follows that the augmented MLE $\hat{\vartheta}_{\delta}^{A}(x_{0})$ satisfies a central limit theorem with rate $\delta$.

5.3 Theorem**.**

Grant Assumptions 3.2, 3.7($z$;$\beta$) Assumption($K$;$2$), and for $d=1$ assume $\vartheta\in C^{1+\alpha^{\prime}}(\overline{\Lambda})$ for $\alpha^{\prime}>1/2$ and $\int_{\mathbb{R}}K(x)dx=0$. Then for $\delta\rightarrow 0$

[TABLE]

and with $\mu^{A}$ from Proposition 5.2.

Proof.

In terms of $Y_{t}^{(\delta)}:=X_{\delta,x_{0}}^{\Delta}(t)/\mathbb{E}[\mathcal{I}_{\delta}^{A}]^{1/2}$ we obtain $M_{\delta}^{A}/\mathbb{E}[\mathcal{I}_{\delta}^{A}]^{1/2}=\int_{0}^{T}Y_{t}^{(\delta)}d\bar{W}(t)$, the quadratic variation of which satisfies $\int_{0}^{T}(Y_{t}^{(\delta)})^{2}dt=\mathcal{I}_{\delta}^{A}/\mathbb{E}[\mathcal{I}_{\delta}^{A}]\xrightarrow{\mathbb{P}}1$ by Proposition 5.1. A standard continuous martingale CLT, e.g. Kutoyants, [24, Theorem 1.19], shows $M_{\delta}^{A}/\mathbb{E}[\mathcal{I}_{\delta}^{A}]^{1/2}\xrightarrow{d}N(0,1)$. Moreover,

[TABLE]

due to Assumption 3.2 and $\delta^{-1}(\mathcal{I}_{\delta}^{A})^{-1}R_{\delta}^{A}\xrightarrow{\mathbb{P}}\mu^{A}$ by Proposition 5.2. We conclude by applying Slutsky’s lemma. ∎

5.4 Remarks**.**

1. (i)

Both, bias and standard deviation of $\hat{\vartheta}_{\delta}^{A}(x_{0})$, are of order $\delta$. The asymptotic bias $\mu^{A}$ is independent of $T$, while the variance $\Sigma^{A}$ decays in $T$. 2. (ii)

$B$, $\nabla\vartheta$ and $a$ appear in the limit only via the localized terms $B_{0,x_{0}}^{*}$, $\nabla\vartheta(x_{0})$, $a(x_{0})$, while $b$ does not appear at all. This demonstrates again the universality property of local measurements, in the spirit of Theorem 3.6. 3. (iii)

The estimator and thus also its asymptotic bias and variance are invariant under constant scaling factors in the kernel. In fact, using the scaling such that $\lVert K_{\delta,x_{0}}\rVert=\lVert K\rVert_{L^{2}(\mathbb{R}^{d})}$ is arbitrary, but simplifies the analysis. 4. (iv)

The additional assumptions for the convergence of the remaining bias $R^{A}_{\delta}$ in $d=1$ allow for compensating the slower decay of the heat kernel compared to $d>1$, cf. Lemma A.6(ii,iii). This is not necessary for constant $\vartheta$ and in that case Theorem 5.3 holds without these assumptions. 5. (v)

When we dispose of observations at different locations $x$, then we can estimate $\vartheta(x)$ pointwise at each location $x$. In the case of multiplicative covariance $B=M_{\sigma}$ it can be shown that estimators at different locations become asymptotically independent. The argument relies on a multivariate martingale difference CLT, using that at points $x_{0},x_{1}\in\Lambda$ the corresponding Brownian motions $\overline{W}_{0},\overline{W}_{1}$ in (2.8) are independent whenever $\operatorname{supp}(K_{\delta,x_{0}})\cap\operatorname{supp}(K_{\delta,x_{1}})=\varnothing$.

From Proposition 5.2 we see that $\mu^{A}$ vanishes if $A_{\vartheta}=\vartheta\Delta+b$ for parametric $\vartheta>0$. Another important situation where $\mu^{A}=0$ is given next.

5.5 Example**.**

(Example 3.4(a) ctd.) Let $B=M_{\sigma}$ and recall the identities $\Psi(\Delta K,\Delta K)=\frac{\sigma(x_{0})^{2}}{2}\lVert\nabla K\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$, $\Psi(\Delta K,\beta)=-\frac{\sigma(x_{0})^{2}}{2}\langle K,\beta\rangle_{L^{2}(\mathbb{R}^{d})}$ with $\beta$ from Proposition 5.2. By Lemma A.3 with $z=K$ this means

[TABLE]

and Theorem 5.3 yields

[TABLE]

In particular, if $\nabla K$ is symmetric in the sense $|\nabla K(-x)|=|\nabla K(x)|$, $x\in\mathbb{R}^{d}$, then the asymptotic bias vanishes:

[TABLE]

The rougher $K$ is, the smaller is the asymptotic variance, which bears some similarity with deconvolution problems.

If the asymptotic bias $\mu^{A}$ vanishes, we can construct a simple confidence interval in terms of the augmented MLE. Note that in the setting of Example 5.5, $\Sigma^{A}=2T^{-1}\lVert K\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\lVert\nabla K\rVert_{L^{2}(\mathbb{R}^{d})}^{-2}$ is easily accessible.

5.6 Corollary**.**

Assume the setting of Theorem 5.3, $\mu^{A}=0$ and let $\overline{\alpha}\in(0,1)$. Then the confidence interval for $\vartheta(x_{0})$

[TABLE]

with the standard normal $(1-\overline{\alpha}/2)$-quantile $q_{1-\overline{\alpha}/2}$, has asymptotic coverage $1-\overline{\alpha}$ for $\delta\to 0$.

Proof.

By Theorem 5.3 and Slutsky’s lemma applied for $\hat{\vartheta}_{\delta}^{A}\left(x_{0}\right)\xrightarrow{\operatorname{{\mathbb{P}}}}\vartheta(x_{0})$, we have

[TABLE]

noting $\mu^{A}=0$. This yields $\operatorname{{\mathbb{P}}}(\vartheta(x_{0})\in I_{1-\overline{\alpha}}^{A})\to 1-\overline{\alpha}$. ∎

5.2 Results for the proxy MLE

In the setting described in Section 4.2 we obtain a CLT for the quadratic functional $\mathcal{I}_{\delta}^{P}$. The proof uses very precise asymptotic moment calculations and the fourth moment theorem in Wiener chaos. Fundamental for this analysis is that $X_{\delta,x_{0}}(t)$ and $X_{\delta,x_{0}}(s)$ quickly decorrelate as $\delta^{-2}\lvert t-s\rvert\to\infty$, which is also predicted by the scaling limit in Corollary 3.6. Note that this method of proof might also cover time-discrete observations of $X_{\delta,x_{0}}$ if the sampling frequency increases sufficiently fast as $\delta\to 0$, but this is not pursued here.

The next assumption gathers the conditions required for the analysis of the proxy MLE.

5.7 Assumption**.**

Let $K=\Delta\tilde{K}$ for $\tilde{K}\in H^{4}(\mathbb{R}^{d})$ with compact support and let $B=M_{\sigma}$ with $\sigma\in C^{1}(\mathbb{R}^{d})$. Grant Assumption 3.7($z$;$\beta$) Assumption($\tilde{K}$;$3$), and for $d=1$ assume $\vartheta\in C^{1+\alpha^{\prime}}(\overline{\Lambda})$ for $\alpha^{\prime}>1/2$ and $\int_{\mathbb{R}}\tilde{K}(x)dx=0$.

The following proposition is proved in Appendix A.1.

5.8 Proposition**.**

Grant Assumption 5.7. Then for $\delta\rightarrow 0$:

[TABLE]

This yields asymptotic normality for the proxy MLE $\hat{\vartheta}_{\delta}^{P}(x_{0})$.

5.9 Theorem**.**

Grant Assumption 5.7. Then for $\delta\rightarrow 0$:

[TABLE]

and with $\mu_{1}^{P}$, $\Sigma^{P}$ from Proposition 5.8.

Proof.

Recall the quadratic variation $\langle X_{\delta,x_{0}}\rangle_{T}=T\lVert\sigma K_{\delta,x_{0}}\rVert^{2}$, the constant $C_{T,K}$ from Proposition 5.8 and set

[TABLE]

Write $\hat{\vartheta}_{\delta}^{P}(x_{0})=(\mathcal{I}_{\delta}^{P})^{-1}D_{T,K}\lVert\sigma K_{\delta,x_{0}}\rVert^{2}$ and decompose

[TABLE]

From the compact support of $K$ we infer for $\delta\to 0$

[TABLE]

Proposition 5.8 and the delta method (Ferguson, [13, Theorem 7]) give

[TABLE]

and, in particular, $(\mathcal{I}_{\delta}^{P})^{-1}\xrightarrow{\mathbb{P}}\vartheta(x_{0})C_{T,K}^{-1}$. The theorem follows from Slutsky’s lemma. ∎

The dependencies on $\delta,T,\vartheta,K$ in the CLT are similar as for $\hat{\vartheta}_{\delta}^{A}(x_{0})$. It is interesting to note that the asymptotic bias depends locally at $x_{0}$ on $\sigma^{2},\vartheta$ and their gradients, while $a$, $b$ do not appear at all. The asymptotic bias vanishes when $\frac{\sigma^{2}}{\vartheta}$ and $\sigma^{2}$ are constant, but also similar to Example 5.5 if $|\nabla\tilde{K}(-x)|=|\nabla\tilde{K}(x)|$, $|K(-x)|=|K(x)|$, $x\in\mathbb{R}^{d}$. As for the augmented MLE in Corollary 5.6, we obtain an asymptotic $(1-\overline{\alpha})$-confidence interval.

5.10 Corollary**.**

Grant Assumption 5.7, suppose $\mu_{1}^{P}+\mu_{2}^{P}=0$ and let $\overline{\alpha}\in(0,1)$. Then the confidence interval for $\vartheta(x_{0})$

[TABLE]

with the standard normal $(1-\overline{\alpha}/2)$-quantile $q_{1-\overline{\alpha}/2}$, has asymptotic coverage $1-\overline{\alpha}$ for $\delta\to 0$.

Let us finally compare the variance factor $\Sigma^{P}$ to $\Sigma^{A}$ from Theorem 5.3.

5.11 Lemma**.**

Under Assumption 5.7 the asymptotic variances of $\hat{\vartheta}_{\delta}^{P}$ and $\hat{\vartheta}_{\delta}^{A}$ always satisfy $\vartheta(x_{0})\Sigma^{P}\geqslant\vartheta(x_{0})\Sigma^{A}$.

Proof.

Using the tensor products $\Delta\otimes\Delta$, $f\otimes f$ and $\Delta\oplus\Delta:=I\otimes\Delta+\Delta\otimes I$, we can write for $f\in L^{2}(\mathbb{R}^{d})$, identifying $L^{2}(\mathbb{R}^{d})\otimes L^{2}(\mathbb{R}^{d})=L^{2}(\mathbb{R}^{2d})$,

[TABLE]

provided the last norm is finite, e.g. if $f=(-\Delta)^{1/2}\tilde{K}$. With this $f$ we conclude via two duality arguments, using $\Delta\tilde{K}=K$:

[TABLE]

which yields the result. ∎

Consequently, the proxy MLE has a larger variance than the augmented MLE, but the loss in precision is not severe if $K$ has a well concentrated Fourier spectrum (consider $\Delta$ as a multiplier in the Fourier domain).

Let us point out that in the one-dimensional parametric case with $A_{\vartheta}=\vartheta\partial_{xx}+a\partial_{x}+b$ and $B=M_{\sigma}$ for constant $\sigma$, Bibinger & Trabs, [4] construct least-squares estimators for $\sigma^{2}/\sqrt{\vartheta}$ and $a/\vartheta$ from discrete high frequency observations in time at two spatial points $x_{1},x_{2}$. Compared to this, the proxy MLE uses spatial averages of the solution in infinitesimally small neighbourhoods, observed continuously in time, to estimate $\vartheta$ itself, without having to know $a$ or $\sigma$. A similar phenomenon has been observed by Cialenco & Huang, [7] for discrete observations when $a=0$, but they achieve only consistent estimation of $\vartheta$ and $\sigma$. A more profound comparison of both approaches would be highly desirable.

5.3 Rate optimality

Let us address the question of optimality of the above estimators by providing a minimax lower bound. For minimax lower bounds it suffices to consider a subclass of the original model and we thus assume here that $X_{\delta,x_{0}}$ is observed with $A_{\vartheta}=\vartheta\Delta$, $B=I$ and a stationary initial condition $X_{\delta,x_{0}}$. Then the following result establishes that the rate of convergence $\delta$ is optimal and gives some lower bound for the dependence on $T$, $\vartheta$ and $K$.

5.12 Proposition**.**

Assume $A_{\vartheta}=\vartheta\Delta$, $\vartheta>0$, $B=I$, $K\in H^{1}(\mathbb{R}^{d})$ with compact support and that $X_{\delta,x_{0}}$ is stationary. For $\vartheta_{0}>0$ and $\delta\to 0$ we have the asymptotic local lower bound for the root mean squared error

[TABLE]

where $\Delta$ is the Laplace operator on $L^{2}(\mathbb{R}^{d})$, $\bar{c}>0$ is some constant and the infimum is taken over all estimators $\hat{\vartheta}$ based on observing $X_{\delta,x_{0}}$.

Proof.

The autocovariance function of the stationary process $(\delta^{-1}X_{\delta,x_{0}}(\delta^{2}t))_{t\in\mathbb{R}}$ is given by

[TABLE]

using the scaling in Lemma 3.1 and $\frac{d}{ds}S_{\vartheta,\delta,x_{0}}(s)=A_{\vartheta,\delta,x_{0}}S_{\vartheta,\delta,x_{0}}(s)$ in the last line. The covariance operator for $\delta^{-1}X_{\delta,x_{0}}(\delta^{2}\cdot)$ on $L^{2}(\mathbb{R})$ is obtained by convolution:

[TABLE]

The squared Hellinger distance $H^{2}(\vartheta,\vartheta_{0})$ between two equivalent centered Gaussian measures can be bounded in terms of the Hilbert-Schmidt norm of the covariance operators, see e.g. the proof of the Feldman-Hajek Theorem in Da Prato & Zabczyk, [9, Theorem 2.25]. For the laws of $(\delta^{-1}X_{\delta,x_{0}}(\delta^{2}t))_{t\in[0,T\delta^{-2}]}$ under $\vartheta_{0}$ and $\vartheta$ we can thus bound the corresponding Hellinger distance via

[TABLE]

Since the Hellinger distance is invariant under bi-measurable bijective transformations, $H(\vartheta,\vartheta_{0})$ denotes equally the Hellinger distance between the observation laws of $(X_{\delta,x_{0}}(t))_{t\in[0,T]}$.

Let now $\vartheta_{\delta}=\vartheta_{0}+c\delta$ for some small $c>0$, which we choose below, and assume that we can show $H^{2}(\vartheta_{\delta},\vartheta_{0})\leqslant 1$ for sufficiently small $\delta$. Then we obtain from the general lower bound scheme in Tsybakov, [37], using his Theorem 2.2(ii) and (2.9), that

[TABLE]

From this we will obtain the claimed lower bound.

In order to show $H^{2}(\vartheta_{\delta},\vartheta_{0})\leqslant 1$, denote by $\iota_{1}:H^{1}([0,T\delta^{-2}])\to L^{2}([0,T\delta^{-2}])$ the Sobolev embedding operator. It is known from Maurin’s Theorem, see e.g. the proof of Adams & Fournier, [1, Theorem 6.61], that $\iota_{1}$ is Hilbert-Schmidt with

[TABLE]

for some constant $K_{HS}>0$. By Hilbert-Schmidt norm calculus (in particular, $\lVert AB\rVert_{HS(H_{2},H_{3})}\leqslant\lVert A\rVert_{HS(H_{1},H_{3})}\lVert B\rVert_{H_{2}\to H_{1}}$ with obvious notation for the Hilbert-Schmidt and operator norms between Hilbert spaces $H_{1},H_{2},H_{3}$), the implicit restriction of the covariance operators and by the covariance bound of Lemma A.1 below we conclude for $\vartheta_{\delta}>\vartheta_{0}$ that

[TABLE]

Hence, $H^{2}(\vartheta_{\delta},\vartheta)\leqslant 1$ holds whenever

[TABLE]

Noting the convergence $\lVert(I-A_{1,\delta,x_{0}})^{-1}K\rVert_{L^{2}(\Lambda_{\delta,x_{0}})}\to\lVert(I-\Delta)^{-1}K\rVert_{L^{2}(\mathbb{R}^{d})}$ from Lemma A.1 below, we can thus find a sufficiently small constant $c^{\prime}>0$ such that, with

[TABLE]

(5.4) holds for $\vartheta_{\delta}=\vartheta_{0}+c\delta$. This yields the result. ∎

6 A numerical example

In this section we briefly illustrate the main results from above with simulation results. Let $\Lambda=(0,1)$, $T=1$, and consider the stochastic heat equation

[TABLE]

with Dirichlet boundary conditions and with spatially varying diffusivity $\vartheta$, which is smooth (true diffusivity in Figure 1 (center)). Assume that $X_{0}$ is zero, except for two equally high “peaks” at $x=0.2$ and $x=0.8$. The heat map for a typical realisation is presented in Figure 1 (left) and we see already qualitatively that the heat diffusion is higher for $x\leqslant 1/2$.

An approximate solution $\tilde{X}(t_{k},y_{j})\approx X(t_{k})(y_{j})$ is obtained with respect to a regular time-space grid $\{(t_{k},y_{j}):t_{k}=k/N,y_{j}=j/M,k=0,\dots,N,j=0,\dots,M\}$ by a semi-implicit Euler scheme and a finite difference approximation of $\Delta_{\vartheta}$ (Lord et al., [27, Section 10.5]). Since the solution is tested against functions $K_{\delta,x_{0}}$ and $\Delta K_{\delta,x_{0}}$ with small support, $M$ needs to be relatively large, while it is well-known that accurate simulation requires $N\asymp M^{2}$, see Lord et al., [27, p. 458]. We therefore choose $M=2000$, $N=10^{6}$.

Consider the kernels $K^{(1)}=\varphi^{\prime\prime\prime}$, $K^{(2)}=\varphi^{\prime}$ with a smooth bump function

[TABLE]

For $\delta\in\{0.03,0.05,0.08,0.12,0.2,0.3\}$ and $x_{0}\in(0,1)$ on a regular grid we obtain approximate local measurements $\tilde{X}_{\delta,x_{0}}$, $\tilde{X}_{\delta,x_{0}}^{\Delta}$ for $K^{(1)}$ and $K^{(2)}$, respectively, from which the augmented MLE $\hat{\vartheta}_{\delta}^{A}(x_{0})$ and the proxy MLE $\hat{\vartheta}_{\delta}^{P}(x_{0})$ are computed. For $x_{0}$ near the boundary and $i=1,2$ set

[TABLE]

Figure 1 (center) shows pointwise estimation results for $\vartheta(x_{0})$ at $\delta=0.12$ and for different $x_{0}$, while Figure 1 (right) presents a $\log_{10}$-$\log_{10}$ plot of root mean squared estimation errors at $x_{0}=0.6$ for $\delta\rightarrow 0$, obtained by 5.000 Monte-Carlo runs.

Already at the relatively large resolution $\delta=0.12$ both $\hat{\vartheta}_{\delta}^{A}(x_{0})$ and $\hat{\vartheta}_{\delta}^{P}(x_{0})$ perform surprisingly well. For $K^{(1)}$ both estimators are close together and achieve after a burn-in phase the convergence rate $\delta$, as predicted by Theorems 5.3 and 5.9. Note that $K^{(1)}=\Delta\tilde{K}$ for $\tilde{K}=\varphi^{\prime}$ and $\int_{\mathbb{R}}\tilde{K}(x)dx=0$ such that the assumptions of Theorem 5.9 are satisfied. With respect to $K^{(2)}$ those assumptions are not met and indeed $\hat{\vartheta}_{\delta}^{P}(x_{0})$ deviates considerably from $\vartheta(x_{0})$, but still seems to be consistent with rate of convergence dropping to about $\delta^{3/4}$. Estimation by $\hat{\vartheta}_{\delta}^{A}(x_{0})$ is unaffected by choosing $K^{(2)}$ instead of $K^{(1)}$ (not shown).

Appendix A Proofs

For a better understanding we structure the appendix such that the proofs for the main theorems of Section 5 are given in Section A.1. Only afterwards, we provide the technical tools used for the main proofs. Section A.2 contains analytical results for rescaled semigroups and heat kernels, while Section A.3 assembles precise asymptotics for variance and covariance expressions.

From now on, without loss of generality replace $\Lambda$ with $\Lambda-x_{0}$ and assume $x_{0}=0$. In particular, we estimate $\vartheta(0)$ and ease notation by removing the subindex $x_{0}$ and write $\Lambda_{\delta}=\Lambda_{\delta,x_{0}}$, $z_{\delta}=z_{\delta,x_{0}}$ and $X_{\delta}=X_{\delta,x_{0}}$. Unless stated otherwise, all limits are for $\delta\rightarrow 0$. $C$ always denotes a generic positive constant, which may depend on $T$, if not made explicit otherwise, and changes from line to line. $A\lesssim B$ means $A\leqslant CB$. For $z\in L^{1}(\mathbb{R}^{d})\cap L^{2}(\mathbb{R}^{d})$ define the norm

[TABLE]

and for $z$ with partial derivatives up to second order in $L^{1}(\mathbb{R}^{d})\cap L^{2}(\mathbb{R}^{d})$ set

[TABLE]

We write throughout $\langle X(t),z\rangle=\langle\tilde{X}(t),z\rangle+\langle S_{\vartheta}(t)X_{0},z\rangle$ for $z\in L^{2}(\Lambda)$ with $\langle\tilde{X}(t),z\rangle$ being defined as $\langle X(t),z\rangle$, but with $X_{0}=0$. Note that $\mathbb{E}[\langle\tilde{X}(t),z\rangle]=0$ and $\mathbb{E}[\langle X(t),z\rangle]=\langle S_{\vartheta}(t)X_{0},z\rangle$. Set also $\tilde{X}_{\delta}(t)=\langle\tilde{X}(t),K_{\delta}\rangle$, $\tilde{X}_{\delta}^{\Delta}(t)=\langle\tilde{X}(t),\Delta K_{\delta}\rangle$. We will use frequently, without explicit mention, that $\Delta K_{\delta}=\delta^{-2}(\Delta K)_{\delta}$ by Lemma 3.1.

A.1 Proofs for Section 5

Proof of Proposition 5.1.

We show the result first for $\tilde{\mathcal{I}}_{\delta}^{A}=\int_{0}^{T}\tilde{X}_{\delta}^{\Delta}(t)^{2}dt$. Propositions A.8(ii) and A.9(ii) below with $w^{(\delta)}=\Delta K$, $z=K$ yield

[TABLE]

In particular, $\operatorname{Var}(\tilde{\mathcal{I}}_{\delta}^{A})/\mathbb{E}[\tilde{\mathcal{I}}_{\delta}^{A}]^{2}\xrightarrow{\mathbb{P}}0$ and thus $\tilde{\mathcal{I}}_{\delta}^{A}/\mathbb{E}[\mathcal{\tilde{I}}_{\delta}^{A}]\xrightarrow{\mathbb{P}}1$. To finish the proof, decompose

[TABLE]

Assumption 3.7($z$;$\beta$) Assumption($K$;$2$) gives

[TABLE]

By the Cauchy-Schwarz inequality, the cross-term in (A.1) is therefore also negligible and the result follows. ∎

Proof of Proposition 5.2.

Define $\tilde{R}_{\delta}^{A}$ as $R_{\delta}^{A}$, but with respect to $\tilde{X}(\cdot)$. In terms of $\beta^{(\delta)}:=\delta^{-1}(A_{\vartheta,\delta}^{*}-\vartheta(0)\Delta)K$ we have $\delta R_{\delta}^{A}=\int_{0}^{T}X_{\delta}^{\Delta}(t)\langle X(t),\beta_{\delta}^{(\delta)}\rangle dt$ and $\delta\tilde{R}_{\delta}^{A}=\int_{0}^{T}\tilde{X}_{\delta}^{\Delta}(t)\langle\tilde{X}(t),\beta_{\delta}^{(\delta)}\rangle dt$. $\beta^{(\delta)}$ and $\beta$ correspond to $v^{(\delta)}$ and $v$ from Lemma A.5 below with $z=K$, and therefore $\beta^{(\delta)}\rightarrow\beta$ in $L^{2}(\mathbb{R}^{d})$. Decompose $R_{\delta}^{A}=\tilde{R}_{\delta}^{A}+V_{1,\delta}+V_{2,\delta}$, where

[TABLE]

We infer $\delta V_{2,\delta}\xrightarrow{\mathbb{P}}0$ from $\mathbb{E}[\delta^{2}\tilde{\mathcal{I}}_{\delta}^{A}]=O(1)$ by Proposition 5.1 and from

[TABLE]

by Lemma A.7(i) below with $u=\beta^{(\delta)}$. This, (A.2) and the Cauchy-Schwarz inequality together with

[TABLE]

also imply $\delta V_{1,\delta}\xrightarrow{\mathbb{P}}0$. By Proposition 5.1 it therefore suffices to show

[TABLE]

The convergence of $\mathbb{E}[\delta\tilde{R}_{\delta}^{A}]$ follows for $d\geqslant 2$ from Proposition A.8(iii) below with $w^{(\delta)}=\Delta K$, $z=K$, $u^{(\delta)}=\beta^{(\delta)}$, $u=\beta$. For $d=1$, $\vartheta\in C^{1+\alpha^{\prime}}(\overline{\Lambda})$ for $\alpha^{\prime}>1/2$ and $\int_{\mathbb{R}}K(x)dx=0$, it follows from Lemma A.5(ii) that there is a compactly supported $\tilde{\beta}\in H^{2}(\mathbb{R})$ with $\beta=\Delta\tilde{\beta}$, $\lVert\beta^{(\delta)}-\Delta\tilde{\beta}\rVert_{L^{1}\cap L^{2}(\mathbb{R})}\leqslant C\delta^{\alpha^{\prime}}$. Then, by polarisation and Proposition A.8(ii), $\mathbb{E}[\delta\tilde{R}_{\delta}^{A}]$ converges to

[TABLE]

Next, $\operatorname{Var}(\delta\tilde{R}_{\delta}^{A})=\operatorname{Var}(\int_{0}^{T}X_{\delta}^{\Delta}(t)\langle X(t),\beta_{\delta}^{(\delta)}\rangle dt)\rightarrow 0$ follows for $d\geqslant 2$ by Proposition A.9(i) below with $z=K$, $u^{(\delta)}=\beta^{(\delta)}$, $u=\beta$. If $\vartheta\in C^{1+\alpha^{\prime}}(\overline{\Lambda})$ and $\int_{\mathbb{R}}K(x)dx=0$, then $\operatorname{Var}(\delta\tilde{R}_{\delta}^{A})\rightarrow 0$ by Proposition A.9(ii) with $z=K$, $w^{(\delta)}=\beta^{(\delta)}$, $m=\tilde{\beta}$.∎

Proof of Proposition 5.8.

Define $\tilde{\mathcal{I}}_{\delta}^{P}$ as $\mathcal{I}_{\delta}^{P}$, but with respect to $\tilde{X}(\cdot)$. By Assumption 3.7($z$;$\beta$) Assumption($\tilde{K}$;$3$) and $K_{\delta}=(\Delta\tilde{K})_{\delta}$ we have $\delta^{-3}\int_{0}^{T}\langle S_{\vartheta}(t)X_{0},K_{\delta}\rangle^{2}dt\to 0$ whence $\delta^{-1}(\mathcal{I}_{\delta}^{P}-\tilde{\mathcal{I}}_{\delta}^{P})\to 0$ follows by $\tilde{\mathcal{I}}_{\delta}^{P}=O_{\mathbb{P}}(\delta)$ and the Cauchy-Schwarz inequality.

It remains to prove the result for $\tilde{\mathcal{I}}_{\delta}^{P}$. Note that

[TABLE]

with $Z_{\delta}:=\delta^{-3}\int_{0}^{T}(\tilde{X}_{\delta}(t)^{2}-\mathbb{E}[\tilde{X}_{\delta}(t)^{2}])dt$. $\tilde{X}_{\delta}$ is a centered Gaussian process and $Z_{\delta}$ is an element of the second Wiener chaos. By the fourth moment theorem (Nualart & Peccati, [30, Theorem 1]) it suffices to prove $\text{Var}(Z_{\delta})\xrightarrow{}\Sigma$ and $\mathbb{E}[Z_{\delta}^{4}]\rightarrow 3\Sigma^{2}$ to conclude $Z_{\delta}\xrightarrow{d}N(0,\Sigma)$. Propositions A.9(iv) and A.16 below (with $w^{(\delta)}=\Delta\tilde{K}$) provide exactly these convergences with

[TABLE]

where the last identity is (3.3). The claim follows from applying Proposition A.10 below with $z=\tilde{K}$ to the second term in (A.3) and Slutsky’s lemma. ∎

A.1 Lemma**.**

Assume the setting of Proposition 5.12 and recall the operator $C_{\vartheta,\delta}$ from (5.3). We have for $\vartheta>\vartheta_{0}>0$

[TABLE]

Moreover, we have $\lVert(I-A_{1,\delta})^{-1}K\rVert_{L^{2}(\Lambda_{\delta})}\to\lVert(I-\Delta)^{-1}K\rVert_{L^{2}(\mathbb{R}^{d})}$ for $\delta\to 0$, where $\Delta$ is the Laplace operator on $L^{2}(\mathbb{R}^{d})$.

Proof.

For simplicity write in the following proof $\vartheta\Delta$ and $e^{\vartheta t\Delta}$ instead of $A_{\vartheta,\delta}$ and $S_{\vartheta,\delta}(t)$. In the Fourier domain, the convolution operator $C_{\vartheta,\delta}$ is given by

[TABLE]

The operator $C_{\vartheta_{0},\delta}^{-1}(C_{\vartheta,\delta}-C_{\vartheta_{0},\delta})$ is expressed in the Fourier domain by multiplication with

[TABLE]

Using the description of $H^{1}(\mathbb{R})$ in the Fourier domain and functional calculus for the Laplacian $\Delta$ on $L^{2}(\Lambda_{\delta})$ yields therefore for $\vartheta>\vartheta_{0}$ that

[TABLE]

where we used in the last line $w^{-1/2}{\lambda}(1+w^{-2}\lambda^{2})^{-1}\leqslant{\lambda}^{1/2}$ for all $\lambda,w>0$. For this and similar arguments note that by spectral calculus with a self-adjoint operator $A$, e.g. $-\Delta$, we have $\lVert f(A)K\rVert\leqslant\lVert g(A)K\rVert$ whenever $\lvert f\rvert\leqslant\lvert g\rvert$ for bounded $f,g$ on the spectrum of $A$. Since $\langle-\Delta K,K\rangle_{L^{2}(\Lambda_{\delta})}=\lVert\nabla K\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$, the numerator is independent of $\delta$. For the denominator write again $A_{1,\delta}=\Delta$ and note similarly $(I+A_{1,\delta}^{2})^{-1/2}\geqslant(I-A_{1,\delta})^{-1}$, where we have explicitly, cf. Pazy, [32, Chapter 2.6],

[TABLE]

Proposition 3.5 yields then first, approximating $K$ by continuous functions, that $\lVert S_{1,\delta}(t)K\rVert_{L^{2}(\Lambda_{\delta})}\lesssim\lVert K\rVert_{L^{2}(\mathbb{R}^{d})}$ uniformly in $\delta$, and second, the convergence

[TABLE]

A.2 Analytical results

Recall that the solution of the heat equation $\frac{d}{dt}u(t)=\lambda\Delta u(t)$, $\lambda>0$, on $\mathbb{R}^{d}$ with initial value $w\in L^{2}(\mathbb{R}^{d})$ is given by the convolution

[TABLE]

with the heat kernel $q_{t}(x)=(4\pi t)^{-d/2}\exp(-|x|^{2}/(4t))$, $x\in\mathbb{R}^{d}$.

A.2 Lemma**.**

We have for $u\in L^{2}(\mathbb{R}^{d})$, $t>0$:

1. (i)

$\lVert e^{t\Delta}u\rVert_{L^{2}(\mathbb{R}^{d})}\lesssim(1\wedge t^{-d/4})\lVert u\rVert_{L^{1}\cap L^{2}(\mathbb{R}^{d})}$, $\lVert\Delta e^{t\Delta}u\rVert_{L^{2}(\mathbb{R}^{d})}\lesssim t^{-1}\lVert u\rVert_{L^{2}(\mathbb{R}^{d})}$. 2. (ii)

$xe^{\vartheta(0)t\Delta}u(x)=-2\vartheta(0)t\nabla e^{\vartheta(0)t\Delta}u(x)+e^{\vartheta(0)t\Delta}(xu)(x)$, $x\in\mathbb{R}^{d}$. 3. (iii)

$\lVert|x|^{2}e^{\vartheta(0)t\Delta}u\rVert_{L^{2}(\mathbb{R}^{d})}\lesssim(1\vee t)(1\wedge t^{-d/4})\lVert(1+|x|+|x|^{2})u\rVert_{L^{1}\cap L^{2}(\mathbb{R}^{d})}$.

Proof.

(i). For the second part use functional calculus. The first part follows from

[TABLE]

(ii). Let $i\in\{1,\dots,d\}$. The result follows from

[TABLE]

(iii). Applying the proof in (ii) twice for $i\in\{1,\dots,d\}$ gives

[TABLE]

Summing over $i$ with $v_{i}=e^{\vartheta(0)t\Delta}(x_{i}u)$ obtain from this for $\lVert|x|^{2}e^{\vartheta(0)t\Delta}u\rVert_{L^{2}(\mathbb{R}^{d})}$ up to a constant the upper bound

[TABLE]

Using $e^{\vartheta(0)t\Delta}=e^{\vartheta(0)(t/2)\Delta}e^{\vartheta(0)(t/2)\Delta}$ and the two statements in (i) yield for the first and last terms the claimed bound. For the second term integration by parts implies $\lVert\partial_{i}v_{i}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\leqslant\langle-\Delta v_{i},v_{i}\rangle_{L^{2}(\mathbb{R}^{d})}\leqslant\lVert\Delta v_{i}\rVert_{L^{2}(\mathbb{R}^{d})}\lVert v_{i}\rVert_{L^{2}(\mathbb{R}^{d})}$. The result follows again from applying (i). ∎

A.3 Lemma**.**

If $z\in H^{2}(\mathbb{R}^{d})$ has compact support, then $\langle z,\partial_{i}z\rangle_{L^{2}(\mathbb{R}^{d})}=0$, $\left\langle x_{i}\Delta z,z\right\rangle_{L^{2}(\mathbb{R}^{d})}=-\left\langle x_{i},|\nabla z|^{2}\right\rangle_{L^{2}(\mathbb{R}^{d})}$ for $i=1,\ldots,d$. If $z\in H^{4}(\mathbb{R}^{d})$, then also $\langle\Delta z,e^{t\Delta}\Delta\partial_{i}z\rangle_{L^{2}(\mathbb{R}^{d})}=0$, $t\geqslant 0$.

Proof.

Integration by parts gives $\langle z,\partial_{i}z\rangle_{L^{2}(\mathbb{R}^{d})}=-\langle\partial_{i}z,z\rangle_{L^{2}(\mathbb{R}^{d})}$ (argue with compactly supported $z$ first, then extend by continuity), implying $\langle z,\partial_{i}z\rangle_{L^{2}(\mathbb{R}^{d})}=0$ and $\langle x_{i}\partial_{jj}z,z\rangle_{L^{2}(\mathbb{R}^{d})}=-\langle x_{i},(\partial_{j}z)^{2}\rangle_{L^{2}(\mathbb{R}^{d})}$ for $j=1,\ldots,d$. The last part follows from the first one for $\tilde{z}_{t}=e^{(t/2)\Delta}z\in H^{2}(\mathbb{R}^{d})$ using

[TABLE]

The upper bounds in the next Proposition are well-known for analytic semigroups. The main difficulty is to ensure that they hold for growing domains, uniformly in $\delta>0$.

A.4 Proposition**.**

There exist universal constants $M_{0},M_{1}$ such that for $\delta,t>0$

1. (i)

$\lVert S_{\vartheta,\delta}^{*}(t)\rVert_{L^{2}(\Lambda_{\delta})}\leqslant M_{0}e^{C\delta^{2}t}$, 2. (ii)

$\lVert tA_{\vartheta,\delta}^{*}S_{\vartheta,\delta}^{*}(t)\rVert_{L^{2}(\Lambda_{\delta})}\leqslant M_{1}e^{C\delta^{2}t}$.

Proof.

The claimed bounds in the statement follow from Proposition 2.1.1 of Lunardi, [28], if we can show

[TABLE]

with $w=c_{1}\delta^{2}$ for all $\lambda\in\Sigma_{\sigma,w}:=\{\rho\in\mathbb{C}:|\text{arg}(\rho-w)|<\sigma\}\backslash\{w\}$ and with constants $c_{1},M>0,\sigma\in(\pi/2,\pi)$ independent of $\delta$. Since the self-adjoint operator $\Delta_{\vartheta(\delta\cdot)}$ has strictly negative spectrum for all $\delta>0$ (cf. Evans, [12, Section 6.5]), by functional calculus (A.5) holds indeed for $\Delta_{\vartheta(\delta\cdot)}$ with $w=0$, $M=1$ and any $\sigma\in(\pi/2,\pi)$.

In order to extend this to $A_{\vartheta,\delta}^{*}$, we consider it as a perturbation of $\Delta_{\vartheta(\delta\cdot)}$. We show first that $A_{0,\delta}^{*}$ is $\Delta_{\vartheta(\delta\cdot)}$-bounded, i.e.

[TABLE]

for $\varepsilon>0$, $v\in H_{0}^{1}(\Lambda_{\delta})\cap H^{2}(\Lambda_{\delta})$ and absolute constants $c_{2},c_{3}>0$. For this note that $\lVert A_{0,\delta}^{*}v\rVert_{L^{2}(\Lambda_{\delta})}$ is upper bounded by

[TABLE]

Moreover, $\lVert\delta\langle a(\delta\cdot),\nabla v\rangle\rVert_{L^{2}(\Lambda_{\delta})}$ is upper bounded by

[TABLE]

with $c_{2}:=\frac{d\sup_{i=1,\dots,d}\lVert a_{i}\rVert_{\infty}^{2}}{\min_{x}\vartheta(x)}$, where we used in the last line the basic inequality $xy\leqslant\varepsilon x^{2}+\frac{1}{4\varepsilon}y^{2}$ for $x,y>0$. This shows (A.6) with $c_{3}:=\sum_{i=1}^{d}\lVert\partial_{i}a_{i}\rVert_{\infty}+\lVert b\rVert_{\infty}$.

Choosing $\varepsilon$ sufficiently small, the proof of Lemma III.2.6 in Engel & Nagel, [11] implies (A.5) for all $\lambda\in\Sigma_{\sigma,0}\cap\{\rho\in\mathbb{C}:|\rho|>c_{4}\delta^{2}\}$ with $c_{4}=\frac{(4\varepsilon)^{-1}+c_{3}}{1-2c_{2}\varepsilon}$, $\sigma=3\pi/4$ and $M^{\prime}>0$ instead of $M$. Setting $w=(1+c_{5})c_{4}\delta^{2}$, for a suitable constant $c_{5}>0$ to be determined later, and assuming that for these $\lambda$

[TABLE]

with a universal constant $C$, we can therefore conclude for any $\lambda\in\Sigma_{\sigma,0}\cap\{\rho\in\mathbb{C}:|\rho|>c_{4}\delta^{2}\}$ that

[TABLE]

In order to obtain (A.5) from this let $\lambda\in\Sigma_{\sigma,w}$ such that $\lambda-w\in\Sigma_{\sigma,0}$. Assume that we can also show

[TABLE]

Then the result follows from (A.9) with $c_{1}=(1+c_{5})c_{4}$, $M=M^{\prime}C$, because

[TABLE]

We are left with showing (A.8) and (A.10). For (A.8) note that $\lambda\in\Sigma_{\sigma,0}$ already yields $\lambda+w\in\Sigma_{\sigma,0}$, because $w>0$, while the inequality $|\lambda+w|>c_{4}\delta^{2}$ holds clearly, if $|\text{Im}(\lambda)|>c_{4}\delta^{2}$. On the other hand, $|\text{arg}(\lambda)|<\sigma$ implies $|\text{Re}(\lambda)|<c_{5}|\text{Im}(\lambda)|$ for a constant $c_{5}>0$ and thus, if $|\text{Im}(\lambda)|\leqslant c_{4}\delta^{2}$, then

[TABLE]

In order to find the constant $C$ in (A.8), note that $|\lambda+w|\geqslant|\lambda|$ holds always if $\text{Re}(\lambda)\geqslant 0$, and that $|\lambda+w|\geqslant\frac{1}{2}|\lambda|$ whenever $2w\leqslant|\lambda|$. Let now $\text{Re}(\lambda)<0$ and $|\lambda|<2w$ such that by (A.8) $\left|\lambda+w\right|>c_{4}\delta^{2}=\frac{2w}{2(1+c_{5})}>C\left|\lambda\right|$, with $C:=\frac{1}{2(1+c_{5})}$. Finally, with respect to (A.10), $|\lambda-w|>c_{4}\delta^{2}$ holds always, if $|\text{Im}(\lambda)|>c_{4}\delta^{2}$. On the other hand, $|\text{arg}(\lambda-w)|<\sigma$ implies $|\text{arg}(\lambda)|<\sigma$ and hence for $|\text{Im}(\lambda)|\leqslant c_{4}\delta^{2}$, as in (A.11), $|\lambda-w|\geqslant w-|\text{Re}(\lambda)|>c_{4}\delta^{2}$. ∎

With these preparations we can proceed to proving Proposition 3.5.

Proof of Proposition 3.5.

(i). The proof is based on giving a stochastic representation for $S_{\vartheta,\delta}^{*}(t)z$ via the Feynman-Kac formulas. Without loss of generality let $\vartheta\in C^{1+\alpha}(\mathbb{R}^{d})$, $a\in C^{1+\alpha}(\mathbb{R}^{d};\mathbb{R}^{d})$, $b\in C^{\alpha}(\mathbb{R}^{d})$, $\alpha>0$, with $\min_{x\in\mathbb{R}^{d}}\vartheta(x)>0$. Then for $f\in C_{c}^{2}(\mathbb{R}^{d})$

[TABLE]

where $\tilde{a}_{\delta}=\delta(\nabla\vartheta(\delta\cdot)-a(\delta\cdot))\in C^{\alpha}(\mathbb{R}^{d})$, $\tilde{b}_{\delta}=\delta^{2}(b(\delta\cdot)-\text{div}(a(\delta\cdot))\in C^{\alpha}(\mathbb{R}^{d})$. By Karatzas & Shreve, [23, Theorem 5.4.22] we can find a process $Y^{(\delta)}=(Y_{t}^{(\delta)})_{t\geqslant 0}$ being a weak solution of the $d$-dimensional stochastic differential equation

[TABLE]

on a filtered probability space $(\tilde{\Omega},\tilde{\mathcal{F}},(\tilde{\mathcal{F}}_{t})_{t\geqslant 0},\tilde{\mathbb{P}})$ carrying a scalar Brownian motion $(\tilde{W}_{t})_{t\geqslant 0}$. We show below for $x\in\mathbb{R}^{d}$

[TABLE]

where $\tilde{\mathbb{P}}_{x}$ and $\tilde{\mathbb{E}}_{x}$ indicate the initial value and $\tau_{\delta}(Y^{(\delta)}):=\inf\{t\geqslant 0:Y_{t}^{(\delta)}\notin\Lambda_{\delta}\}$. Assume first this holds true. Denote the transition densities of $Y^{(\delta)}$ by $p_{\delta,t}(x,y)$, $x,y\in\mathbb{R}^{d}$. According to Sheu, [35, Eq. (1.4)] we have $p_{\delta,t}(x,y)\leqslant c_{3}q_{c_{2}t}(x-y)$ for universal constants $c_{2},c_{3}>0$. Then by (A.13), using $\lVert\tilde{b}_{\delta}\rVert_{\infty}\leqslant c_{1}\delta^{2}$ for some constant $c_{1}>0$, it follows

[TABLE]

We are left with showing (A.13). The proof is similar to Friedman, [15, Theorem 6.5.2] and extends Peres & Mörters, [33, Theorem 7.44], which applies only to Brownian motion. It is enough to consider $x\in\overline{\Lambda}_{\delta}$, because otherwise $(S_{\vartheta,\delta}^{*}(t)z)(x)=0$ and $\mathbf{1}_{\{t<\tau_{\delta}(Y_{\delta})\}}=0$ $\tilde{\mathbb{P}}_{x}$-a.s. and so (A.13) holds trivially. The function $u(t)=u(t,\cdot)$ with $u(t,x):=(S_{\vartheta,\delta}^{*}(t)z)(x)$ for $t\geqslant 0$, $x\in\overline{\Lambda}_{\delta}$, is the unique solution in $L^{2}(\Lambda_{\delta})$ of

[TABLE]

where the derivative is taken in $L^{2}(\Lambda_{\delta})$. Classical PDE theory yields $u\in C([0,\infty),\overline{\Lambda}_{\delta})\cap C^{1,2}([\varepsilon,\infty),\overline{\Lambda}_{\delta})$ for any $\varepsilon>0$, see for example Friedman, [15, Theorem 6.3.6] (here we use the regularity assumptions on $\vartheta,a,b$). Set $h(t)=\exp(\int_{0}^{t}\tilde{b}_{\delta}(Y_{s}^{(\delta)})ds)$ and let $\rho=\inf\{t\geqslant 0:Y_{t}^{(\delta)}\notin U\}$ for a compact set $U\subseteq\Lambda_{\delta}$. Set $g(t^{\prime},x)=u(t-t^{\prime},x)$, $0\leqslant t^{\prime}\leqslant t$. Noting that $A_{\vartheta,\delta}^{*}-\tilde{b}_{\delta}$ generates the transition semigroup of $Y^{(\delta)}$ and $h^{\prime}(t)=\tilde{b}_{\delta}(Y_{t}^{(\delta)})h(t)$, Itô’s formula shows for any $0\leqslant t^{\prime}<t$

[TABLE]

Using the previous display, the second term vanishes. Taking expectations and letting $t^{\prime}\rightarrow t$ yields therefore

[TABLE]

If $U$ exhausts $\Lambda_{\delta}$, then $\rho\rightarrow\tau_{\delta}(Y^{(\delta)})$ and $Y_{\rho}^{(\delta)}\rightarrow 0$ such that $u(t-\rho,Y_{\rho}^{(\delta)})\rightarrow 0$. This implies (A.13).

(ii). We can assume $z\in C(\overline{\Lambda}_{\delta})$ for sufficiently small $\delta$. Indeed, for $z\in L^{2}(\mathbb{R}^{d})$ let $z^{(\varepsilon)}\in C_{c}(\mathbb{R}^{d})$ converge to $z$ in $L^{2}(\mathbb{R}^{d})$ as $\varepsilon\rightarrow 0$. For small $\delta$ we have $z^{(\varepsilon)}\in C(\overline{\Lambda}_{\delta})$. Applying Proposition A.4(i) to $S_{\vartheta,\delta}^{*}\left(t\right)(z|_{\Lambda_{\delta}}-z^{(\varepsilon)})$, Lemma A.2(i) to $e^{\vartheta(0)t\Delta}(z-z^{(\varepsilon)})$ we have

[TABLE]

Using the statement with respect to $z^{(\varepsilon)}$, and letting first $\delta\rightarrow 0$ and then $\varepsilon\rightarrow 0$, the last line tends to zero.

For $z\in C(\overline{\Lambda}_{\delta})$ it is enough to show $(S_{\vartheta,\delta}^{*}(t)z)(x)\rightarrow(e^{\vartheta(0)\Delta t}z)(x)$ pointwise for $x\in\mathbb{R}^{d}$. $L^{2}(\mathbb{R}^{d})$-covergence follows then from (i) and dominated convergence. Using the notation from (i) we have the representation $(e^{\vartheta(0)t\Delta}z)\left(x\right)=\tilde{\mathbb{E}}_{x}[z(Y_{t}^{(0)})]$ for $Y_{t}^{(0)}=x+\sqrt{2}\vartheta(0)^{1/2}\tilde{W}_{t}$. (A.13) therefore allows us to write $S_{\vartheta,\delta}^{*}\left(t\right)z-e^{\vartheta(0)t\Delta}z=:T_{1}+T_{2}+T_{3}$ with

[TABLE]

We shall show that $T_{i}\rightarrow 0$, $i=1,2,3$. The transition semigroup of $(Y_{t}^{(0)})_{t\geqslant 0}$ is generated by $\mathcal{A}^{(0)}=\vartheta(0)\Delta$. Since $\mathcal{A}^{(\delta)}f\rightarrow\mathcal{A}^{(0)}f$ uniformly on $\mathbb{R}^{d}$ for $f\in C_{c}^{\infty}(\mathbb{R}^{d})$ as $\delta\rightarrow 0$, it follows from Kallenberg, [22, Theorem 19.25] that $Y^{(\delta)}\xrightarrow{d}Y^{(0)}$ with respect to the uniform topology on compacts in $\mathbb{R}_{+}$. This yields $T_{1}\rightarrow 0$. As $z$ is bounded and $\sup_{s>0}|\tilde{b}_{\delta}(Y_{s}^{(\delta)})|\lesssim\delta^{2}$, we also have $|T_{2}|\lesssim e^{Ct\delta^{2}}\delta^{2}$ and $|T_{3}|\lesssim\tilde{\mathbb{P}}_{x}(\tau_{\delta}(Y^{(\delta)})\leqslant t)$. To see why $\tilde{\mathbb{P}}_{x}(\tau_{\delta}(Y^{(\delta)})\leqslant t)\rightarrow 0$ holds let $Z_{s}^{(\delta)}=Y_{s}^{(\delta)}-x-\int_{0}^{s}\tilde{a}_{\delta}(Y_{s^{\prime}}^{(\delta)})ds^{\prime}$ and observe that $|\int_{0}^{s}\tilde{a}_{\delta}(Y_{s^{\prime}}^{(\delta)})ds^{\prime}|\lesssim\delta t$ such that

[TABLE]

where $Z^{(\delta)}=(Z^{(\delta,i)})_{1\leqslant i\leqslant d}$. Since each $Z^{(\delta,i)}$ is a continuous martingale vanishing at [math] such that $\langle Z^{(\delta,i)}\rangle_{s}=2\int_{0}^{s}\vartheta(\delta Y_{s^{\prime}}^{(\delta)})ds^{\prime}\leqslant cs$, $c>0$, uniformly in $i=1,\dots,d$, we find for some scalar Brownian motion $(\tilde{B}_{s})_{s\geqslant 0}$ and $\tilde{c}>0$

[TABLE]

because the density of the running maximum of a Brownian motion decays exponentially (Karatzas & Shreve, [23, Chapter 2.8]). This yields $T_{3}\rightarrow 0$. ∎

A.5 Lemma**.**

Let $z\in H^{2}(\mathbb{R}^{d})$ have compact support in $\Lambda_{\delta^{\prime}}$ for some $\delta^{\prime}>0$. For $0<\delta\leqslant\delta^{\prime}$ set $v^{(\delta)}:=\delta^{-1}(A_{\vartheta,\delta}^{*}-\vartheta(0)\Delta)z$ and define

[TABLE]

Then the following holds:

1. (i)

$\lVert v^{(\delta)}\rVert_{L^{1}\cap L^{2}(\mathbb{R}^{d})}\leqslant C\lVert z\rVert_{W_{1,2}^{2}(\mathbb{R}^{d})}$* and $v^{(\delta)}\rightarrow v$ in $L^{2}(\mathbb{R}^{d})$ for $\delta\rightarrow 0$.* 2. (ii)

If $\int_{\mathbb{R}^{d}}z(x)dx=0$, then $v=\Delta m$ for $m\in H^{2}(\mathbb{R}^{d})$ with compact support. Moreover, if $\vartheta\in C^{1+\alpha^{\prime}}(\overline{\Lambda})$ for $0\leqslant\alpha^{\prime}\leqslant 1$, then $\lVert v^{(\delta)}-v\rVert_{L^{1}\cap L^{2}(\mathbb{R}^{d})}\leqslant C\delta^{\alpha^{\prime}}\lVert z\rVert_{W_{1,2}^{2}(\mathbb{R}^{d})}$. 3. (iii)

If $\int_{\mathbb{R}^{d}}z(x)dx=0$ and $\int_{\mathbb{R}^{d}}xz(x)dx=0$, then $z=\Delta m$ for $m\in H^{4}(\mathbb{R}^{d})$ with compact support.

Proof.

(i). Without loss of generality let $\vartheta,a,b$ as well as the partial derivatives of $\vartheta,a$ be bounded on $\mathbb{R}^{d}$. Then for $x\in\mathbb{R}^{d}$

[TABLE]

From this obtain the upper bound on $v^{(\delta)}$ and the convergence in $L^{2}(\mathbb{R}^{d})$ to

[TABLE]

(ii). In order to find $m$, as $z$ has compact support, it suffices to find a compactly supported function $g\in H^{2}(\mathbb{R}^{d})$ with $\Delta g=\langle\nabla\vartheta(0)+a(0),\nabla z\rangle_{\mathbb{R}^{d}}$ in $L^{2}(\mathbb{R}^{d})$ and to set $m:=\langle\nabla\vartheta(0),x\rangle_{\mathbb{R}^{d}}z(x)-g(x)$. Using the Fourier transform ${\cal F}g(\omega)=\int_{\mathbb{R}^{d}}g(x)e^{i\langle x,\omega\rangle}dx$ this means by usual Fourier calculus

[TABLE]

By the compact support of $z$ and $\int_{\mathbb{R}^{d}}z(x)dx=0$ the Fourier transform ${\cal F}z$ is analytic with ${\cal F}z(0)=0$. We can thus define

[TABLE]

as the inverse Fourier transform of the $L^{2}$-function $u$. Noting $z\in H^{2}(\mathbb{R}^{d})$ and $\lvert u(\omega)\rvert\lesssim\lvert{\cal F}z(\omega)\rvert$ for $\lvert\omega\rvert\to\infty$, we see $g\in H^{2}(\mathbb{R}^{d})$ and $\Delta g=\langle\nabla\vartheta(0)+a(0),\nabla z\rangle_{\mathbb{R}^{d}}$ in $L^{2}(\mathbb{R}^{d})$.

For compactness of $g$ we use the Paley-Wiener Theorem (Rudin, [34, Theorem II.7.22]) to deduce from the compact support of $z$ that ${\cal F}z$ can be extended to an entire function on $\mathbb{C}^{d}$, satisfying the exponential growth condition $\lvert{\cal F}z(\omega)\rvert\leqslant\gamma_{N}(1+\lvert\omega\rvert)^{-N}\exp(r\lvert\operatorname{Im}(\omega)\rvert)$, $\omega\in\mathbb{C}^{d}$, for all $N\in\mathbb{N}$ and suitable positive constants $\gamma_{N}$, $r$. Hence, $u$ is the quotient of an entire function and $\lvert\omega\rvert^{2}$, which is also entire. A meromorphic function with removable singularity extends continuously to an entire function. Consequently, we can work with an entire function $u$, which by definition satisfies the same exponential growth condition. A reverse application of the Paley-Wiener Theorem shows that $g$ has compact support.

Finally, we can assume that $\nabla\vartheta,a$ are uniformly $\alpha^{\prime}$-Hölder continuous on $\mathbb{R}^{d}$. The upper bound on $\lVert v^{(\delta)}-v\rVert_{L^{1}\cap L^{2}(\mathbb{R}^{d})}$ follows then for $x\in\mathbb{R}^{d}$ using (A.14) from

[TABLE]

(iii). The argument is similar to (ii). As above, the Fourier transform $\mathcal{F}z$ is analytic with $\mathcal{F}z(0)=0$. The assumption $\int_{\mathbb{R}^{d}}x_{i}z(x)dx=0$ gives also $\partial_{i}(\mathcal{F}z)(0)=0$, $i=1,\dots,d$. It follows for

[TABLE]

that $m\in H^{4}(\mathbb{R}^{d})$ and $\Delta m=z$. A Paley-Wiener argument as in (ii) shows that $m$ has compact support. ∎

The following heat kernel bounds will be used frequently. The conditions in (iii) are essential for $d=1$ to improve on (ii).

A.6 Lemma**.**

Let the functions $u,w\in L^{2}(\mathbb{R}^{d})$, $z\in H^{2}(\mathbb{R}^{d})$ have compact support in $\Lambda_{\delta}$ for some $\delta>0$. Then for $0<t\leqslant T\delta^{-2}$:

1. (i)

$\lVert S_{\vartheta,\delta}^{*}\left(t\right)u\rVert_{L^{2}(\Lambda_{\delta})}\leqslant e^{CT}(1\wedge t^{-d/4})\lVert u\rVert_{L^{1}\cap L^{2}(\mathbb{R}^{d})}.$** 2. (ii)

If $\lVert w-\Delta z\rVert_{L^{1}\cap L^{2}(\mathbb{R}^{d})}\leqslant C\delta^{\alpha^{\prime}}\lVert z\rVert_{W_{1,2}^{2}(\mathbb{R}^{d})}$ for $0\leqslant\alpha^{\prime}\leqslant 1$, then

[TABLE] 3. (iii)

If $\vartheta\in C^{1+\alpha^{\prime}}(\overline{\Lambda})$ for $0\leqslant\alpha^{\prime}\leqslant 1$ and $\int_{\mathbb{R}^{d}}z(x)dx=0$, then

[TABLE]

Proof.

(i). The semigroup bound in Proposition 3.5(i), applied to a sequence of continuous functions approximating $u$, and Lemma A.2(i) show for $t\leqslant T\delta^{-2}$

[TABLE]

(ii). Write $w=u^{(\delta)}+\vartheta(0)^{-1}A_{\vartheta,\delta}^{*}z$ for $u^{(\delta)}=(w-\Delta z)-\vartheta(0)^{-1}\delta v^{(\delta)}$ and $v^{(\delta)}=\delta^{-1}(A_{\vartheta,\delta}^{*}-\vartheta(0)\Delta)z$ such that

[TABLE]

The second term is up to a constant bounded by $e^{CT}(1\wedge t^{-1})\lVert S_{\vartheta,\delta}^{*}(t/2)z\rVert_{L^{2}(\Lambda_{\delta})}$, using Proposition A.4(ii) for $t\leqslant T\delta^{-2}$ and $S_{\vartheta,\delta}^{*}(t)=S_{\vartheta,\delta}^{*}(t/2)S_{\vartheta,\delta}^{*}(t/2)$. Applying (i) to $u=z$ gives the upper bound $e^{CT}(1\wedge t^{-1-d/4})\lVert z\rVert_{L^{2}(\mathbb{R}^{d})}$. The result follows from applying (i) to $u=u^{(\delta)}$ in the last display and noting that $\lVert u^{(\delta)}\rVert_{L^{1}\cap L^{2}(\mathbb{R}^{d})}\leqslant e^{CT}(1\wedge t^{-\alpha^{\prime}/2})\lVert z\rVert_{W^{2}_{1,2}(\mathbb{R}^{d})}$ by Lemma A.5(i) and $\delta\leqslant(T/t)^{1/2}$, as well as adjusting the constant $C$.

(iii). Following the proof of (ii) for $w=\Delta z$ it is enough to show the improved upper bound $\lVert S_{\vartheta,\delta}^{*}\left(t\right)u^{(\delta)}\rVert_{L^{2}(\Lambda_{\delta})}\lesssim e^{CT}(1\wedge t^{-1/2-\alpha^{\prime}/2-d/4})\lVert z\rVert_{W_{1,2}^{2}(\mathbb{R}^{d})}$. Lemma A.5(ii) shows the existence of a compactly supported $m\in H^{2}(\mathbb{R}^{d})$ such that $\lVert v^{(\delta)}-\Delta m\rVert_{L^{1}\cap L^{2}(\mathbb{R}^{d})}\leqslant C\delta^{\alpha^{\prime}}\lVert z\rVert_{W_{1,2}^{2}(\mathbb{R}^{d})}$. With $\tilde{u}^{(\delta)}=v^{(\delta)}-\Delta m$ write $u^{(\delta)}=\vartheta(0)^{-1}\delta\tilde{u}^{(\delta)}+\vartheta(0)^{-1}\delta\Delta m$. Applying (i) to $u=\tilde{u}^{(\delta)}$ and (ii) to $z=m$ yields

[TABLE]

For $\delta\leqslant(T/t)^{1/2}$ the order in $t$ is $1\wedge t^{-1/2-\alpha^{\prime}/2-d/4}$, as claimed. ∎

A.7 Lemma**.**

Let $u\in L^{2}(\mathbb{R}^{d})$, $z\in H^{2}(\mathbb{R}^{d})$ have compact support in $\Lambda_{\delta}$ for some $\delta>0$. Using $\ell_{d,2}(\delta)$ as in Assumption 3.7($z$;$\beta$) Assumption, we have:

1. (i)

$\int_{0}^{T}\left\langle S_{\vartheta}\left(t\right)X_{0},u_{\delta}\right\rangle^{2}dt\leqslant e^{CT}\lVert X_{0}\rVert^{2}\ell_{d,2}(\delta)\delta^{2\wedge d}\lVert u\rVert_{L^{1}\cap L^{2}(\mathbb{R}^{d})}^{2}$. 2. (ii)

If $X_{0}\in L^{p}(\Lambda)$, $p\geqslant 2$, $1/p+1/p^{\prime}=1$, then, with $\gamma(d,p):=2\frac{1+d/p}{1+d/2}+d(1-\frac{2}{p})$,

[TABLE] 3. (iii)

If $X_{0}\in\mathcal{D}(A_{\vartheta})$, then

[TABLE]

Proof.

(i). By Lemma A.6(i) and the scaling in Lemma 3.1 we find

[TABLE]

The claim follows, because the integral has order $O(1)$ for $d\geqslant 3$, order $O(\log(T\delta^{-2}))$ for $d=2$ and order $O(T^{1/2}\delta^{-1})$ for $d=1$.

(ii). It is enough to consider continuous $X_{0}$. Using the Hölder inequality and Proposition 3.5(i), we obtain

[TABLE]

Here, $\lVert q_{c_{2}t}*\left|X_{0}\right|\rVert_{L^{p}(\mathbb{R}^{d})}\leqslant\lVert X_{0}\rVert_{L^{p}(\Lambda)}$. For $\varepsilon>0$, Lemmas 3.1 and A.6(ii) show

[TABLE]

Splitting up the integral and adjusting the constant $C$ yields thus

[TABLE]

The claim follows with $\lVert X_{0}\rVert\lesssim\lVert X_{0}\rVert_{L^{p}(\Lambda)}$ and $\varepsilon=\delta^{2\frac{1+d/p}{1+d/2}}$.

(iii). With $v^{(\delta)}:=\delta^{-1}(A_{\vartheta,\delta}^{*}-\vartheta(0)\Delta)z$ as in Lemma A.5 and using the scaling in Lemma 3.1, write $\vartheta(0)(\Delta z)_{\delta}=-\delta v_{\delta}^{(\delta)}+\delta^{2}A_{\vartheta}^{*}z_{\delta}$. Then

[TABLE]

and the claim follows from applying (i) with $u=v$ and $u=z$ (with $A_{\vartheta}X_{0}\in L^{2}(\Lambda)$ instead of $X_{0}$). ∎

A.3 Asymptotic results for the covariances

The general idea for the proofs in this section is to apply the scaling in Lemma 3.1 to the covariance function as in Section 3.3 and to deduce a limit for the integral using the heat kernel bounds and the convergence of the semigroups from the last section.

A.8 Proposition**.**

Grant Assumption 3.2. Consider functions $z\in H^{2}(\mathbb{R}^{d})$, $u\in L^{2}(\mathbb{R}^{d})$, $(w^{(\delta)})_{\delta>0}$, $(u^{(\delta)})_{\delta>0}\subseteq L^{2}(\mathbb{R}^{d})$ with compact support in $\Lambda_{\delta^{\prime}}$ for some $\delta^{\prime}>0$. Assume for $0<\delta\leqslant\delta^{\prime}$ that $\lVert w^{(\delta)}-\Delta z\rVert_{L^{1}\cap L^{2}(\mathbb{R}^{d})}\leqslant C\delta^{\alpha^{\prime}}$ for $\alpha^{\prime}>1/2$, $\lVert u^{(\delta)}-u\rVert_{L^{2}(\mathbb{R}^{d})}\rightarrow 0$ as $\delta\rightarrow 0$. Then with $\Psi$ from (3.2):

1. (i)

$\delta^{-2}\operatorname{Var}(\langle\tilde{X}(t),w_{\delta}^{(\delta)}\rangle)\rightarrow\vartheta(0)^{-1}\Psi(\Delta z,\Delta z)$, $t>0$. 2. (ii)

$\delta^{-2}\int_{0}^{T}\operatorname{Var}(\langle\tilde{X}(t),w_{\delta}^{(\delta)}\rangle)dt\rightarrow\ T\vartheta(0)^{-1}\Psi(\Delta z,\Delta z)$. 3. (iii)

If $d\geqslant 2$, then $\delta^{-2}\int_{0}^{T}\operatorname{Cov}(\langle\tilde{X}(t),w_{\delta}^{(\delta)}\rangle,\langle\tilde{X}(t),u_{\delta}^{(\delta)}\rangle)dt\rightarrow\ T\vartheta(0)^{-1}\Psi(\Delta z,u)$.

Proof.

(i). By (3.4) for $t=t^{\prime}$ we have

[TABLE]

with $f_{\delta}(s)=\lVert B_{\delta}^{*}S_{\vartheta,\delta}^{*}\left(s\right)w^{(\delta)}\rVert_{L^{2}(\Lambda_{\delta})}^{2}\mathbf{1}_{\{s\leqslant t\delta^{-2}\}}$. Set $f(s)=\lVert B_{0}^{*}e^{\vartheta(0)s\Delta}\Delta z\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$ and note $\int_{0}^{\infty}f(s)\vartheta(0)ds=\Psi(\Delta z,\Delta z)$, substituting $ds^{\prime}=\vartheta(0)ds$. By assumption $w^{(\delta)}\xrightarrow{}\Delta z$ in $L^{2}(\Lambda_{\delta})$ and by Proposition 3.5(ii) above $S_{\vartheta,\delta}^{*}(s)\Delta z\rightarrow e^{\vartheta(0)s\Delta}\Delta z$ in $L^{2}(\mathbb{R}^{d})$. From Proposition A.4(i) above we have $\sup_{0\leqslant s\leqslant t/\delta^{2}}\lVert S_{\vartheta,\delta}^{*}(s)\rVert_{L^{2}(\Lambda_{\delta})}<\infty$, as well as $\sup_{0<\delta\leqslant 1}\lVert B_{\delta}^{*}\rVert_{L^{2}(\mathbb{R}^{d})}<\infty$ by Assumption 3.2 and the uniform boundedness principle. We deduce

[TABLE]

which implies $f_{\delta}(s)\xrightarrow{}f(s)$ pointwise. Lemma A.6(ii) yields $\left|f_{\delta}\left(s\right)\right|\lesssim 1\wedge s^{-\alpha^{\prime}-d/2}$. Since $\alpha^{\prime}>1/2$, $\sup_{0<\delta\leqslant 1}f_{\delta}(\cdot)\in L^{1}([0,\infty))$, for any fixed $t$, and the result follows from the dominated convergence theorem.

(ii). By (i) and Fatou’s lemma we obtain

[TABLE]

On the other hand, $\operatorname{Var}(\langle\tilde{X}(t),w_{\delta}^{(\delta)}\rangle)$ is increasing in $t$, cf. (2.5). The result follows from

[TABLE]

(iii). Revisiting the derivations in (i) and (ii), we obtain

[TABLE]

Putting $f(s):=\langle B_{0}^{*}e^{\vartheta(0)s\Delta}\Delta z,B_{0}^{*}e^{\vartheta(0)s\Delta}u\rangle$, we obtain as in (i), (iii) that $f_{\delta}(s)\to f(s)$ holds pointwise for $\delta\to 0$ by the $L^{2}$-continuity of the scalar product. Furthermore, the Cauchy-Schwarz inequality and Lemma A.6(i,ii) yield the bound

[TABLE]

for $d\geqslant 2$. Since this bound is integrable in $s\geqslant 0$, we conclude that

[TABLE]

meaning in particular that $\Psi(\Delta z,u)$ is well defined. What is more, the bound (A.15) also shows that the covariance is uniformly bounded in $t\in[0,T]$ so that another application of the dominated convergence theorem shows that the integral over $t\in[0,T]$ converges to $T\vartheta(0)^{-1}\Psi(\Delta z,u)$. ∎

A.9 Proposition**.**

Grant Assumption 3.2. Consider functions $z,m\in H^{2}(\mathbb{R}^{d})$, $u\in L^{2}(\mathbb{R}^{d})$, $(w^{(\delta)})_{\delta>0}$, $(u^{(\delta)})_{\delta>0}\subseteq L^{2}(\mathbb{R}^{d})$ with compact support in $\Lambda_{\delta^{\prime}}$ for some $\delta^{\prime}>0$. Assume for $0<\delta\leqslant\delta^{\prime}$ that $\lVert w^{(\delta)}-\Delta m\rVert_{L^{1}\cap L^{2}(\mathbb{R}^{d})}\leqslant C\delta^{\alpha^{\prime}}$ for $\alpha^{\prime}>1/2$, $\lVert u^{(\delta)}-u\rVert_{L^{2}(\mathbb{R}^{d})}\rightarrow 0$ as $\delta\rightarrow 0$. Then:

1. (i)

For $d\geqslant 2$, $\operatorname{Var}(\int_{0}^{T}\langle\tilde{X}(t),(\Delta z)_{\delta}\rangle\langle\tilde{X}(t),u_{\delta}^{(\delta)}\rangle dt)=O(\delta^{6}\ell_{d,2}(\delta)^{3})$. 2. (ii)

$\operatorname{Var}(\int_{0}^{T}\langle\tilde{X}(t),(\Delta z)_{\delta}\rangle\langle\tilde{X}(t),w_{\delta}^{(\delta)}\rangle dt)=o(\delta^{4})$. 3. (iii)

For $d\geqslant 2$, $\operatorname{Var}(\int_{0}^{T}\langle\tilde{X}(t),u_{\delta}^{(\delta)}\rangle^{2}dt)=O(\delta^{4}\ell_{d,2}(\delta)^{2})$. 4. (iv)

Let $d\geqslant 2$, or let $\vartheta\in C^{1+\alpha^{\prime}}(\overline{\Lambda})$ and $\int_{\mathbb{R}^{d}}m(x)dx=0$. Then with $\Psi$ from (3.2)

[TABLE]

Proof.

We first make some preliminary remarks. For $v,\tilde{v}\in L^{2}(\Lambda_{\delta})$ set $\xi(t)=\langle\tilde{X}(t),v_{\delta}\rangle$, $\tilde{\xi}(t)=\langle\tilde{X}(t),\tilde{v}_{\delta}\rangle$. The random variables $\{\xi(t)\,|\,t\geqslant 0\}\cup\{\tilde{\xi}(t)\,|\,t\geqslant 0\}$ are jointly Gaussian and centered and so it follows from Wick’s formula (Janson, [21, Theorem 1.28]) that

[TABLE]

with $V_{1}=V(v,v,\tilde{v},\tilde{v})$, $V_{2}=V(v,\tilde{v},\tilde{v},v)$, where for $k,\tilde{k}\in L^{2}(\Lambda_{\delta})$

[TABLE]

and $V(v):=V(v,v,v,v)$. It is thus enough to study $V_{1},V_{2}$. Set

[TABLE]

Then by (3.4), $V(v,\tilde{v},k,\tilde{k})$ equals

[TABLE]

substituting $ds^{\prime}=s\delta^{-2}-dr^{\prime}$, $ds^{\prime\prime}=s\delta^{-2}-dr^{\prime\prime}$ and $ds^{\prime\prime\prime}=\delta^{-2}(t-ds)$, but writing again $s$ for $s^{\prime\prime\prime}$. With this we prove now the Proposition.

(i). Let $v=\Delta z$, $\tilde{v}=u^{(\delta)}$. By the rescaling with $\delta^{-6}$ it is enough to show $V_{i}=O(\ell_{d,2}(\delta)^{3})$, $i=1,2$. Observe by Lemma A.6(i,ii) that

[TABLE]

These bounds yield in (A.18) when $d\geqslant 2$ for $V(\Delta z,\Delta z,u^{(\delta)},u^{(\delta)})$ up to a constant the upper bound

[TABLE]

Similarly, $\left|f_{\delta}((s+s^{\prime\prime},\Delta z),(s^{\prime\prime},u^{(\delta)}))\right|\lesssim(1\wedge s{}^{-d/4})(1\wedge(s^{\prime\prime})^{-1/2-d/4})$, $\left|f_{\delta}\left((s+s^{\prime},u^{(\delta)}),(s^{\prime},\Delta z)\right)\right|\lesssim(1\wedge s{}^{-d/4})(1\wedge(s^{\prime})^{-1/2-d/4})$, implying the upper bound $\ell_{d,2}(\delta)^{3}$ also for $V(\Delta z,u^{(\delta)},u^{(\delta)},\Delta z)$. In all, we find $|V_{1}|$, $|V_{2}|\lesssim\ell_{d,2}(\delta)^{3}$.

(ii). Let $v=\Delta z$, $\tilde{v}=w^{(\delta)}$. By the rescaling of $V$ it is enough to show $\delta^{2}V_{i}\rightarrow 0$, $i=1,2$. We have by Lemma A.6(ii) for any $v_{1},v_{2}\in\{v,\tilde{v}\}$

[TABLE]

Therefore, as in (A.21) but this time for all $d\geqslant 1$, $|V(v_{1},v_{2},v_{3},v_{4})|\lesssim\delta^{4(\alpha^{\prime}-1)}$, $v_{3},v_{4}\in\{v,\tilde{v}\}$.

(iii). Let $v=\tilde{v}=u^{(\delta)}$. The claim is a direct consequence of (A.19) and (A.21) for $d\geqslant 2$, with the $ds$-integral of order $O(\delta^{-2})$ this time.

(iv). Let $v=\tilde{v}=w^{(\delta)}$. Since $V_{1}=V_{2}=V(w^{(\delta)})$, it is enough to show

[TABLE]

We argue by dominated convergence. Set

[TABLE]

Exactly as in the proof of Proposition A.8(i) we get pointwise by polarisation

[TABLE]

In order to conclude observe for $d\geqslant 2$ by (A.20) (with $w^{(\delta)}$ instead of $\Delta z$) that

[TABLE]

If $\vartheta\in C^{1+\alpha^{\prime}}(\overline{\Lambda})$ for $\alpha^{\prime}>1/2$ and $\int_{\mathbb{R}^{d}}m(x)dx=0$, the improved bound in Lemma A.6(iii) for (A.20) gives

[TABLE]

In both cases (A.22) follows from dominated convergence, noting

[TABLE]

The next result improves on Proposition A.8(ii) when $B$ is a multiplication operator, by making lower order terms explicit. This is necessary for the proof of Theorem 5.9. The main difficulty is to work around not having a rate of convergence in Proposition 3.5(ii).

A.10 Proposition**.**

Let $z\in H^{4}(\mathbb{R}^{d})$ have compact support in $\Lambda_{\delta^{\prime}}$ for some $\delta^{\prime}>0$ and suppose that $B=M_{\sigma}$ with $\sigma\in C^{1}(\overline{\Lambda})$. If $d=1$, then assume $\vartheta\in C^{1+\alpha^{\prime}}(\overline{\Lambda})$ for $\alpha^{\prime}>1/2$ and $\int_{\mathbb{R}}z(x)dx=0$. Then for $0<\delta\leqslant\delta^{\prime}$ and $\delta\rightarrow 0$

[TABLE]

Proof.

Let $\langle\overline{X}(t),\cdot\rangle$ be defined as $\langle\tilde{X}(t),\cdot\rangle$ in (2.4), but with semigroup $(\overline{S}_{\vartheta}(t))_{t\geqslant 0}$ on $L^{2}(\Lambda)$ generated by $A_{\vartheta}=\Delta_{\vartheta}$. As before, $(\overline{S}_{\vartheta,\delta}(t))_{t\geqslant 0}$ is the corresponding semigroup on $L^{2}(\Lambda_{\delta})$ generated by $\Delta_{\vartheta(\delta\cdot)}$. Note that the $\overline{S}_{\vartheta,\delta}(t)$ are self-adjoint. With $v^{(\delta)}:=\delta^{-1}(\Delta_{\vartheta(\delta\cdot)}-\vartheta(0)\Delta)z$ set for $0\leqslant t\leqslant T$

[TABLE]

and introduce the decompositions

[TABLE]

where we use for the remainder terms that $\int_{0}^{T}R_{i}(t)dt=o(\delta)$, $i=1,2$, by Lemmas A.14 and A.15 below. The claim follows from Lemmas A.11, A.12 and A.13 below, which show

[TABLE]

∎

A.11 Lemma**.**

In Proposition A.10 we have

[TABLE]

Proof.

Lemma A.5 above with $A_{\vartheta}=\Delta_{\vartheta}$ yields $v^{(\delta)}\rightarrow v:=\Delta(\langle\nabla\vartheta(0),x\rangle_{\mathbb{R}^{d}}z)-\langle\nabla\vartheta(0),\nabla z\rangle_{\mathbb{R}^{d}}$ in $L^{2}$. Moreover, since $\vartheta\in C^{1}(\overline{\Lambda})$, we have $\lVert\Delta_{\vartheta(\delta\cdot)}z-\vartheta(0)\Delta z\rVert_{L^{1}\cap L^{2}(\mathbb{R}^{d})}\leqslant C\delta$. If $d\geqslant 2$, then Proposition A.8(iii) with $w^{(\delta)}=\vartheta(0)^{-1}\Delta_{\vartheta(\delta\cdot)}z$, $u^{(\delta)}=v^{(\delta)}$ already implies $\delta^{-1}\int_{0}^{T}T_{1}(t)dt\rightarrow-2T\vartheta^{-2}(0)\Psi(\Delta z,v)$, and the claim follows from Lemma A.3 above, recalling the identity $\Psi(\Delta z,v)=-\frac{\sigma^{2}(0)}{2}\langle z,v\rangle_{L^{2}(\mathbb{R}^{d})}$ from (3.3). For $d=1$, on the other hand, the properties $\vartheta\in C^{1+\alpha^{\prime}}(\overline{\Lambda})$ for $\alpha^{\prime}>1/2$ and $\int_{\mathbb{R}}z(x)dx=0$ ensure by Lemma A.5(ii) that $v=\Delta m$ for a compactly supported $m\in H^{2}(\mathbb{R})$ with $\lVert v^{(\delta)}-\Delta m\rVert_{L^{1}\cap L^{2}(\mathbb{R})}\leqslant C\delta^{\alpha^{\prime}}$. By polarisation and Proposition A.8(ii) (with $w^{(\delta)}=\vartheta(0)^{-1}\Delta_{\vartheta(\delta\cdot)}z$ and $w^{(\delta)}=v^{(\delta)}$) $\delta^{-1}\int_{0}^{T}T_{1}(t)dt$ converges again to the claimed limit. ∎

A.12 Lemma**.**

In Proposition A.10 we have

[TABLE]

Proof.

Using (3.4), we have $\vartheta^{2}(0)T_{2}(t)=\int_{0}^{t\delta^{-2}}f_{\delta}(s)ds$ for

[TABLE]

By the Cauchy-Schwarz inequality and the semigroup bounds in Proposition A.4(i,ii) above this means

[TABLE]

where we used Proposition 3.5(i) for an approximating sequence of $\Delta_{\vartheta(\delta\cdot)}z$ with continuous functions and Lemma A.2(iii) in the last two lines. We conclude from Proposition 3.5(ii) that

[TABLE]

Combining this with (A.23), the dominated convergence theorem shows $\vartheta^{2}(0)\delta^{-1}\int_{0}^{T}T_{2}(t)dt\rightarrow T\int_{0}^{\infty}f(s)ds$. For the result note that by Lemmas A.2(ii) and A.3 (here we need $z\in H^{4}(\mathbb{R}^{d})$) $\vartheta^{-2}(0)\int_{0}^{\infty}f(s)ds$ equals

[TABLE]

A.13 Lemma**.**

In Proposition A.10 we have

[TABLE]

Proof.

Since $\overline{S}_{\vartheta,\delta}(s)$ is self-adjoint, we have

[TABLE]

Integrating over $0\leqslant t\leqslant T$ and using the semigroup bound in Proposition A.4(i) the first term is of order $O(\delta^{2})$. Since $\vartheta\in C^{1}(\overline{\Lambda})$, the result follows from Lemma A.3 and

[TABLE]

A.14 Lemma**.**

In Proposition A.10 we have $\delta^{-1}\int_{0}^{T}R_{1}(t)dt\rightarrow 0$.

Proof.

By (3.4) write $R_{1}(t)=\int_{0}^{t\delta^{-2}}f_{\delta}(s)ds$ with $f_{\delta}(s)=\langle g_{\delta}(s),h_{\delta}(s)\rangle_{L^{2}(\Lambda_{\delta})}$ for $s\geqslant 0$, where

[TABLE]

An application of the dominated convergence theorem then proves the result, if

[TABLE]

In order to show (A.24) and (A.25) we use the *variation of parameters formula *(Engel & Nagel, [11, p. 162]): The function $[0,s]\ni s^{\prime}\mapsto S_{\vartheta,\delta}^{*}\left(s^{\prime}\right)\overline{S}_{\vartheta,\delta}(s-s^{\prime})\Delta z$ has derivative $S_{\vartheta,\delta}^{*}\left(s^{\prime}\right)(A_{\vartheta,\delta}^{*}-\Delta_{\vartheta(\delta\cdot)})\overline{S}_{\vartheta,\delta}\left(s-s^{\prime}\right)\Delta z$, implying

[TABLE]

Since the operator $A_{\vartheta,\delta}^{*}-\Delta_{\vartheta(\delta\cdot)}=A_{0,\delta}^{*}$ is not bounded, a careful analysis is required. Decomposing it into first and zero order terms we have

[TABLE]

The semigroup bounds in Proposition A.4(ii) and in Lemma A.6(i,ii,iii) above, subject to $d\geqslant 2$ or $d=1$ with $\vartheta\in C^{1+\alpha^{\prime}}(\overline{\Lambda})$ for $\alpha^{\prime}>1/2$ and $\int_{\mathbb{R}}z(x)dx=0$, show for sufficiently small $\delta$ and $0\leqslant s^{\prime}<s\leqslant t\delta^{-2}$ that

[TABLE]

Hence, the computations in (A.7) above show

[TABLE]

Because of Proposition A.4(i) this yields for $0<s\leqslant t\delta^{-2}$

[TABLE]

In all, this is of order $1\wedge s^{-1/2}$. Since also $\lVert h_{\delta}(s)\rVert_{L^{2}(\Lambda_{\delta})}\lesssim 1\wedge s^{-1}$ by Lemma A.6(ii,iii), subject to $d\geqslant 2$ or the conditions in $d=1$, we obtain (A.24). With respect to (A.25) fix $s$ and observe by the convergence of the semigroups in Proposition 3.5(ii) above and $\sigma^{2}(\delta\cdot)\rightarrow\sigma^{2}(0)$ that $h_{\delta}(s)\rightarrow 2\sigma^{2}(0)e^{\vartheta(0)s\Delta}\Delta z$ in $L^{2}(\mathbb{R}^{d})$. Therefore, $f_{\delta}\left(s\right)=-2\delta\sigma^{2}(0)f_{\delta}^{(1)}\left(s\right)+o\left(\delta\right)$, uniformly in $s$, for

[TABLE]

In the same way, since $a_{i}(\delta\cdot)S_{\vartheta,\delta}\left(s^{\prime}\right)(v|_{\Lambda_{\delta}})\rightarrow a_{i}(0)e^{\vartheta(0)s^{\prime}\Delta}v$ for $v\in L^{2}(\mathbb{R}^{d})$ by Proposition 3.5(ii), we have $f_{\delta}^{(1)}(s)=f_{\delta}^{(2)}(s)+o(\delta)$ for

[TABLE]

Noting that $\overline{S}_{\vartheta,\delta}\left(s-s^{\prime}\right)\Delta z\rightarrow e^{\vartheta(0)(s-s^{\prime})\Delta}\Delta z$ in $L^{2}(\mathbb{R}^{d})$, we finally obtain

[TABLE]

which vanishes according to Lemma A.3. ∎

A.15 Lemma**.**

In Proposition A.10 we have $\delta^{-1}\int_{0}^{T}R_{2}(t)dt\rightarrow 0$.

Proof.

By (3.4) we have for $0\leqslant t\leqslant T$

[TABLE]

Recall from Lemma A.11 that $v^{(\delta)}$ converges in $L^{2}(\mathbb{R}^{d})$. For $d\geqslant 2$, Lemma A.6(i) then shows uniformly in $0\leqslant t\leqslant T$ that

[TABLE]

implying the claim in this case. For $d=1$ it is enough to recall from Lemma A.11 that $\lVert v^{(\delta)}-\Delta m\rVert_{L^{1}\cap L^{2}(\mathbb{R})}\leqslant C\delta^{\alpha^{\prime}}$, and so the claim follows from Lemma A.6(iii). ∎

A.16 Proposition**.**

Grant Assumption 3.2. Let $z\in H^{2}(\mathbb{R}^{d})$ have compact support in $\Lambda_{\delta^{\prime}}$ for some $\delta^{\prime}>0$ and for $d=1$ assume $\vartheta\in C^{1+\alpha^{\prime}}(\overline{\Lambda})$ for $\alpha^{\prime}>1/2$, $\int_{\mathbb{R}}z(x)dx=0$. For $0<\delta\leqslant\delta^{\prime}$ set $\xi_{\delta}(t)=\langle\tilde{X}(t),(\Delta z)_{\delta}\rangle$. Then the fourth moment of $\delta^{-3}\int_{0}^{T}(\xi_{\delta}(t)^{2}-\mathbb{E}[\xi_{\delta}(t)^{2}])dt$ converges, with $\Psi$ from (3.2), for $\delta\rightarrow 0$ to

[TABLE]

Proof.

In view of Proposition A.9(iv) it is enough to show

[TABLE]

Abbreviating $c_{\delta}(t,s)=\text{Cov}(\langle\tilde{X}(t),(\Delta z)_{\delta}\rangle,\langle\tilde{X}(s),(\Delta z)_{\delta}\rangle)$, recall from (A.16) that

[TABLE]

Wick’s formula (Janson, [21, Theorem 1.28]) for 8th centered Gaussian moments $\mathbb{E}[\prod_{i=1}^{8}Z_{i}]=\sum_{\pi\in\Pi_{2}(8)}\prod_{(i,j)\in\pi}\mathbb{E}[Z_{i}Z_{j}]$ applied to $Z_{i}=\xi_{\delta}(t_{i})$ for $0\leqslant t_{i}\leqslant T$, with $\Pi_{2}(8)$ being the set of all partitions $\pi$ of $\{1,\ldots,8\}$ into 2-tuples, therefore yields, using the symmetry of the integrand in $(t_{1},t_{2},t_{3},t_{4})$,

[TABLE]

The calculations in the proof of Proposition A.9(iv) show for $s\leqslant t$, both when $d\geqslant 2$ and when $d=1$, $\vartheta\in C^{1+\alpha^{\prime}}(\overline{\Lambda})$ for $\alpha^{\prime}>1/2$, $\int_{\mathbb{R}}z(x)dx=0$, that

[TABLE]

so that as in (A.18), substituting $s_{i}=\delta^{-2}(t_{i+1}-t_{i})$,

[TABLE]