Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions

TL;DR

This paper establishes the existence and uniqueness of solutions for a class of nonlinear stochastic evolution equations with singular drifts and gradient noise on a torus, using advanced mathematical techniques under specific geometric and commutation conditions.

Contribution

It introduces a novel approach to handle singular drifts and gradient noise in stochastic evolution equations through curvature and commutation conditions, expanding the theoretical understanding of such equations.

Findings

Proved well-posedness of the stochastic evolution equations under certain conditions.

Developed a framework using resolvent, Dirichlet forms, and an approximative Itô formula.

Extended the theory to include singular p-Laplace and total variation flows.

Abstract

We prove existence and uniqueness of solutions to a nonlinear stochastic evolution equation on the $d$-dimensional torus with singular $p$-Laplace-type or total variation flow-type drift with general sublinear doubling nonlinearities and Gaussian gradient Stratonovich noise with divergence-free coefficients. Assuming a weak defective commutator bound and a curvature-dimension condition, the well-posedness result is obtained in a stochastic variational inequality setup by using resolvent and Dirichlet form methods and an approximative It\^{o}-formula.

Equations

$$d{X}_t$$

$$X_0$$

$$\mathcal{E}(u,v):={\int }_{{\mathbb{T}}^d}⟨\nabla u,\nabla v⟩d\xi ,u,v\in D(\mathcal{E}),$$

$$d{X}_t$$

$$X_0$$

$$d{X}_t$$

$$X_0$$

$$\tilde{\Psi }:u\mapsto \{\begin{array}{ll}{\int }_{{\mathbb{T}}^d}\psi (a\nabla u)d\xi ,& u\in H^1({\mathbb{T}}^d),\\ +\infty ,& u\in L^2({\mathbb{T}}^d)\setminus H^1({\mathbb{T}}^d),\end{array}$$

$$\Psi (u):=\mathrm{cl}\tilde{\Psi }(u):=\inf \left\{\underset{n\to \infty }{\mathrm{liminf}}\text{leavevmode}\tilde{\Psi }(u_n)|\text{leavevmode}{u}_n\to u\in L^2({\mathbb{T}}^d)\text{strongly}\right\},$$

$$⟨\eta ,\zeta ⟩\le K\psi (\zeta )\forall \eta \in \varphi (\zeta )\forall \zeta \in {\mathbb{R}}^d.$$

$$\theta (s)={\int }_0^s{\theta }_{+}^{\prime }(r)dr,s\ge 0.$$

$$K\theta (s)\ge \theta (2s)={\int }_0^{2s}{\theta }_{+}^{\prime }(r)dr\ge {\int }_s^{2s}{\theta }_{+}^{\prime }(r)dr\ge s{\theta }_{+}^{\prime }(s).$$

$${\psi }^{\lambda }(\zeta ):=\underset{\eta \in {\mathbb{R}}^d}{\inf }\left[\psi (\eta )+\frac{1}{2\lambda }|\zeta -\eta {|}^2\right],\zeta \in {\mathbb{R}}^d,\lambda >0,$$

$$|\psi (\zeta )-{\psi }^{\lambda }(\zeta )|\le C\lambda (1+\psi (\zeta ))\forall \zeta \in {\mathbb{R}}^d.$$

$$|\psi (\zeta )-{\psi }^{\lambda }(\zeta )|\le \lambda \underset{\eta \in \varphi (\zeta )}{\sup }|\eta {|}^2\forall \zeta \in {\mathbb{R}}^d.$$

$$|\eta {|}^2\le C(1+⟨\eta ,\zeta ⟩)\forall \eta \in \varphi (\zeta )\forall \zeta \in {\mathbb{R}}^d.$$

$$⟨\eta ,\zeta ⟩\le C\psi (\zeta )\forall \eta \in \varphi (\zeta )\forall \zeta \in {\mathbb{R}}^d.$$

$$|a(\xi )\zeta {|}^2\ge \kappa |\zeta {|}^2\text{ for every }\zeta \in {\mathbb{R}}^d\text{ and every }\xi \in {\mathbb{T}}^d.$$

$$\mathrm{div}{b}_i=0\text{on Td for every}1\le i\le N.$$

$${\mathcal{L}}^{\ell }(u,v):={\int }_{{\mathbb{T}}^d}⟨\ell \nabla u,\ell \nabla v⟩d\xi ,u,v\in H^1({\mathbb{T}}^d),$$

$$L^a|a\nabla f{|}^2-2⟨a\nabla f,a\nabla L^{a}f⟩\ge \frac{K}{2}|a\nabla f{|}^2,\forall f\in {\Lambda }^a.$$

$$\sum _{k=1}^d\sum _{q=1}^d\left[a_{qj}{\partial }_k{a}_{qi}+a_{qi}{\partial }_k{a}_{qj}\right]=0\text{on }{\mathbb{T}}^d,$$

$$|a\nabla P_t^{a}f|\le e^{-2Kt}{P}_t^a|a\nabla f|\forall t\ge 0\forall f\in C^1(M^a).$$

$$\tilde{\Psi }(J_{\delta }^{0}u)\le \tilde{\Psi }(u)$$

$$\left|a\nabla J_{\delta }^{0}u\right|=$$

$$\le$$

$${\int }_{{\mathbb{T}}^d}\theta (J_{\delta }^{a}u)d\xi \le {\int }_{{\mathbb{T}}^d}\theta (u)d\xi .$$

$$\tilde{\Psi }(J_{\delta }^{0}u)=$$

$$\le$$

$$\tilde{\Psi }(J_{\delta }^{a}u)\le \tilde{\Psi }(u)$$

$$\sum _{k=1}^d\sum _{p=1}^d\left[b_{ik}(a_{pl}{\partial }_k{a}_{pj}+a_{pj}{\partial }_k{a}_{pl})-a_{pk}(a_{pj}{\partial }_k{b}_{il}+a_{pl}{\partial }_k{b}_{ij})\right]=0\text{on }{\mathbb{T}}^d,$$

$$\beta {\int }_{{\mathbb{T}}^d}⟨\beta G_{\beta }^{a}b\nabla f-\beta b\nabla G_{\beta }^{a}f,b\nabla f⟩d\xi \ge c\mathcal{A}(f,f),\forall f\in S.$$

$$(b\nabla (\mathrm{div}(\mathbb{A}\nabla f){)}_i$$

Full text

The author would like to thank Max von Renesse, Hiroshi Kawabi and Lisa Beck for stimulating discussions on the topic of this work. The author is indebted to the anonymous referees for their suggestions of several improvements. Continuous support by the M.O.P.S. program is gratefully acknowledged.

1. Introduction

We shall study the nonlinear stochastic partial differential equation (nonlinear SPDE) with singular drift and gradient-type multiplicative Stratonovich noise

[TABLE]

In order to eliminate boundary or curvature effects from the underlying space, we shall consider the SPDE on $\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$, $d\geq 1$. Here, $a:\mathbb{T}^{d}\to\mathbb{R}^{d\times d}$, $b:\mathbb{T}^{d}\to\mathbb{R}^{N\times d}$ are $C^{1}$-coefficient fields (where $a^{\ast}$ denotes the transpose of $a$) and $\{W_{t}\}_{t\geq 0}$ is an $\mathbb{R}^{N}$-valued standard Wiener process, $N\geq 1$. In this work, we shall prove well-posedness for initial data $x\in L^{2}(\mathbb{T}^{d})$ or, more generally, for $x\in L^{2}(\Omega,{\mathcal{F}}_{0},\mathbb{P};L^{2}(\mathbb{T}^{d}))$. We assume a linear growth condition on $\phi:\mathbb{R}^{d}\to\mathbb{R}^{d}$, where we also may consider the multi-valued case $\phi:\mathbb{R}^{d}\to 2^{\mathbb{R}^{d}}$. Furthermore, a geometric curvature-dimension condition on the Riemannian metric $g_{a}:=(a^{\ast}a){}^{-1}$ and a (defective) commutator estimate for the first order operator $u\mapsto b\nabla u$ are assumed.

The drift operator $u\mapsto\operatorname{div}(a^{\ast}\phi(a\nabla u))$ is a distorted $p$-Laplace-type operator, when $\phi(\zeta)=|\zeta|^{p-2}\zeta$, $p\in[1,2]$, which reduces to the linear diffusion operator $u\mapsto\operatorname{div}(a^{\ast}a\nabla u)$ for $p=2$ and includes multi-valued examples as the total variation flow operator $u\mapsto\operatorname{div}(\operatorname{sgn}(\nabla u))$ for $p=1$, $a=1$, which is also called $1$-Laplace. The equation is perturbed by independent Brownian motions $\{W_{t}^{i}\}_{t\geq 0}$, $1\leq i\leq N$, driven by family of divergence-free $C^{1}$-vector fields $b_{i}$, $1\leq i\leq N$ acting in gradient direction, which may be degenerate, so that the “deterministic” PDE case is covered for $b\equiv 0$.

Equations of the type (1.1), have previously been studied in [6] (for the case $p>1$), in [17] (for the case of a domain with symmetries and Neumann boundary conditions), in [42] (for Dirichlet boundary conditions), and in [7] (by an approach with weak solutions in the sense of distributions). As the coefficient field of the noise term is given by an unbounded operator in space, the Itô-analogue of (1.1) is ill-posed in general. The equation is discussed in [11, Section 7] as an example for an approach via a transformation by a group of random multipliers. Note that our approach does not rely on a transformation of (1.1) to a random PDE — rather than that, we obtain the unique solutions in terms of (stochastic) variational inequalities by a multi-step approximation procedure and a perturbation argument which relies on a weak defective commutator bound formulated in terms of Dirichlet forms, see [25, 39] for this notion. The conditions are discussed in Section 2 below. In particular, we need a curvature bound related to heat kernel estimates and lower bounds for Ricci curvature in order to derive a priori estimates for the singular equation. The higher order a priori estimates, which we need for the approximation procedure in the proof of the main result, are then derived from the defective commutator bounds. In [52], some previous results and the rough idea, which this work is based on, have been proposed by the author. In the future, equations involving more general Dirichlet operators, e.g. of nonlocal type, could be considered.

As the drift term in the SPDE (1.1) is singular, it is a known issue that solutions to the SPDE do not satisfy an Itô-equation in general (on these lines, see e.g. [28]), when, for instance, there is no Sobolev embedding for the energy of the drift available (as is e.g. in [37] for $p>1\vee\frac{2d}{d+2}$). See [41] for an approach for nonlinearities that satisfy an superlinearity condition at infinity and see [54] for the case that the drift is perturbed with a first-order transport term. In our situation, the Itô-Stratonovich correction and the special linear structure of the noise adds some regularity to the equation, however, even for initial data in $H^{1}(\mathbb{T}^{d})$ and for $p\approx 1$, to the best of our knowledge, the (limit) solutions to (1.1) can merely be characterized in terms of stochastic variational inequalities (SVI), see e.g. [10]. Among others, SVI-solutions to stochastic evolution equations have also been discussed in [45, 12, 27, 26, 29]. As seen in [30], the SVI-approach is quite robust under perturbations of the convex-subpotential-type drift with respect to the Mosco-topology. In this work, we prove existence and uniqueness for initial data in $L^{2}(\mathbb{T}^{d})$, see our main result Theorem 4.1 in Section 4 below.

Let us point out that we generalize the previously known results on solutions to (1.1) in the following aspects. As we assume periodic boundary conditions, we are able to dispense with the requirement of the boundary to be assumed to be $C^{3}$ and coefficient fields to be assumed to be perpendicular to the boundary and $C^{2}$. We do merely require that the driving vector fields $b_{i}$, $1\leq i\leq N$ are $C^{1}$, divergence-free and pointwise linearly independent — we do not need to assume commutation111We remark that there is another way to dispense with commutation via a transformation to a flow of diffeomorphisms, see e.g. [15]. here, as is done e.g. in the classical linear SPDE case in [18, 19]. The introduction of the deformed diffusivity is new and may possibly be applied to more general compact Riemannian manifolds than the torus in the future. Apart from the geometric deformation, we also include more general nonlinearities (in the spirit of [28, Section 7]), extending the results of [17, 30] from homogeneous nonlinearities to ones satisfying a doubling condition, see Example 2.2 below. In fact, the nonlinearities might become multi-valued so that equation (1.1) and its Itô-Stratonovich corrected form (2.1) below are actually given by the more general subpotential equation (2.2) below, which involves the relaxed convex potentials. In the sequel, all of our analysis is generally referring to equation (2.2) below rather than to the formal equation (1.1).

Possible future topics for equations of this type include ergodicity and uniqueness of invariant measures for Markov semigroups (compare also [29, 38, 8, 22, 23] for additive noise), stability under perturbations of the drift or noise (cf. [30, 16]) and regularity (see e.g. [14, 13]) or an approach via entropy solutions [20] or renormalized solutions [48]. We also refer to [21, 53, 24] for porous media type stochastic equations with gradient noise.

Notation

Denote by \mathbb{T}^{d}:=\mathbb{R}^{d}\big{/}\mathbb{Z}^{d} the standard flat torus of dimension $d\geq 1$, equipped with the $d$-dimensional Lebesgue measure $d\xi$ on its Borel $\sigma$-algebra $\mathcal{B}(\mathbb{T}^{d})$. On $\mathbb{R}^{d}$, the Euclidean norm and inner product are denoted by $|\cdot|$ and $\langle\cdot,\cdot\rangle$ respectively. For $\alpha\in\mathbb{R}$, we denote the linear operator $u\mapsto\alpha u$ (on some vector space) simply by $\alpha$. Denote $H:=L^{2}(\mathbb{{T}}^{d})$, $S:=H^{1}(\mathbb{{T}}^{d})$. Let $S^{\ast}$ denote the topological dual of $S$. Note that the embedding $S\hookrightarrow H$ is dense and compact. Let $(\mathcal{E},D(\mathcal{E}))$ be the Dirichlet form of the Laplace-Beltrami operator $L:=\Delta:=\operatorname{div}(\nabla\cdot)$ on $H$, that is, $D(\mathcal{{E}}):=S$ and

[TABLE]

see [25, 39]. Let $(G_{\alpha})_{\alpha>0}$ be the resolvent of $L$, i.e., for $\alpha>0$, $G_{\alpha}u:=(\alpha-L)^{-1}u$, $u\in H$. For convenience, we shall also introduce the alternative resolvent $J_{\delta}u:=(1-\delta L)^{-1}u$, where $u\in H$ and $\delta>0$. Clearly, $J_{\delta}=\frac{1}{\delta}G_{1/\delta}$ for every $\delta>0$. The resolvent $J_{\delta}$, when considered both as a map from $H$ to $H$ or as a map from $S$ to $S$, is a contraction. Denote the Yosida-approximation of $L$* *by $L^{(\delta)}u:=LJ_{\delta}u=\frac{{1}}{\delta}(J_{\delta}-1)u$, $u\in H$. $L^{(\delta)}:H\to H$ is a negative definite bounded linear operator with operator norm bounded by $\frac{1}{\delta}$. Furthermore, for $\alpha>0$, $u,v\in D(\mathcal{E})$, let $\mathcal{E}_{\alpha}(u,v):=\mathcal{E}(u,v)+\alpha(u,v)_{H}$ be inner products for $S$ with the property that ${\mathcal{E}}_{\alpha_{1}}$, ${\mathcal{E}}_{\alpha_{2}}$ are mutually equivalent for $\alpha_{1},\alpha_{2}>0$. For $\beta>0$, define also approximate forms \mathcal{E}^{(\beta)}(u,v):=\beta(u-\text{\beta}G_{\text{\beta}}u,v)_{H}, $u,v\in H$, see e.g. [39, Chapter I, p. 20]. Set also $\mathcal{E}_{\alpha}^{(\beta)}(u,v):=\mathcal{E}^{(\beta)}(u,v)+\alpha(u,v)_{H}$, $\alpha,\beta>0$, $u,v\in H$. Denote by $(P_{t})_{t\geq 0}$ the *heat semigroup *associated to $L$. We shall denote the trace of a square matrix by $\operatorname{Tr}$.

Organization of the paper

In Section 2, which is subdivided into several subsections, we shall discuss the different terms of equation (1.1) and the main assumptions on those. Here, also the main hypotheses of this work are stated and discussed. In Section 3, we shall introduce and discuss the concept of so-called stochastic variational inequality (SVI) solutions to our equation. Section 4, which is subdivided into several subsections, contains the statement and the proof of our main result, including the a priori estimates, the passage to the limit of the approximating equations, the existence and the uniqueness result. In Appendix A, we recall the conditions from [28] which are needed to guarantee the existence of approximating solutions to equation (1.1). The concluding Appendix B contains the postponed proof of Proposition 2.11.

2. Main hypotheses

Let $\{W_{t}\}_{t\geq 0}$ be a Wiener process on $\mathbb{R}^{N}$, modeled on a filtered probability space $(\Omega,{\mathcal{F}},\{{\mathcal{F}}_{t}\}_{t\geq 0},\mathbb{P})$ that satisfies the usual conditions.

Consider the following Itô-SPDE in $H$,

[TABLE]

Equation (2.1) is the formal analogue to equation (1.1), after passing over to an (merely formal) Itô-Stratonovich correction, see e.g. [34, 33] and compare with the previous works [7, 17, 6, 52].

The precise assumptions on $a,b$ and $\phi$ will be stated below. The operator $\frac{1}{2}L^{b}$ is a realization of $u\mapsto\frac{1}{2}\operatorname{div}[b^{\ast}b\nabla u]$, which equals the Itô-Stratonovich correction $u\mapsto\frac{1}{2}\operatorname{Tr}[b\nabla(b\nabla u)]$ of the noise term $u\mapsto\langle b\nabla u,\circ dW_{t}\rangle$, whenever $\operatorname{div}b_{i}=0$ for all $1\leq i\leq N$.

2.1. The nonlinearity $\phi$

To improve precision once more, instead of (2.1), we shall in fact study the following Itô-SPDE (stochastic evolution equation / inclusion222The set membership symbol is to be understood as an equality whenever the r.h.s. is single-valued.)

[TABLE]

where $\partial\Psi$ denotes the subdifferential333The subdifferential $\partial F:H\to 2^{H}$ of a convex, lower semi-continuous, proper map $F:H\to[0,+\infty]$ is defined by $y\in\partial F(x)$, $x,y\in H$, whenever $(y,z-x)_{H}\leq F(z)-F(x)$ for every $z\in H$, see e.g. [9]. of $\Psi$, which, in turn, is defined to be the lower semi-continuous (l.s.c.) envelope of the map

[TABLE]

where $\psi:\mathbb{R}^{d}\to[0,\infty)$ is some convex potential of $\phi$, that is, $\langle\eta,\zeta-\xi\rangle\leq\psi(\zeta)-\psi(\xi)$ for every $\eta\in\phi(\xi)$ and all $\xi,\zeta\in\mathbb{R}^{d}$. The l.s.c. envelope is defined by

[TABLE]

see e.g. [3] for further details.

Assumption 2.1**.**

Let us assume that there exist constants $C,K>0$ and a map $\theta:[0,\infty)\to[0,\infty)$ such that

1. (N)

$\psi(\zeta)=\theta(|\zeta|)$, for all $\zeta\in\mathbb{R}^{d}$ and $\theta$ is convex, continuous, and satisfies $\theta(0)=0$, $\lim_{r\to\infty}\theta(r)=\infty$, $\theta(r)\leq C(1+|r|^{2})$, $\theta(2r)\leq K\theta(r)$ for $r\geq 0$.

Note that the doubling condition $\theta(2r)\leq K\theta(r)$, $r\geq 0$ is also known as $\Delta_{2}$-condition in the literature, see e.g. [44]. Our assumptions on $a$ shall be made precise further below.

Example 2.2**.**

Condition (N) is e.g. satisfied for the following choices of $\psi$. For $\zeta\in\mathbb{R}^{d}$, let

$p$**-Laplace, total variation flow: **

$\psi(\zeta):=\frac{1}{p}|\zeta|^{p}$, $p\in[1,2]$,

**Logarithmic diffusion: **

$\psi(\zeta):=(1+|\zeta|)\log(1+|\zeta|)-|\zeta|$,

**Minimal surface flow: **

$\psi(\zeta):=\sqrt{1+|\zeta|^{2}}$,

**Curve shortening flow: **

$\psi(\zeta):=|\zeta|\arctan(|\zeta|)-\frac{1}{2}\log(|\zeta|^{2}+1)$.

Remark 2.3*.*

Note that e.g. for $\psi(\zeta)=|\zeta|$, the subdifferential $\phi=\partial\psi$ becomes multi-valued, and the l.s.c. envelope $\Psi$ is a Radon measure on the space of functions of bounded variation, see [3] for details. At this point it is enough to recall that the above l.s.c. envelope exists in $L^{2}(\mathbb{T}^{d})$.

We continue with two technical lemmas needed later.

Lemma 2.4**.**

Assume that (N) holds. Then

[TABLE]

Proof.

By (N) and [51, Chapter II, Example 8.A], there exists a function $\theta_{+}^{\prime}:[0,\infty)\to[0,\infty)$ such that $\theta_{+}^{\prime}$ is non-decreasing, right-continuous and satisfies

[TABLE]

Also, for $r\geq 0$, $\theta_{+}^{\prime}(r)\geq\sup_{s\in\partial\theta(r)}|s|$. Using (N) again, for $s\geq 0$,

[TABLE]

However, it is easy to see that from $\psi=\theta(|\cdot|)$, it follows that $\langle\eta,\zeta\rangle\leq|\zeta|\theta_{+}^{\prime}(|\zeta|)$, for all $\zeta\in\mathbb{{R}}^{d}$ and $\eta\in\phi(\zeta)$, cf. [51, Chapter II, Proposition 8.6]. Hence the claim follows. ∎

For the statement of the following lemma and for use further below, we shall introduce the notion of the so-called Moreau-Yosida approximation $\{\psi^{\lambda}\}_{\lambda>0}$ of $\psi$, that is, the family of continuous convex functions defined by the following variational formula

[TABLE]

see [2, p. 266] or [9, p. 97] for further details.

Lemma 2.5**.**

Assume that (N) holds and let $\psi^{\lambda}$, $\lambda>0$ be the Moreau-Yosida approximation of $\psi$. Then, there exists a constant $C>0$ (not depending on $\lambda$) such that

[TABLE]

Proof.

As above, denote $\phi=\partial\psi$. By the arguments preceding [30, Eq. (A.4) in Appendix A], we get in this slightly more general situation that

[TABLE]

By [28, Proof of Proposition 7.1] and (N), there exists a constant $C>0$, such that

[TABLE]

Again, by (N) and by Lemma 2.4 there exists another constant $C>0$ such that

[TABLE]

Combining these inequalities finishes the proof. ∎

2.2. Basic hypotheses on the coefficient fields $a$ and $b$

We shall formulate the main assumptions on $a$ and $b$.

Assumption 2.6**.**

Assume that $a\in C^{1}(\mathbb{T}^{d};\mathbb{R}^{d\times d})$ and that there exists a constant $\kappa>0$ such that

[TABLE]

Assumption 2.7**.**

Assume that $b\in C^{1}(\mathbb{T}^{d};\mathbb{R}^{N\times d})$ such that the rows $b_{i}$, $1\leq i\leq N$ of $b$ satisfy

[TABLE]

Let $\ell\in\{a,b\}$. Let $L^{\ell}$ denote the Dirichlet operator associated to the Dirichlet form

[TABLE]

where $\ell z$ denotes the application of matrix-multiplication for $z\in\mathbb{R}^{d}$. For smooth functions $u\in D(L^{\ell})\cap C^{\infty}(\mathbb{T}^{d})$, by $C^{1}$-regularity of the coefficients, we have that $L^{\ell}u=\operatorname{div}(\ell^{\ast}\ell\nabla u)$, where $\ell^{\ast}$ denotes the matrix-adjoint of $\ell$. Denote the associated Dirichlet operators by $L^{a}$, $L^{b}$, respectively, with Dirichlet forms ${\mathcal{A}}:={\mathcal{L}}^{a}$, ${\mathcal{B}}:={\mathcal{L}}^{b}$, respectively.

Remark 2.8*.*

Note that Assumption 2.6 implies that $D({\mathcal{L}}^{a})=D({\mathcal{A}})=D(\mathcal{E})=S=H^{1}(\mathbb{T}^{d})$ and that $-L^{a}$ is uniformly elliptic with domain equal to $D(-L^{a})=H^{2}(\mathbb{T}^{d})$. Note also that $b$ and thus $L^{b}$ may be degenerate.

Lemma 2.9**.**

Suppose that $a\in C^{1}(\mathbb{T}^{d};\mathbb{R}^{d\times d})$. Then condition (E) is equivalent to $a_{1}(\xi),\ldots,a_{d}(\xi)$ being linearly independent in $\mathbb{R}^{d}$ for each $\xi\in\mathbb{T}^{d}$.

Proof.

First note that $a^{\ast}a$ is precisely the Gram matrix of the vectors $a_{1},\ldots,a_{d}$. Consider the statement of $a_{1},\ldots,a_{d}$ being pointwise linearly independent. It is well-known that this is equivalent to the Gram matrix being positive definite, see e.g. [35, Chapter 10], which in turn is equivalent to (E) with $\kappa:=\min_{\xi\in\mathbb{T}^{d}}$ $\kappa(\xi)$, $\kappa(\xi)$ being the smallest eigenvalue of $a^{\ast}(\xi)a(\xi)$ (all eigenvalues are strictly positive and real). By continuous dependence of the eigenvalues of $a^{\ast}a$ on the space variable $\xi\in\mathbb{T}^{d}$, noting that $\mathbb{T}^{d}$ is compact, we argue by contradiction to see that $\kappa>0$. ∎

2.3. Curvature-dimension condition

In all of this subsection, assume condition (E) and (N). Set $M:=\mathbb{T}^{d}$ and set $g_{a}:=(a^{\ast}a)^{-1}$. By condition (E), $(M,g_{a})$ is a Riemannian manifold which is quasi-isometric444See [32, p. 93] for this notion. to $M$, when equipped with the flat metric. We write $M^{a}$ if we want to emphasize the choice of the metric. We denote the volume measure on $M^{a}$ by $d\nu:=\sqrt{\det g_{a}}\,d\xi$. Note that $(M^{a},g_{a})$, equipped with the Lebesgue measure, is thus a weighted manifold $(M^{a},g_{a},d\xi)$ with density $\rho_{a}:=\sqrt{\det(a^{\ast}a)}$ with respect to $\nu$. Note that by [32, Exercise 3.12], $L^{a}=\Delta^{a}$ on the weighted manifold $(M^{a},g_{a},d\xi)$, where $\Delta^{a}$ denotes the weighted Laplace-Beltrami operator of $M^{a}$.

Definition 2.10**.**

Let $\Lambda^{a}:=\{f\in S\;:\>L^{a}f\in S\}$. We say that $(M^{a},g_{a},d\xi)$ satisfies a *Bakry-Émery curvature-dimension condition *$BE(K,\infty)$ if there exists $K\in\mathbb{R}$ with

[TABLE]

Proposition 2.11**.**

Suppose that (E) holds. Suppose that $a\in C^{2}(\mathbb{T}^{d};\mathbb{R}^{d\times d})$ and that for all $1\leq i,j\leq d$,

[TABLE]

where $a=(a_{ij})$. Then condition (BE) holds for some $K\leq 0$.

Proof.

See Appendix B. ∎

Note that condition (2.3) is symmetric in the sense that interchanging the indices $i$ and $j$ yields the same condition.

Theorem 2.12**.**

Let $(P_{t}^{a})_{t\geq 0}$ be the heat semigroup associated to ${\mathcal{A}}$. Then the following condition is equivalent to condition $BE(K,\infty)$.

There exists a constant $K\in\mathbb{R}$, such that

[TABLE]

Proof.

See e.g. [1, 5]. ∎

See [1, 5, 56, 57, 31, 58, 55] for the terminology and further results on equivalent curvature-dimension conditions as well as Ricci curvature bounds in weighted Riemannian manifolds.

Lemma 2.13**.**

Let condition (N) and (E) hold. Let $(P_{t}^{a})_{t\geq 0}$ be the heat semigroup associated to ${\mathcal{A}}$. Assume that $BE(K,\infty)$ holds for some $K\leq 0$. Let $J_{\delta}^{0}:=(1-\delta(L^{a}+2K))^{-1}$, $\delta>0$ be the resolvent associated to ${\mathcal{A}}_{-2K}={\mathcal{A}}-2K(\cdot,\cdot)_{H}$. Then

[TABLE]

for any $u\in S$ and any $\delta>0$, where $\tilde{\Psi}$ is as in (2.3).

Proof.

Compare with [28, Proof of Example 7.11]. Let $(P_{t}^{0})_{t\geq 0}$ be the $C_{0}$-semigroup associated to $J_{\delta}^{0}$, $\delta>0$. Then by the Trotter product formula [46, Ch. VIII.8], $P_{t}^{0}=e^{2Kt}P_{t}^{a}$, $t\geq 0$. We get for $u\in C^{1}(\mathbb{T}^{d})$ and $\delta>0$ that

[TABLE]

where we have used Theorem 2.12. Since $J_{\delta}^{a}$, $\delta>0$ is Markovian symmetric on $L^{2}(\mathbb{T}^{d})$, we get by an application of [40, Theorem 3] and condition (N) that for $u\in L^{2}(\mathbb{T}^{d})$,

[TABLE]

Altogether,

[TABLE]

density of $C^{1}(\mathbb{T}^{d})\subset S$ and Lebesgue’s dominated convergence theorem completes the proof. ∎

Remark 2.14*.*

Note that

1. (i)

the above statement remains true if $BE(K,\infty)$ holds for $K>0$. If one investigates the proof carefully, one sees that then it even holds that

[TABLE]

for any $u\in S$ and any $\delta>0$; 2. (ii)

the above statement remains true if $\tilde{\Psi}$