Central Limit Theorem for a Self-Repelling Diffusion

TL;DR

This paper proves a Central Limit Theorem for the displacement of a 1D self-repelling diffusion process, extending known results from higher dimensions to one dimension under certain conditions.

Contribution

It establishes a CLT for the 1D self-repelling diffusion, filling a gap in the understanding of such processes in low dimensions.

Findings

CLT proven for 1D self-repelling diffusion displacement

Extends results previously known only for dimensions ≥ 3

Supports conjecture for CLT in 1D under integrability conditions

Abstract

We prove a Central Limit Theorem for the finite dimensional distributions of the displacement for the 1D self-repelling diffusion which solves \begin{equation*} dX_t =dB_t -\big(G'(X_t)+ \int_0^t F'(X_t-X_s)ds\big)dt, \end{equation*} where $B$ is a real valued standard Brownian motion and $F(x)=\sum_{k=1}^n a_k \cos(kx)$ with $n<\infty$ and $a_1,\cdots ,a_n >0$. In dimension $d\geq 3$, such a result has already been established by Horv\'ath, T\'oth and Vet\"o in \cite{HTV} in 2012 but not for $d=1,2$. Under an integrability condition, Tarr\`es, T\'oth and Valk\'o conjectured in \cite{TTV} that a Central Limit Theorem result should also hold in dimension $d=1$.

Equations

$$dX_{t}=dB_{t}-\big{(}G^{\prime}(X_{t})+\int_{0}^{t}F^{\prime}(X_{t}-X_{s})ds\big{)}dt,$$

$$dX_{t}=dB_{t}-\big{(}G^{\prime}(X_{t})+\int_{0}^{t}F^{\prime}(X_{t}-X_{s})ds\big{)}dt,\quad X_{0}=0,$$

$$\underset{t\to \infty }{\lim }\frac{X_t}{t}=0a.s.$$

$${\phi }_{2\pi }(b)(x)=\sum _{n=-\infty }^{\infty }b(x+2\pi n)$$

$$\varphi_{2\pi}(b)(x)=\frac{1}{2\pi}\sum_{k\in\mathbb{Z}}\hat{b}\big{(}\frac{k}{2\pi}\big{)}e^{ikx}.$$

$$\varphi_{2\pi}(b)(x)=\frac{\hat{b}(0)}{2\pi}+\frac{1}{\pi}\sum_{k\geqslant 1}\hat{b}\big{(}\frac{k}{2\pi}\big{)}\cos(kx).$$

$$\left\{\begin{array}[]{l}dX_{t}=dB_{t}+\sum_{j=1}^{n}ja_{j}\Big{(}\sin(jX_{t})U_{j}(t)-\cos(jX_{t})V_{j}(t)\Big{)}dt\vskip 6.0pt plus 2.0pt minus 2.0pt\\ dU_{j}(t)=\cos(jX_{t})dt,\;j=1,\ldots,n.\vskip 6.0pt plus 2.0pt minus 2.0pt\\ dV_{j}(t)=\sin(jX_{t})dt,\;j=1,\ldots,n.\end{array}\right.$$

$$U_{t}=\big{(}U_{1}(t),\cdots,U_{n}(t)\big{)}\qquad\text{ and }\qquad V_{t}=\big{(}V_{1}(t),\cdots,V_{n}(t)\big{)}.$$

$$\mu (d\theta dudv)=\nu (d\theta )\otimes \frac{e^{-\Phi (u,v)}}{C}dudv,$$

$$\Vert P_{t}g-\int g(\theta ,u,v)\mu (d\theta dudv){\Vert }_{L^2(\mu )}⩽\sqrt{1+2\eta }\Vert g-\int g(\theta ,u,v)\mu (d\theta dudv){\Vert }_{L^2(\mu )}{e}^{-\lambda t},$$

$$\lambda =\frac{\eta }{1+\eta }\frac{K_1}{1+K_2+K_3},$$

$$C_{j}(t)=U_{j}(t)\cos(jX_{t})+V_{j}(t)\sin(jX_{t})=\Big{\langle}\begin{pmatrix}U_{j}(t)\\ V_{j}(t)\end{pmatrix},\begin{pmatrix}\cos(jX_{t})\\ \sin(jX_{t})\end{pmatrix}\Big{\rangle}$$

$$S_{j}(t)=\sin(jX_{t})U_{j}(t)-\cos(jX_{t})V_{j}(t)=\Big{\langle}\begin{pmatrix}U_{j}(t)\\ V_{j}(t)\end{pmatrix},\begin{pmatrix}\sin(jX_{t})\\ -\cos(jX_{t})\end{pmatrix}\Big{\rangle}.$$

$${\eta }_t(x)=F_t(x+X_t)={\int }_0^t\sum _{k=1}^n{a}_k\cos (k(x+X_t-X_s))ds+G(x+X_t),$$

$$\eta_{t}(x)=\sum_{k=1}^{n}a_{k}\Big{(}C_{k}(t)\cos(kx)-S_{k}(t)\sin(kx)\Big{)}.$$

$$X_t=B_t+{\int }_0^t\sum _{k=1}^{n}k{a}_k{S}_k(u)du=B_t+{\int }_0^t\phi ({\eta }_u^{\prime })du,$$

$$((C_{t},S_{t}))_{t\geqslant 0}:=\Big{(}\big{(}C_{1}(t),S_{1}(t),\cdots,C_{n}(t),S_{n}(t)\big{)}\Big{)}_{t\geqslant 0}$$

$$Gf(c,s)$$

$$\pi (dcds)=\frac{e^{-\Phi (c,s)}}{C}dcds$$

$$\Big{\|}T_{t}f-\int_{{\mathbb{R}}^{2n}}f(c,s)\pi(dcds)\Big{\|}_{L^{2}(\pi)}\leq\sqrt{3}\Big{\|}f-\int_{{\mathbb{R}}^{2n}}f(c,s)\pi(dcds)\Big{\|}_{L^{2}(\pi)}e^{-\lambda t},$$

$$\displaystyle\pi\Big{(}A_{1}\times\cdots\times A_{n}\Big{)}$$

$$g(\theta ,u,v)=f(c,s),$$

$$d\begin{pmatrix}C_{1}(t)\\ S_{1}(t)\\ C_{2}(t)\\ S_{2}(t)\\ \vdots\\ C_{n}(t)\\ S_{n}(t)\end{pmatrix}=\begin{pmatrix}-S_{1}(t)\\ C_{1}(t)\\ -2S_{2}(t)\\ 2C_{2}(t)\\ \vdots\\ -nS_{n}(t)\\ nC_{n}(t)\end{pmatrix}\big{(}dB_{t}+(\sum_{k=1}^{n}ka_{k}S_{k}(t))dt\big{)}-\begin{pmatrix}C_{1}(t)\\ S_{1}(t)\\ 4C_{2}(t)\\ 4S_{2}(t)\\ \vdots\\ n^{2}C_{n}(t)\\ n^{2}S_{n}(t)\end{pmatrix}dt+\begin{pmatrix}1\\ 0\\ 1\\ 0\\ \vdots\\ 1\\ 0\end{pmatrix}dt.$$

$$T_{t}f(c,s)$$

$$1\leq\sigma^{2}\leq 1+2\Big{(}\sum_{j=1}^{n}\frac{a_{j}}{j^{2}}\Big{)}.$$

$$\Big{(}\frac{X_{t_{1}/\varepsilon}}{\sigma^{2}t_{1}/\varepsilon},\cdots,\frac{X_{t_{n}/\varepsilon}}{\sigma^{2}t_{n}/\varepsilon}\Big{)}\overset{\varepsilon\rightarrow 0}{\Longrightarrow}\Big{(}W_{t_{1}},\cdots,W_{t_{n}}\Big{)}$$

$$g(c,s)=\sum _{k=1}^{n}k{a}_k{s}_k\text{ and }h(c,s)=\sum _{k=1}^n{a}_k{c}_k.$$

$${\mathbb{E}}_{\pi }(X_t^2)$$

$$=$$

$$\lim_{t\rightarrow\infty}\frac{\mathbb{E}_{\pi}(X_{t}^{2})}{t}=1+\int_{0}^{\infty}\Big{\langle}T_{u}g,g\Big{\rangle}_{L^{2}(\pi)}du:=\sigma^{2}.$$

$$\big{\langle}Gf,f\big{\rangle}_{L^{2}(\pi)}=\big{\langle}Sf,f\big{\rangle}_{L^{2}(\pi)}=-\frac{1}{2}\int\Big{(}\sum_{j=1}^{n}j(s_{j}\partial_{c_{j}}f-c_{j}\partial_{s_{j}}f)\Big{)}^{2}d\pi.$$

Full text

[TABLE]

where $B$ is a real valued standard Brownian motion and $F(x)=\sum_{k=1}^{n}a_{k}\cos(kx)$ with $n<\infty$ and $a_{1},\cdots,a_{n}>0$.

In dimension $d\geq 3$, such a result has already been established by Horváth, Tóth and Vetö in [3] in 2012 but not for $d=1,2$. Under an integrability condition, Tarrès, Tóth and Valkó conjectured in [6] that a Central Limit Theorem result should also hold in dimension $d=1$.

Keywords: Central Limit Theorem, Self-Repelling Diffusion

MSC2010: 60F05, 60K35, 60H10, 60J60

1 Introduction

In this short note, we aim to prove the Central Limit Theorem (denoted by CLT in the sequel) for the finite dimensional distribution of the displacement for the 1D self-repelling diffusion solving

[TABLE]

where $B$ is a real valued standard Brownian motion, G(x)=\sum_{k=1}^{n}\big{(}u_{k}\cos(kx)+v_{k}\sin(kx)\big{)} and $F(x)=\sum_{k=1}^{n}a_{k}\cos(kx)$ with $n<\infty$ and $a_{1},\cdots,a_{n}>0$.

Roughly speaking, self-repelling diffusions (as considered here) are time continuous stochastic processes which solve an inhomogeneous stochastic differential equation whose drift part is evolving in time according to the whole past history of the process in such a way that it tends to push the diffusing particle away from the most visited sites.

Under the assumptions made on $F$ and $G$, the Law of Large Number has already been established in [1, Theorem 2 and Remark 1], namely

[TABLE]

A question that one may then ask is whether or not a CLT result holds. The purpose of this note is to provide a positive answer to it.

To the author’s knowledge it is the first time that such a result is obtained for a 1D self-repelling diffusion, despite being conjectured in 2012 by Tarrès, Tóth and Valkó ([6, Theorem 2 and its remark]) under a positivity222The Fourier transform of $F$ is nonnegative and an integrability333$\rho^{2}:=\int_{-\infty}^{\infty}p^{-2}\hat{F}(p)dp<\infty$ condition. Nevertheless, Horváth, Tóth and Vetö were able to prove in 2012 a CLT in dimensions $d\geq 3$ ([3, Theorem 2]) by using the Kipnis-Varadhan’s CLT result for additive functionals via a graded sector condition which turns out to fail in dimension $1$.

Before turning to the presentation of the result, let us briefly make a link between the positivity condition from [6] and the positivity of the coefficients $a_{1},\cdots,a_{n}$. Let $b\in L^{1}(\mathbb{R})\cap C^{\infty}({\mathbb{R}})$ be a function satisfying the positivity condition from [6] and let

[TABLE]

be the $2\pi-$periodization transform of $b$. It is an exercise in Fourier analysis to show that

[TABLE]

Because $b$ is even, we have

[TABLE]

The paper is organised as follows. In Section 2, we state the CLT result and present the tools and concepts involved while its proof is presented in Section 3.

2 Tools, notation and results

In this section, we introduce the mathematical background that is necessary to present the main result.

Following the same idea as in [1], set $U_{j}(t)=u_{j}+\int_{0}^{t}\cos(jX_{s})ds$ and $V_{j}(t)=v_{j}+\int_{0}^{t}\sin(jX_{s})ds$. With these new variables, we obtain the following system

[TABLE]

Since for all $j=1\cdots,n$, the functions $x\mapsto\cos(jx)$ and $x\mapsto\sin(jx)$ are $2\pi$-periodic, we can replace $X_{t}$ by $\Theta_{t}=X_{t}$ (mod $2\pi$ ) $\in S^{1}$, where $S^{1}={\mathbb{R}}/2\pi\mathbb{Z}$ denotes the 1-dimensional flat torus. This replacement allows us to use the framework from [1].

In order to shorten the notation, we let $U_{t}$ and $V_{t}$ denote the vectors

[TABLE]

Summarizing the main results from [1], we have

Theorem 1** (Theorem 5 and Theorem 6, [1]).**

) Let $(P_{t})_{t\geqslant 0}$ be the semi-group associated to the process \Big{(}(\Theta_{t},U_{t},V_{t})\Big{)}_{t\geq 0} and $P_{t}((\theta_{0},u_{0},v_{0}),d\theta dudv)$ its transition probability. Then

The unique invariant probability measure is

[TABLE]

where $\Phi(u,v)=\frac{1}{2}\sum_{k=1}^{n}a_{k}k^{2}(u_{k}^{2}+v_{k}^{2})$, $C$ is a normalization constant and $\nu(d\theta)$ is the uniform probability measure on $S^{1}={\mathbb{R}}/2\pi{\mathbb{Z}}$. 2. 2)

Let $\mu_{t}=\mathcal{L}(\Theta_{t},U_{t},V_{t})$ denote the law of $(\Theta_{t},U_{t},V_{t})$. Then for any initial distribution $\mu_{0}$, $\mu_{t}$ converges to $\mu$ in total variation. 3. 3)

For every $\eta>0$ and $g\in L^{2}(\mu)$

[TABLE]

where

[TABLE]

with explicit constants $K_{1},K_{2}$ and $K_{3}$.

In this paper, we will adopt the same point of view as in [6]: the environment seen from the particle. For that purpose, introduce the following new variables

[TABLE]

and

[TABLE]

So, if we denote by $\eta_{t}$ the potential viewed from the particle’s position, i.e

[TABLE]

then

[TABLE]

Moreover, this allows us to rewrite $X_{t}$ as

[TABLE]

where $\varphi:\Omega\rightarrow{\mathbb{R}}$ is defined by $\varphi(\omega)=\omega(0)$ and $\Omega$ is the vector space spanned by the functions $\cos(kx)$ and $\sin(kx)$ for $k=0,1,\cdots n$.

Before turning to the results, let us introduce the following notation. We denote by $T_{t}$ the semigroup induced by the process

[TABLE]

and by $G$ its infinitesimal generator. For an operator $R$, we denote its domain by $D(R)$.

Given a probability measure $\pi$ over ${\mathbb{R}}^{2n}$, we denote by $L^{2}(\pi)$ the space $L^{2}({\mathbb{R}}^{2n},\pi)$, by \Big{\langle}.,.\Big{\rangle}_{L^{2}(\pi)} the associated inner product and by \Big{\|}.\Big{\|}_{L^{2}(\pi)} the induced $L^{2}-$norm. Finally, we denote by $\mathbb{P}_{\pi}$ the law of the process with initial distribution \pi=\mathcal{L}\big{(}(C_{0},S_{0})\big{)} and by $\Longrightarrow$ the convergence in distribution.

Proposition 1**.**

For any smooth function $f$ having compact support, we have

[TABLE] 2. 2.

The process $((C_{t},S_{t}))_{t\geqslant 0}$ admits a unique invariant probability measure of the form

[TABLE]

where $\Phi(c,s)=\frac{1}{2}\sum_{k=1}^{n}a_{k}k^{2}(c_{k}^{2}+s_{k}^{2})$ and $C$ is the normalizing constant. 3. 3.

For any function $f\in L^{2}(\pi)$, we have

[TABLE]

where $\lambda=\frac{1}{2}\frac{K_{1}}{1+K_{2}+K_{3}}$ and the constants $K_{1},K_{2}$ and $K_{3}$ are those from Theorem 1.

Proof.

It follows from Itô’s formula and [5, Propositions VII.1.6 and VII.1.7]. 2. 2.

The fact that $\pi(dcds)$ is an invariant probability measure follows from Theorem 1 as well as from the equations (3) and (4). Indeed, for any $A_{j}\in\mathcal{B}({\mathbb{R}}^{2})$, we have by rotation invariance of the Gaussian measure

[TABLE]

Concerning the uniqueness, let $\nu$ be an invariant probability measure for the process $((C_{t},S_{t}))_{t\geqslant 0}$. Then define on $S^{1}\times{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$ the probability measure $\mu_{0}(d\theta dudv)=\delta_{0}\otimes\nu(dudv)$ and sample $(\Theta_{0},U_{0},V_{0})$ according to $\mu_{0}$.

By Theorem 1, $\mu_{t}$ converges to $\mu$ in total variation. In particular the marginal law of $\mu_{t}$ corresponding to $(U_{t},V_{t})$ converges to $\pi$. Thus $\nu=\pi$. 3. 3.

Let $f:{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}$ and define a function $g:S^{1}\times{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}$ by

[TABLE]

where the pairs $(c_{j},s_{j})$ are defined as in (3) and (4). Applying Itô’s formula to (3) and (4) yields

[TABLE]

Thus the evolution of $(C_{t},S_{t})$ does not depend on the dynamic of $\Theta_{t}$. Therefore

[TABLE]

where the pairs $(u_{k},v_{k})$ are such that $c_{k}=u_{k}\cos(k\theta)+v_{k}\sin(k\theta)$ and $s_{k}=u_{k}\sin(k\theta)-v_{k}\cos(k\theta)$.

The statement follows then from Theorem 1 with $\eta=1$.

∎

Our CLT result is the following

Theorem 2**.**

$\sigma^{2}:=\lim_{t\rightarrow\infty}\frac{\mathbb{E}_{\pi}(X_{t}^{2})}{t}$* exists and it satisfies*

[TABLE] 2. 2.

For any $0<t_{1}<\cdots<t_{n}<\infty$, we have

[TABLE]

under $\mathbb{P}_{\pi}$, where $W$ is a real valued standard Brownian motion.

3 Proof of Theorem 2

Throughout this section, we let $g,h:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}$ denote the functions defined by

[TABLE]

3.1 Proof of Part 1

First of all, by repeating the arguments of [6], we have as at the beginning of [6, Section 4]

[TABLE]

By the Cauchy-Schwarz inequality and the third part of Proposition 1, \Big{\langle}T_{u}g,g\Big{\rangle}_{L^{2}(\pi)} decreases exponentially fast to [math]. Hence \int_{0}^{\infty}\Big{\langle}T_{u}g,g\Big{\rangle}_{L^{2}(\pi)}du exists and, therefore, it yields

[TABLE]

Now that the existence of $\sigma^{2}$ is established, let us prove the bounds in (5). The lower bound is trivial since it follows from (3.1). In order to establish the upper bound, we follow the arguments presented in [4] based on the Kipnis-Varadhan’s CLT theorem.

Letting $G^{\ast}$ denote the adjoint of $G$ in $L^{2}(\pi)$, $S=\frac{1}{2}(G+G^{\ast})$ and $A=\frac{1}{2}(-G+G^{\ast})$ denote the symmetric and the skew-symmetric part of $G$, we have for any smooth function $f$ having compact support

[TABLE]

Hence, using the notation of [4], we have

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Because

[TABLE]

it follows from the Cauchy-Schwarz inequality that

[TABLE]

Hence, with the notation of [4],

[TABLE]

Thus, the upper bound comes from Eq. (2.1.7) in [4].

3.2 Proof of Part 2

Set $\mathcal{S}_{t}=\int_{0}^{t}g(C_{u},S_{u})du$ and denote by $\mathfrak{s}_{t}^{2}$ its variance under $\pi$.

Since $\|T_{t}g\|_{L^{2}(\pi)}$ decreases exponentially fast to [math] by Proposition 1 and $\int gd\pi=0$, then, by [2, Corollary 3.2.i], there exists a function $f\in D(G^{-1})$ such that $Gf=g$ and

[TABLE]

Because \mathbb{E}_{\pi}\Big{(}\mathcal{}S_{t}\Big{)}=0 for all $t\geq 0$, then $\|f\|_{1}=0$ implies that the process $t\mapsto\mathcal{S}_{t}$ is the null process $\mathbb{P}_{\pi}$-almost surely and the result is therefore trivial in that case.

If $\|f\|_{1}>0$, then by [2, Theorem 3.1], we have that for any $0<t_{1}<\cdots<t_{n}<\infty$

[TABLE]

under $\mathbb{P}_{\pi}$, where $W$ is a real valued standard Brownian motion.

Thus, the second part of Theorem 2 follows from the martingale approximation in the Kipnis-Varadhan approach as presented in [2].

Acknowledgement

I acknowledge financial support from the Swiss National Foundation for Research through the grants $200021\_163072$ and P$2$NEP$2\_171951$.

I am grateful to Bálint Tóth for asking me wether or not a CLT result holds in the periodic case considered in this work.