Central Limit Theorem for a Self-Repelling Diffusion

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 $d=1$.