Central Limit Theorem for a Self-Repelling Diffusion

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 $n<\mathrm{\infty }$ and $a_1,\cdots ,a_n>0$. In dimension $d\ge 3$, 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 $d=1$.