Central Limit Theorem for one and two dimensional Self-Repelling Diffusions

Abstract

We prove a Central Limit Theorem for the finite dimensional distributions of the displacement for the 1D self-repelling diffusion which solveswhere B is a real valued standard Brownian motion and F (x) = n k=1 a k cos(kx) with n < ∞ and a 1 , · · · , a n > 0.In dimension d ≥ 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.

Inverting the Ray-Knight identity on the line

“…Our self-repelling diffusion is different from the Brownian polymer models studied in [10,8,9,24,32,5,17], as here the interaction of the moving particle with the occupation profile occurs locally, at zero range, and is not an average over positive ranges. In other words, we do not mollify the occupation profile prior to taking its derivative.…”

Convergence of jump processes with stochastic intensity to Brownian motion with inert drift

“…BMID is just one example where a process with memory has a Gaussian stationary distribution. See [6], where Gauthier studies diffusions whose drift is also dependent on the diffusions past through a linear combination of sine and cosine functions. He shows the average displacement across time obtains a Gaussian stationary distribution as time approaches infinity.…”