Convergence in distribution of some self-interacting diffusions (Q2260579)

Summary: The present paper is concerned with some self-interacting diffusions \((X_t,t \geq 0)\) living on \(\mathbb R^d\). These diffusions are solutions to stochastic differential equations: \(dX_t=dB_t-g(t)\nabla V(X_t-\overline\mu_t)dt\), where \(\overline\mu_t\) is the empirical mean of the process \(X\), \(V\) is an asymptotically strictly convex potential, and \(g\) is a given positive function. We study the asymptotic behaviour of \(X\) for three different families of functions \(g\). If \(g(t)=k \log t\) with \(k\) small enough, then the process \(X\) converges in distribution towards the global minima of \(V\), whereas if \(g(t) \to c \in ]0,+\infty]\) or if \(g(t) \to g(\infty) \in [0,+\infty[\), then \(X\) converges in distribution if and only if \(\int xe^{-2V(x)}dx=0\).