Rates of convergence of a transient diffusion in a spectrally negative Lévy potential

Abstract

We consider a diffusion process $X$ in a random Lévy potential $\mathbb{V}$ which is a solution of the informal stochastic differential equation \begin{eqnarray*}\cases{dX_t=dβ_t-{1/2}\mathbb{V}&#39;(X_t) dt,\cr X_0=0,}\end{eqnarray*} ($β$ B. M. independent of $\mathbb{V}$). We study the rate of convergence when the diffusion is transient under the assumption that the Lévy process $\mathbb{V}$ does not possess positive jumps. We generalize the previous results of Hu--Shi--Yor for drifted Brownian potentials. In particular, we prove a conjecture of Carmona: provided that there exists $0<κ<1$ such that $\mathbf{E}[e^{κ\mathbb{V}_1}]=1$, then $X_t/t^κ$ converges to some nondegenerate distribution. These results are in a way analogous to those obtained by Kesten--Kozlov--Spitzer for the transient random walk in a random environment.