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 $$\begin{eqnarray*}\cases{dX_{t}=d\beta_{t}-\frac{1}{2}\mathbb{V}'(X_{t})\,dt,\cr X_{0}=0,}\end{eqnarray*}$$ (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 $\mathbf{E}[e^{\kappa\mathbb{V}_{1}}]=1$, 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.