The mean velocity of a Brownian motion in a random Lévy potential

Abstract

A Brownian motion in a random Lévy potential V is the informal solution of the stochastic differential equation

$$dX_t = dB_t - 1/2 V'(X_t)dt,$$

where $B$ is a Brownian motion independent of $V$.

We generalize some results of Kawazu-Tanaka, who considered for $V$ a Brownian motion with drift, by proving that $X_t /t$ converges almost surely to a constant, the mean velocity, which we compute in terms of the Lévy exponent $\phi$ of $V$, defined by $\mathbb{E}[e^{mV(t)}]=e{-t\phi(m)}$.