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 $\mathbb{E}[e^{mV(t)}]=e{-t\phi(m)}$.
Citation
Philippe Carmona. "The mean velocity of a Brownian motion in a random Lévy potential." Ann. Probab. 25 (4) 1774 - 1788, October 1997. https://doi.org/10.1214/aop/1023481110
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