The Annals of Probability

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

Philippe Carmona

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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 $\mathbb{E}[e^{mV(t)}]=e{-t\phi(m)}$.

Article information

Ann. Probab., Volume 25, Number 4 (1997), 1774-1788.

First available in Project Euclid: 7 June 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60J30 60J15 60J65

Diffusion process random walk random media


Carmona, Philippe. The mean velocity of a Brownian motion in a random Lévy potential. Ann. Probab. 25 (1997), no. 4, 1774--1788. doi:10.1214/aop/1023481110.

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