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October 1997 The mean velocity of a Brownian motion in a random Lévy potential
Philippe Carmona
Ann. Probab. 25(4): 1774-1788 (October 1997). DOI: 10.1214/aop/1023481110

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

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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

Information

Published: October 1997
First available in Project Euclid: 7 June 2002

zbMATH: 0903.60065
MathSciNet: MR1487435
Digital Object Identifier: 10.1214/aop/1023481110

Subjects:
Primary: 60J60
Secondary: 60J15, 60J30, 60J65

Rights: Copyright © 1997 Institute of Mathematical Statistics

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Vol.25 • No. 4 • October 1997
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