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January 2008 Rates of convergence of a transient diffusion in a spectrally negative Lévy potential
Arvind Singh
Ann. Probab. 36(1): 279-318 (January 2008). DOI: 10.1214/009117907000000123


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 Xt/tκ 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.


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Arvind Singh. "Rates of convergence of a transient diffusion in a spectrally negative Lévy potential." Ann. Probab. 36 (1) 279 - 318, January 2008.


Published: January 2008
First available in Project Euclid: 28 November 2007

zbMATH: 1130.60084
MathSciNet: MR2370605
Digital Object Identifier: 10.1214/009117907000000123

Primary: 60J60
Secondary: 60J55

Keywords: diffusion with random potential , generalized Ornstein–Uhlenbeck process , Lévy process with no positive jumps , rates of convergence

Rights: Copyright © 2008 Institute of Mathematical Statistics

Vol.36 • No. 1 • January 2008
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