The Annals of Applied Probability

Stochastic functional differential equations driven by Lévy processes and quasi-linear partial integro-differential equations

Xicheng Zhang

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In this article we study a class of stochastic functional differential equations driven by Lévy processes (in particular, $\alpha$-stable processes), and obtain the existence and uniqueness of Markov solutions in small time intervals. This corresponds to the local solvability to a class of quasi-linear partial integro-differential equations. Moreover, in the constant diffusion coefficient case, without any assumptions on the Lévy generator, we also show the existence of a unique maximal weak solution for a class of semi-linear partial integro-differential equation systems under bounded Lipschitz assumptions on the coefficients. Meanwhile, in the nondegenerate case (corresponding to $\Delta^{\alpha/2}$ with $\alpha\in(1,2]$), based upon some gradient estimates, the existence of global solutions is established too. In particular, this provides a probabilistic treatment for the nonlinear partial integro-differential equations, such as the multi-dimensional fractal Burgers equations and the fractal scalar conservation law equations.

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Ann. Appl. Probab., Volume 22, Number 6 (2012), 2505-2538.

First available in Project Euclid: 23 November 2012

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Zentralblatt MATH identifier

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 35R09: Integro-partial differential equations [See also 45Kxx]

Lévy processes Feyman–Kac formula fractal Burgers equation


Zhang, Xicheng. Stochastic functional differential equations driven by Lévy processes and quasi-linear partial integro-differential equations. Ann. Appl. Probab. 22 (2012), no. 6, 2505--2538. doi:10.1214/12-AAP851.

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