Stochastic functional differential equations driven by L\'{e}vy processes and quasi-linear partial integro-differential equations

TL;DR

This paper investigates stochastic functional differential equations driven by Lévy processes, establishing local and global solutions, and connects these to quasi-linear and semi-linear partial integro-differential equations, including applications to fractal Burgers and conservation laws.

Contribution

It introduces new existence and uniqueness results for solutions of stochastic differential equations driven by Lévy processes and links these to nonlinear partial integro-differential equations.

Findings

Existence and uniqueness of Markov solutions for stochastic equations.

Global solutions established under nondegeneracy conditions.

Probabilistic approach applied to nonlinear PDEs like fractal Burgers.

Abstract

In this article we study a class of stochastic functional differential equations driven by L\'{e}vy 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 L\'{e}vy 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.