Stochastic Functional Differential Equations Driven by Levy Processes and Quasi-Linear Partial Integro-Differential Equations

Details Stochastic Functional Differential Equations Driven by Levy Processes and Quasi-Linear Partial Integro-Differential Equations Stochastic Functional Differential Equations Driven by Levy Processes and Quasi-Linear Partial Integro-Differential Equations

2011-06-18

arxiv.org/abs/1106.3601v1

Mathematics

Probability

25pp

Scientific paper

In this article we study a class of stochastic functional differential equations driven by L\'evy processes (in particular, $\alpha$-stable processes), and obtain the existence and uniqueness of Markov solutions in small time interval. 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\'evy 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 and Lipschitz assumptions on the coefficients. Meanwhile, in the non-degenerate case (corresponding to $\Delta^{\alpha/2}$ with $\alpha\in(1,2]$), basing upon some gradient estimates, the existence of global solutions is established too. In particular, this provides a probabilistic treatment for non-linear partial integro-differential equations such as the multi-dimensional fractal Burgers equations and the fractal scalar conservation law equations.

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