On a stochastic nonlinear equation arising from 1D integro-differential scalar conservation laws

In this paper we study the initial problem for a stochastic nonlinear equation arising from 1D integrodifferential scalar conservation laws. The equation is driven by Lévy space–time white noise in the following form: (∂t − A)u+∂xq(u) = f (u) +g(u)Ft,x for u: (t, x) ∈ (0,∞)×R →u(t, x) ∈ R, where A is an integro-differential operator associated with a symmetric, nonlocal, regular Dirichlet form, and Ft,x stands for a Lévy space–time white noise. The problem is interpreted as a stochastic integral equation of jump type involving certain convolution kernels. Existence of a unique local (in time) L2(R)-valued solution is obtained. © 2006 Elsevier Inc. All rights reserved.