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 integro-differential 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(∂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,xFt,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)L2(R)-valued solution is obtained.