Solving a non-linear stochastic pseudo-differential equation of Burgers type

In this paper, we study the initial value problem for a class of non-linear stochastic equations of Burgers type of the following form [not partial differential]tu+q(x,D)u+[not partial differential]xf(t,x,u)=h1(t,x,u)+h2(t,x,u)Ft,x for , where q(x,D) is a pseudo-differential operator with negative definite symbol of variable order which generates a stable-like process with transition density, are measurable functions, and Ft,x stands for a Lévy space-time white noise. We investigate the stochastic equation on the whole space in the mild formulation and show the existence of a unique local mild solution to the initial value problem by utilising a fixed point argument.