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 ∂tu+q(x,D)u+∂xf(t,x,u)=h1(t,x,u)+h2(t,x,u)Ft,x∂tu+q(x,D)u+∂xf(t,x,u)=h1(t,x,u)+h2(t,x,u)Ft,x for u:(t,x)∈(0,∞)×R↦u(t,x)∈Ru:(t,x)∈(0,∞)×R↦u(t,x)∈R, where q(x,D)q(x,D) is a pseudo-differential operator with negative definite symbol of variable order which generates a stable-like process with transition density, f,h1,h2:[0,∞)×R×R→Rf,h1,h2:[0,∞)×R×R→R are measurable functions, and Ft,xFt,x stands for a Lévy space-time white noise. We investigate the stochastic equation on the whole space RR 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.