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

Abstract

In this paper, we study the initial value problem for a class of non-linear stochastic equations of Burgers type of the following form ∂ t u + q ( x , D ) u + ∂ x f ( t , x , u ) = h 1 ( t , x , u ) + h 2 ( t , x , u ) F t , x for u : ( t , x ) ∈ ( 0 , ∞ ) × R ↦ u ( t , x ) ∈ R , 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, f , h 1 , h 2 : [ 0 , ∞ ) × R × R → R are measurable functions, and F t , x stands for a Lévy space-time white noise. We investigate the stochastic equation on the whole space R 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.