A modified Kardar-Parisi-Zhang model

Abstract

A one dimensional stochastic differential equation of the form \[dX=A X dt+\tfrac12 (-A)^{-\alpha}\partial_\xi[((-A)^{-\alpha}X)^2]dt+\partial_\xi dW(t),\qquad X(0)=x\] is considered, where $A=\tfrac12 \partial^2_\xi$. The equation is equipped with periodic boundary conditions. When $\alpha=0$ this equation arises in the Kardar-Parisi-Zhang model. For $\alpha\ne 0$, this equation conserves two important properties of the Kardar-Parisi-Zhang model: it contains a quadratic nonlinear term and has an explicit invariant measure which is gaussian. However, it is not as singular and using renormalization and a fixed point result we prove existence and uniqueness of a strong solution provided $\alpha>\frac18$.