Viscosity limit of stationary distributions for the random forced Burgers equation

We prove convergence of stationary distributions for the randomly forced Burgers and Hamilton–Jacobi equations in the limit when viscosity tends to zero. It turns out that for all values of the viscosity $\nu$ there exists a unique (up to an additive constant) global stationary solution to the randomly forced Hamilton–Jacobi equation. The main result follows from the convergence of these solutions in a limit when $\nu$ tends to zero without changing its sign. The two limiting solutions (for different signs of the viscosity term) correspond to unique backward and forward viscosity solutions. Our approach, which is an extension of the previous work, is based on the stochastic version of Lax formula for solutions to the initial and final value problems for the viscous Hamilton–Jacobi equation.