On stochastic diffusion equations and stochastic Burgers’ equations

In this paper we construct a strong solution for the stochastic Hamilton Jacobi equation by using stochastic classical mechanics before the caustics. We thereby obtain the viscosity solution for a certain class of inviscid stochastic Burgers’ equations. This viscosity solution is not continuous beyond the caustics of the corresponding Hamilton Jacobi equation. The Hopf–Cole transformation is used to identify the stochastic heat equation and the viscous stochastic Burgers’ equation. The exact solutions for the above two equations are given in terms of the stochastic Hamilton Jacobi function under a no-caustic condition. We construct the heat kernel for the stochastic heat equation for zero potentials in hyperbolic space and for harmonic oscillator potentials in Euclidean space thereby obtaining the stochastic Mehler formula.