The authors construct a strong solution for the stochastic Hamilton-Jacobi equation by using classical mechanics before the caustics. The authors 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 solution for the above two equations is given in terms of the stochastic Hamilton-Jacobi function under a no-caustic condition. The authors construct the heat kernel for harmonic oscillator potentials in hyperbolic space and for harmonic oscillator potentials in Euclidean spaces thereby obtaining the stochastic Mehler formula.
In this paper, the authors restrict the Stratonovich white noise term to be white noise in time. The authors are currently investigating whether our treatment can be generalized to include white noise in space and the (much more difficult) white noise in space-time.