The variational approach to Hamilton–Jacobi equations driven by a Gaussian noise

The global existence and uniqueness of viscosity solutions to the Cauchy problem for the Hamilton–Jacobi equations in RNRN driven by additive and multiplicative Wiener processes are studied for convex Hamiltonians via variational techniques. The finite speed of propagation is also established in the multiplicative noise case for equations with Lipschitzian Hamiltonians.