THE BURGERS-EQUATION WITH A NOISY FORCE AND THE STOCHASTIC HEAT-EQUATION

We consider the multidimensional Burgers equation with a viscosity ter m and a random force modelled by a functional of time-space white nois e, {w(k)(t, x)}: (B) partial derivative u(k)/partial derivative t + la mbda Sigma(j=1)(n) u(j) partial derivative u(k)/partial derivative xj = nu Delta u(k) + w(k)(t, x); 1 less than or equal to k less than or e qual to n, (t, x) is an element of R(n+1) We discuss the equation in t he framework of a class of distribution valued stochastic processes ca lled functional processes, and interpret the products u(j) partial der ivative u(k)/partial derivative xj as Wick products. Then we show that the nonlinear equation (B) can be transformed into a linear, stochast ic heat equation with a noisy potential. This heat equation is solved explicitly in the following two cases a) For a white noise potential b ) For a positive noise potential.