Stochastic PDEs: convergence to the continuum? Original Research Article

We examine the convergence properties of stochastic PDEs discretized using finite differences. In one space dimension, where the continuum solution is a stochastic process whose values are continuous functions in space, the transfer integral allows exact calculation of steady state properties, including the corrections due to finite grid spacing. The method applies to arbitrarily nonlinear PDEs, provided they have a stationary density. In two or more space dimensions, however, solution configurations are not continuous functions but only distributions. The stochastic PDE can still be solved on a finite grid of points in space, but the mean squared value at a grid point does not approach a finite limit as the grid spacing is decreased.