Semi-discretization of stochastic partial differential equations on ℝ¹ by a finite-difference method

Source abstract

The paper concerns finite-difference scheme for the approximation of partial differential equations in R 1 \mathbb {R}^1 , with additional stochastic noise. By replacing the space derivatives in the original stochastic partial differential equation (SPDE, for short) with difference quotients, we obtain a system of stochastic ordinary differential equations. We study the difference between the solution of the original SPDE and the solution to the corresponding equation obtained by discretizing the space variable. The need to approximate the solution in R 1 \mathbb {R}^1 with functions of compact support requires us to introduce a scale of weighted Sobolev spaces. Employing the weighted L p L_p -theory of SPDE, a sup-norm error estimate is derived and the rate of convergence is given.