Weak Approximation of a Nonlinear Stochastic Partial Differential Equation

Abstract

Consider the following formal initial-boundary value problem

$$[tex]\left\{ {\begin{array}{*{20}{c}}{\frac{\partial }{{\partial t}}u(t,x) = \frac{{{\partial ^2}}}{{\partial {x^2}}}u(t,x) + f(u(t,x)) + \sigma \zeta (t,x),{\kern 1pt} \quad t > 0,x \in (0,1),} \\{u(0,x) = {u_0}(x),\quad x \in [0,1]} \\{u(t,0) = u(t,1) = 0,\quad t \ge 0,} \\\end{array}} \right.[/tex]$$

((D))

which was investigated by many authors from quite different viewpoints (cf.[1],[3],[6]-[10]). Let (Ω𝔉,ℙ) denote the basic complete probability space and suppose that u₀:Ω×[0,1] → ℝ is pathwise continuous. If ξ is a space-time GAUSSian white noise, i.e. a centered GAUSSian 𝔇 (ℝ₊ ×[0,1])-valued random element with covariance functional

$$[tex]E\zeta (\varphi )\zeta (\psi ) = \int_{{_ + }} {\int_0^1 {\varphi (t,x)\psi (t,x)dxdt,} } [/tex]$$

then (D) represents only a symbol (𝔇′ denotes the space of SCHWARTZ distributions.). Indeed, in the considered case of a one-dimensional space parameter x∈[0, 1] it is possible to give (D) a precise mathematical meaning as follows. Let W denote a BROWNian sheet, i.e. a pathwise continuous centered GAUSSian random field defined on (math) with covariance function

$$[tex]EW(t,x)W(s,y) = (t \wedge s)(x \wedge y).[/tex]$$