Weak Approximation of a Nonlinear Stochastic Partial Differential Equation

摘要: Consider the following formal initial-boundary value problem $$[tex]\left\{ {\begin{array}{*{20}{c}}{\frac{\partial }{{\partial t}}u(t,x) = \frac{{{\partial ^2}}}{{\partial {x^2}}}u(t,x) + f(u(t,x)) \sigma \zeta (t,x),{\kern 1pt} \quad t > 0,x \in (0,1),} \\{u(0,x) {u_0}(x),\quad x [0,1]} \\{u(t,0) u(t,1) 0,\quad \ge 0,} \\\end{array}} \right.[/tex]$$ (D) which was investigated by many authors from quite different viewpoints (cf.[1],[3],[6]-[10]). Let (Ω𝔉,ℙ) denote basic complete probability space and suppose that u0:Ω×[0,1] → ℝ is pathwise continuous. If ξ a space-time GAUSSian white noise, i.e. centered 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 symbol (𝔇′ denotes of SCHWARTZ distributions.). Indeed, in considered case one-dimensional parameter x∈[0, 1] it possible to give precise mathematical meaning as follows. W BROWNian sheet, continuous field defined on (math) function $$[tex]EW(t,x)W(s,y) (t \wedge s)(x y).[/tex]$$