Stochastic heat equation with random coefficients
Given a two-parameter Wiener process \(W=\{W(t,x), (t,x)\in [0,T] \times[0,1]\}\), the authors prove the existence of a unique adapted solution of the one-dimensional stochastic parabolic partial differential equation (SPDE) \[ \begin{aligned}\partial_tu= & {\mathcal A}_tu+ f\bigl(t,x,u(t,x)\bigr)+ g\bigl(t,x,u(t,x) \bigr)\partial^2_{t,x}W(t,x), \quad (t,x)\in [0,T]\times[0,1],\\ & u(t,0)= u(t, 1)=0, \quad u(0,x)= u_0(x),\end{aligned} \] with \({\mathcal A}_t=a(t,x) \partial^2_{xx}+ b(t,x) \partial_x\), where \(a(t,x)\), \(b(t,x)\), \(f(t,x,r)\), \(g(t,x,r)\) are predictable processes, \(f,g\) are globally Lipschitz and of at most linear growth in \(r\). In their approach the authors use the Green kernel \(\Gamma=\{\Gamma_{t, s} (x,y),\;0\leq s\leq t\leq T,\;x,y\in[0,1]\}\), to rewrite the SPDE into an evolution equation containing the stochastic integral \(\int^t_0 \int^1_0 \Gamma_{t,0}(s,y) f(s,y,u(s,y))dW^-(s,y)\). Since \(\Gamma_{t,s}(s,y)\) depends on \(W(s,z)\), \((s,z) \in[0,t] \times[0,1]\), the integrand anticipates \(W\). As shown by the authors, the choice of \(dW^-(s,y)\) as anticipating forward integral [see \textit{F. Russo} and \textit{P. Vallois}, Probab. Theory Relat. Fields 97, No. 3, 403-421 (1993; Zbl 0792.60046)] guarantees equivalence between the SPDE and the evolution equation. To solve the evolution equation, maximal estimates of the \(L^p\)-norms are needed. To obtain them the forward integral is decomposed into the anticipating Skorokhod integral which is treated by means of the Malliavin calculus [see \textit{D. Nualart} and \textit{E. Pardoux}, ibid. 78, No. 4, 535-581 (1988; Zbl 0629.60061)], and into a correcting Lebesgue integral.
stochastic partial differential equation
scientific article; zbMATH DE number 1371487