## Stochastic heat equation with random coefficients.(English)Zbl 0939.60065

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 F. Russo and 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 D. Nualart and E. Pardoux, ibid. 78, No. 4, 535-581 (1988; Zbl 0629.60061)], and into a correcting Lebesgue integral.
Reviewer: R.Buckdahn (Brest)

### MSC:

 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60H07 Stochastic calculus of variations and the Malliavin calculus

### Citations:

Zbl 0792.60046; Zbl 0629.60061
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