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Stochastic partial differential equations with Dirichlet white-noise boundary conditions. (English) Zbl 0998.60065
The authors study the one-dimensional stochastic partial differential equation with boundary noise \[ d_tu(t,x) = \frac{\partial^2}{\partial^2x} u(t,x)dt+\Bigl(b(x) \frac{\partial}{\partial x} u(t,x) +F(t,x,u(t,x))\Bigr)dW_t, \]
\[ u(t,0)=\dot V_t,\quad u(0,x)=0,\quad (t,x)\in [0,T]\times \mathbb{R}_+, \] where \(W\) is an \(n\)-dimensional Brownian motion and \(V\) a 1-dimensional Brownian motion adapted to the filtration generated by \(W\). They study this equation by rewriting it in the anticipative evolution equation \[ u(t,x)=\int^t_0\frac{\partial p}{\partial y} (s,t,0,x)dV_s + \int^t_0\int^{+\infty}_0 p(s,t,y,x)F(s,y,u(s,y))dydW_s, \] where \(p\) is the fundamental solution of the linear homogeneous part of the original equation. Following the approach of D. Nualart and F. Viens (2000) and E. Alòs, D. Nualart and F. Viens [Ann. Inst. Henri Poincaré, Probab. Stat. 36, No. 2, 181-218 (2000; Zbl 0970.60068)] who have studied the evolution equation related to the above problem on \([0, T]\times R\), the authors prove the existence of a unique solution \(u(t,.)\) taking its values in appropriate weighted Sobolev space and study its regularity properties. Their main results are based on estimates of Skorokhod integrals of the form \(\int^t_0\int^{+\infty}_0p(s,t,y,x) F(s,y)dydW_s\).

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