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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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