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Stability for stochastic partial differential equations with Dirichlet white-noise boundary conditions. (English) Zbl 1052.60046

The authors study a stochastic partial differential equation with boundary noise: \[ d_tu(t,x)={\partial^2\over \partial x^2}u(t,x)dt+ \sum_{j=1}^n\left[ b_j(x){\partial\over \partial x}u(t,x)+F_j(t,x,u(t,x))\right]dW^j_t, \]
\[ u(t,0)={\dot V}_t,\qquad u(0,x)=u_0(x), \] where \(W^1,\dots,W^n,V\) are independent real Wiener processes. The solution is understood in a mild form, involving anticipating (Skorokhod) stochastic integrals. The authors [Ann. Inst. Henri Poincaré, Probab. Stat. 38, 125–154 (2002; Zbl 0998.60065)] have shown that under some regularity assumptions on \(b_j, F_j\) a unique solution exists. In the present paper, the existence of a unique invariant measure (on an appropriate function space) and the exponential mean-square stability are proved.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
35B35 Stability in context of PDEs

Citations:

Zbl 0998.60065
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References:

[1] Alòs E., Probab. Statist. 38 pp 125– (2002)
[2] Alós E., Probab. Statist. 36 pp 181– (2000)
[3] DOI: 10.1007/s00245-001-0020-z · Zbl 0995.60056 · doi:10.1007/s00245-001-0020-z
[4] Maslowski B., Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 pp 55– (1995)
[5] DOI: 10.1214/aop/1019160111 · Zbl 1044.60052 · doi:10.1214/aop/1019160111
[6] DOI: 10.1214/aop/1176988495 · Zbl 0834.60067 · doi:10.1214/aop/1176988495
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