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Exponentially stable stationary solutions for stochastic evolution equations and their perturbation. (English) Zbl 1066.60058

For the abstract evolution equation \(dX=\bigl(AX+f(X)\bigr)\,dt+B(X)\,dW_t\), where \(A\) generates a strongly continuous semigroup with growth bound \(a\) on a separable Hilbert space \(H\), \(f:H\to H\) is globally Lipschitz with Lipschitz constant \(L_f\), \(W\) is a trace class Wiener process on some Hilbert space \(U\), and \(B\) is globally Lipschitz with Lipschitz constant \(L_B\) from \(H\) to the space of linear operators from \(U\) to \(H\), it is assumed that \[ a+L_f+L_B/2<0.\tag{1} \] Under this condition, existence of an exponentially attracting stationary solution is proved. Assuming a finite-dimensional Wiener process with \(B\) linear and commuting with \(A\), it is shown that exponential attraction holds uniformly in bounded sets of initial conditions. Finally, it is shown that the stationary solution depends continuously on perturbations of the nonlinearity \(f\), provided (1) holds uniformly.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
37L55 Infinite-dimensional random dynamical systems; stochastic equations
93E15 Stochastic stability in control theory
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