## 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

### Keywords:

random dynamical systems; random attractors
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