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Exponential stability of non-linear stochastic evolution equations. (English) Zbl 0933.60072

Let \(K,H\) be real separable Hilbert spaces, \(V\) a uniformly rotund Banach space with a uniformly rotund dual space \(V^*\), let \(V\subseteq H\subseteq V^*\) with continuous dense embeddings. Let \(W\) be a Wiener process in \(K\) with a nuclear covariance operator, defined on a filtered probability space \((\Omega , \mathcal F, (\mathcal F_{t}),P)\). A stochastic differential equation \[ X_{t} = x_{0} + \int ^{t}_{0} f(t,X_{s}) ds + \int ^{t}_{0} g(s,X_{s}) dW_{s}, \quad t\geq 0, \tag{1} \] in \(V^*\) is considered, where \(f(t,\cdot)\) is a family of nonlinear operators from \(V\) into \(V^*\) and \(g(t,\cdot):V\to \mathcal L(K,H)\) are locally bounded mappings. Assume that there exists a unique strong solution \(X_{t}(x_{0})\in L^{2}((0,T)\times \Omega ;V) \cap L^{2}(\Omega ;\mathcal C([0,T];H))\) for any \(T>0\) and any \(\mathcal F_0\)-measurable \(V\)-valued random function \(x_{0}\) with \(E\|x_0\|^{2}_{V}<\infty \). Sufficient conditions for the equation (1) to be almost surely exponentially stable (that is, for the existence of a \(\gamma >0\) such that \(\limsup_{t\to \infty} t^{-1}\log \|X_t(x_0)\|_{H} \leq -\gamma \) almost surely for any initial condition \(x_{0}\)) are given, their proofs being based on Lyapunov functions techniques. Analogous results are established also for stochastic evolution equations with time delays.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
93E03 Stochastic systems in control theory (general)
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