Exponential stability of non-linear stochastic evolution equations
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.
stochastic partial differential equations
stochastic evolution equations with delays
scientific article; zbMATH DE number 1364520