Asymptotic exponential stability for diffusion processes driven by stochastic differential equations in duals of nuclear spaces (Q1596542)

The authors consider a nonlinear stochastic diffusion equation described by \[ X_t= X_0+ \int^t_0 A(s, X_s) ds+ \int^t_0 B(s, X_s) dW_s\tag{1} \] where \(X_t\in \mathbb{R}^n\) is the state, \(A: \mathbb{R}_+\times \Phi'\to \Phi'\), \(B: \mathbb{R}_+\times \Phi'\to{\mathcal L}(\Phi',\Phi')\) are two measurable mappings, \(W_t\) is a \(\Phi'\)-valued Wiener process, \(\Phi'\) is the dual space of a certain countably Hilbertian nuclear space and \({\mathcal L}(\Phi',\Phi')\) is the space of all bounded linear operators from \(\Phi'\) into itself. The authors derive sufficient conditions of mean square exponential stability of strong solutions of equation (1) and exponential stability of paths with probability one. The obtained results are illustrated by an example.