Stabilization of stationary solutions of evolution equations by noise
Existence, uniqueness, and exponential stability of non-constant stationary solutions of some semilinear evolution equation are studied. The equation under consideration is a mild formulation corresponding to an SPDE of the type \[ dX=AX\;dt+F(X)\; dt +B(X)\; dW \] on some separable Hilbert space, where \(A\) generates a \(C_0\)-semigroup, \(F\) and \(B\) are Lipschitz and \(W\) is some Wiener process. The main results show under some assumptions on \(B\) a stabilization effect on the deterministic dynamics. Here an exponentially stable solution for the stochastic equation exists, even if the equation without noise does not have any exponentially stable solution. The main tools in the proof are pullback techniques for random dynamical systems combined with exponential martingale inequalities. The stabilization relies on Ito-Stratonovic corrections.
scientific article; zbMATH DE number 5124798