The reduction principle in stability theory of invariant sets of Itô stochastic systems (Q1874940)

The reduction principle in (deterministic) stability theory is known as the problem for an invariant set assumed to be stable if the initial data belong to a manifold containing this set. This paper aims at presenting a similar result with respect to a.s. stochastic stability for \(n\)-dimensional real-valued systems of ordinary Itô stochastic differential equations of the form \[ dX(t) = b(X(t)) \, dt+ \sum^k_{r=1} \sigma_r(t,X(t)) \, dW_r(t) , \tag{1} \] where \(t \geq 0\) and \(W_r=\{W_r(t):t\geq 0\} \), \(r=1,\dots,k\), are independent Wiener processes defined on a complete probability space \((\Omega,{\mathcal F},P)\). It is assumed that the coefficients \(b\) and \(\sigma_r\) are at least continuous functions in all variables. The authors prove a fairly general theorem on the uniform stochastic stability with respect to forward invariant sets \(S_t\) with probability one (i.e., \(P (\forall t \geq t_0 : X(t| t_0,x_0) \in S_t) = 1\)) for system (1). For this purpose, \(b\) is supposed to be global Lipschitz-continuous, \(\sigma_r\) are local Lipschitz-continuous with respect to \(x\)-variable (uniformly in time \(t\)) and linear-polynomially bounded (in \(x\), uniformly in time \(t\)) functions on every cylinder \(\{t \geq 0\} \times \{| | x| | <R\}\) and it is supposed the existence of a Lyapunov function \(V=V(x)\) with \({\mathcal L} V \leq - c_1 V\) and \(| | \sigma_r (t,x)| | ^2 \leq c_2 V\), where \({\mathcal L}\) is the infinitesimal generator of (1) and \(c_1, c_2 >0\) are positive real constants. The proof is based on Khasminskii's approach to Lyapunov-Krasovskii functions applied to discuss the asymptotic stability of singleton invariant sets for ordinary stochastic differential equations. The stability analysis of the stochastic system (1) is reduced to that of a related deterministic system by incorporation of a larger invariant set \(N_t\) containing \(S_t\) and on which (1) degenerates into that deterministic system.