Abstract
Consider the following infinite dimensional stochastic evolution equation over some Hilbert space H with norm ∣·∣: X t = x 0 + ∫ 0 t f ( X s , s ) d s + ∫ 0 t g ( X s , s ) d W s , t ⩾ 0 , P almost surely It is proved that under certain mild assumptions, the strong solution Xt(x0)∈V↪H↪V*, t ⩾ 0, is mean square exponentially stable if and only if there exists a Lyapunov functional Λ(·, ·):H×R+→R1 which satisfies the following conditions: (i) c 1 | x | 2 − k 1 e − μ 1 t ⩽ Λ ( x , t ) ⩽ c 2 | x | 2 + k 2 e − μ 2 t ; (ii) ℒ Λ ( x , t ) ⩽ − c 3 Λ ( x , t ) + k 3 e − μ 3 t , ∀ x ∈ V , t ⩾ 0 ; where L is the infinitesimal generator of the Markov process Xt and ci, ki, μi, i = 1, 2, 3, are positive constants. As a by-product, the characterization of exponential ultimate boundedness of the strong solution is established as the null decay rates (that is, μi = 0) are considered.
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