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Lyapunov functionals and asymptotic stability of stochastic delay evolution equations. (English) Zbl 0947.93037
The author studies almost sure decay rates for mild solutions to stochastic evolution equations with delay on a Hilbert space \(H\), by applying the method he recently has developed for strong solutions to a sequence of approximating evolution equations together with a limiting procedure. The investigation yields results of the form \[ \log|\text{mild solution }(t)|/L(t)\leq\text{const}, \] almost surely, \(L(t)\) being an increasing function. Examples are given for \(H= L^2([a, b])\) and \(H= L^2(D)\), \(D\) a bounded domain in a Euclidean space.

MSC:
93E15 Stochastic stability in control theory
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
93E03 Stochastic systems in control theory (general)
93C25 Control/observation systems in abstract spaces
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