The Existence,Exponential Behavior And Upper Noise Excitation Index Of Solutions To Stochastic Evolution Equations With Unbounded Delay And A Tempered Fractional Brownian Motion

In this paper we investigate stochastic unbounded delay evolution equations in fractional power spaces perturbed by a tempered fractional Brownian motion BQ?,?-(t)with-1/2<?<0 and ?>0.Generally speaking,a tempered fractional Brownian motion(TFBM)is defined by exponentially tempering the power law kernel in the moving average representation of a frac-tional Bownian motion(FBM).In particular,TFBM can provide a useful stochastic process model for wind speed data.We first recall the stochastic integral with respect to TFBM and introduce a technical lemma which is crucial in our stability analysis.Then we prove the exis-tence and uniqueness of mild solutions by using semigroup methods.Specifically speaking,we obtain that the global existence and uniqueness of mild solutions to stochastic delay evolution equations in fractional power spaces.The upper nonlinear noise excitation index of the energy solutions at any finite time t is also obtained.In other words,we show an upper bound of the upper excitation index of the mild solution to stochastic delay evolution equations at time t.Finally,we consider the exponential asymptotic behavior of mild solutions in mean square,that is,we establish some sufficient conditions ensuring the exponential decay to zero of the mild solution in mean square.