Exponential behavior and upper noise excitation index of solutions to evolution equations with unbounded delay and tempered fractional Brownian motions

Abstract

In this paper we investigate stochastic evolution equations with unbounded delay in fractional power spaces perturbed by a tempered fractional Brownian motion B σ,λ Q (t) with −1/2 < σ < 0 and λ > 0. We first introduce a technical lemma which is crucial in our stability analysis. Then we prove the existence and uniqueness of mild solutions by using semigroup methods. The upper nonlinear noise excitation index of the energy solutions at any finite time t is also obtained. Finally, we consider the exponential asymptotic behavior of mild solutions in mean square.

The continuity, regularity and polynomial stability of mild solutions for stochastic 2D-Stokes equations with unbounded delay driven by tempered fractional Gaussian noise

“…For the stochastic integrals with respect to Brownian motion, FBM and TFBM, we have the following properties; for the particular case of p = 2 see, e.g., 7,11,19,54 .…”

“…In spite of the fast growth of the literature on tempered fractional processes, there has been little mention of stochastic differential equations driven by tempered fractional Gaussian noise even in the nondelay case. Very recently, we proved the existence, uniqueness and exponential stability of mild solutions for stochastic delay evolution equations driven by tempered fractional Gaussian noise in mean square 54 .…”

Nontrivial Equilibrium Solutions and General Stability for Stochastic Evolution Equations with Pantograph Delay and Tempered Fractional Noise

“…where the parameter > 0 [31]. In this paper, because of the presence of pantograph delay and tfBm, we first introduce a novel estimate of stochastic integrals with respect to tfBm (see Lemma 2.6 for more details).…”

Local Strong Solutions to the Full Compressible Navier--Stokes System with Temperature-Dependent Viscosity and Heat Conductivity

“…However, to the best of our knowledge, there has been little mention of stochastic differential equations driven by tfBm even in the nondelay case. Very recently, we have established the existence, uniqueness, Hölder regularity, exponential and polynomial stability of mild solutions for stochastic delay evolution equations driven by tfBm [20,31].…”

“…The first part is devoted to the global existence, uniqueness and mean square stability with general decay rate of mild solutions for problem (1.2) by using the Banach fixed point theorem, the fractional power of operators and the semigroup theory. It has been pointed out in [31] that the stochastic integral with respect to tfBm is bounded by…”