Existence, uniqueness, and stability of mild solutions for second-order neutral stochastic evolution equations with infinite delay and Poisson jumps

2012 ◽  
Vol 53 (7) ◽  
pp. 073517 ◽  
Author(s):  
Yong Ren ◽  
R. Sakthivel
Keyword(s):  
Evolution Equations ◽  
Infinite Delay ◽  
Mild Solutions ◽  
Second Order ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Poisson Jumps

scholarly journals Fractional Measure-dependent Nonlinear Second-order Stochastic Evolution Equations with Poisson Jumps

2018 ◽  
Vol 5 (1) ◽  
pp. 59-75 ◽  
Author(s):  
Mark A. McKibben ◽  
Micah Webster
Keyword(s):  
Evolution Equations ◽  
Mild Solutions ◽  
Stability Result ◽  
Growth Conditions ◽  
Second Order ◽  
Probability Measures ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Poisson Jumps ◽  
Fractional Measure

Abstract This paper focuses on a nonlinear second-order stochastic evolution equations driven by a fractional Brownian motion (fBm) with Poisson jumps and which is dependent upon a family of probability measures. The global existence of mild solutions is established under various growth conditions, and a related stability result is discussed. An example is presented to illustrate the applicability of the theory.

THE EXISTENCE AND ASYMPTOTIC BEHAVIOUR OF MILD SOLUTIONS TO STOCHASTIC EVOLUTION EQUATIONS WITH INFINITE DELAYS DRIVEN BY POISSON JUMPS

2009 ◽  
Vol 09 (02) ◽  
pp. 217-229 ◽  
Author(s):  
TAKESHI TANIGUCHI ◽  
JIAOWAN LUO
Keyword(s):  
Initial Function ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Jump Processes ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Poisson Jumps ◽  
Infinite Delays ◽  
Poisson Jump ◽  
Exponentially Stable

In this paper we consider a sufficient condition for mild solutions to exist and to be almost surely exponentially stable or exponentially ultimate bounded in mean square for the following stochastic evolution equation with infinite delays driven by Poisson jump processes: [Formula: see text] with an initial function X(s) = φ (s), -∞ < s ≤ 0, where φ : (-∞, 0] → H is a càdlàg function with [Formula: see text].

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