The existence and asymptotic behaviour of solutions to non-Lipschitz stochastic functional evolution equations driven by Poisson jumps

摘要

In this paper, we consider the existence and uniqueness of the energy solutions to the following non-Lipschitz stochastic functional evolution equation driven both by Brownian motion and by Poisson jumps: dX(t)=[A(t,X(t)) + f(t,X_t)]dt + g(t,X_t)dW(t) + ∫(k(t,X_t,y)q(dtdy), y=U, t≥0, X_0=φ∈D([-r,0],H), where A(t,·): V→V~* is a linear or nonlinear bounded operator, f:[0, ∞) × D([-r,0]) →H, g: [0, ∞)×D([-r,0], H) →(L_2)~0(K,H) and k: [0, ∞)×D([-r,0], H)×F→H are measurable functions. We also investigate the almost sure exponential stability of energy solutions by using the energy equality for this equation.