Exponential stability of second-order stochastic evolution equations with Poisson jumps. (English) Zbl 1273.60077

The paper formulates conditions for mild solutions of certain second order stochastic differential equations with infinite delay and Poisson jumps to exist and to be exponentially stable in second mean.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
34K20 Stability theory of functional-differential equations
34K50 Stochastic functional-differential equations
35B35 Stability in context of PDEs
35R60 PDEs with randomness, stochastic partial differential equations
93E15 Stochastic stability in control theory
Full Text: DOI


[1] Balasubramaniam, P.; Ntouyas, S.K.; Vinayagam, D., Existence of solutions of semilinear stochastic delay evolution inclusions in a Hilbert space, J math anal appl, 305, 438-451, (2005) · Zbl 1067.60035
[2] Boufoussi, B.; Hajji, S., Successive approximation of neutral functional stochastic differential equations with jumps, Statist probab lett, 80, 324-332, (2010) · Zbl 1196.60114
[3] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press Cambridge · Zbl 0761.60052
[4] Da Prato, G.; Zabczyk, J., Second-order partial differential equations in hilber spaces, (2002), Cambridge University Press Cambridge
[5] Fattorini, H.O., Second-order linear differential equations in Banach spaces, (1985), North Holland Mathematics Studies 108, Elsevier Science North Holland · Zbl 0564.34063
[6] Govindan, T.E., Stability of mild solutions of stochastic evolution equations with variable delay, Stoch anal appl, 21, 1059-1077, (2003) · Zbl 1036.60052
[7] Liu, K., Stability of infinite dimensional stochastic differential equations with applications, (2006), Chapman & Hall CRC, London
[8] Luo, J.; Liu, K., Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps, Stoch process appl, 118, 864-895, (2008) · Zbl 1186.93070
[9] Mahmudov, N.I., Existence and uniqueness results for neutral SDEs in Hilbert spaces, Stoch anal appl, 24, 79-95, (2006) · Zbl 1110.60063
[10] Mahmudov, N.I.; McKibben, M.A., Abstract second-order damped mckean – vlasov stochastic evolution equations, Stoch anal appl, 24, 303-328, (2006) · Zbl 1102.35044
[11] Meng, X.; Li, Z.; Nieto, J.J., Dynamic analysis of michaelis – menten chemostat-type competition models with time delay and pulse in a polluted environment, J math chem, 47, 123-144, (2010) · Zbl 1194.92075
[12] Nieto, J.J.; Regan, D.O., Variational approach to impulsive differential equations, Nonlinear anal: real world appl, 10, 680-690, (2009) · Zbl 1167.34318
[13] Nieto, J.J.; Rodriguez-Lopez, R., New comparison results for impulsive integro-differential equations and applications, J math anal appl, 328, 1343-1368, (2007) · Zbl 1113.45007
[14] Chang, Y.-K.; Zhao, Z.-H.; N’Guérékata, G.M., Square-Mean almost automorphic mild solutions to non-autonomous stochastic differential equations in Hilbert spaces, Comput math appl, 61, 384-391, (2011) · Zbl 1211.60025
[15] Taniguchi, T.; Luo, J., The existence and asymptotic behaviour of mild solutions to stochastic evolution equations with infinite delays driven by Poisson jumps, Stoch dyn, 9, 217-229, (2009) · Zbl 1181.60102
[16] Ren, Y.; Sun, D.D., Second-order neutral impulsive stochastic evolution equations with delay, J math phys, 50, 102709, (2009) · Zbl 1236.60057
[17] Sakthivel, R.; Ren, Y.; Kim, H., Asymptotic stability of second-order neutral stochastic differential equations, J math phys, 51, 052701, (2010) · Zbl 1310.35248
[18] Sakthivel, R.; Luo, J., Asymptotic stability of impulsive stochastic partial differential equations with infinite delays, J math anal appl, 356, 1-6, (2009) · Zbl 1166.60037
[19] Samoilenko, A.M.; Perestyuk, N.A., Impulsive differential equations, (1995), World Scientific Singapore · Zbl 0837.34003
[20] Travis, C.C.; Webb, G.F., Cosine families and abstract nonlinear second order differential equations, Acta math acad sci hungaricae, 32, 76-96, (1978) · Zbl 0388.34039
[21] Wan, L.; Duan, J., Exponential stability of non-autonomous stochastic partial differential equations with finite memory, Statist probab lett, 78, 490-498, (2008) · Zbl 1141.37030
[22] Zhao, H., On existence and uniqueness of stochastic evolution equation with Poisson jumps, Statist probab lett, 79, 2367-2373, (2009) · Zbl 1182.60018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.