Stochastic evolution equations with multiplicative Poisson noise and monotone nonlinearity

Document Type : Research Paper

Authors

1 School of Mathematics‎, ‎Institute for Research in Fundamental Sciences (IPM)‎, ‎P.O‎. ‎Box 19395-5746‎, ‎Tehran‎, ‎Iran ‎ ‎Faculty of Mathematics and Computer Science‎, ‎Amirkabir University of Technology‎, ‎Tehran‎, ‎Iran.

2 Department of Mathematics‎, ‎Sharif University of Technology‎, ‎Tehran‎, ‎Iran.

Abstract

Semilinear stochastic evolution equations with multiplicative Poisson noise and monotone nonlinear drift in Hilbert spaces are considered‎. ‎The coefficients are assumed to have linear growth‎. ‎We do not impose coercivity conditions on coefficients‎. ‎A novel method of proof for establishing existence and uniqueness of the mild solution is proposed‎. ‎Examples on stochastic partial differential equations and stochastic delay differential equations are provided to demonstrate the theory developed‎.

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References

S. Albeverio, V. Mandrekar and B. Rudiger, Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Levy noise, Stochastic Process. Appl. 119 (2009), no. 3, 835--863.
F.E. Browder, Non-linear equations of evolution, Ann. of Math. 80 (1964), no. 3, 485--523.
R.F. Curtain and A.J. Pritchard, Infinite Dimensional Linear Systems Theory, Springer-Verlag, 1978.
H. Dadashi-Arani and B.Z. Zangeneh, Large deviation principle for semilinear stochastic evolution equations with monotone nonlinearity and multiplicative noise, Differential Integral Equations 23 (2010), no. 7-8, 747--772.
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, 1992.
I. Gyongy, On stochastic equations with respect to semimartingales III, Stochastics 7 (1982) 231-254.
R. Jahanipur, Stochastic functional evolution equations with monotone nonlinearity: existence and stability of the mild solutions, J. Differential Equations 248 (2010), no. 5, 1230--1255.
R. Jahanipur and B.Z. Zangeneh, Stability of semilinear stochastic evolution equations with monotone nonlinearity, Math. Inequal. Appl. 3 (2000), no. 4, 593--614.
T. Kato, Nonlinear evolution equations in Banach spaces, in: Proc. Sympos. Appl. Math. 17, pp. 50--67, Amer. Math. Soc. Providence, RI, 1965.
P. Kotelenez, A stopped Doob inequality for stochastic convolution integrals and stochastic evolution equations, Stoch. Anal. Appl. 2 (1984), no. 3, 245--265.
M.A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Translated by A.H. Armstrong; translation edited by J. Burlak, A Pergamon Press Book, The Macmillan Co. New York, 1964.
N.V. Krylov and B.L. Rozovskii, Stochastic evolution equations, J. Soviet Math. 16 (1981) 1233--1277.
C. Marinelli and M. Rockner, Well-posedness and asymptotic behavior for stochastic reaction-diffusion equations with multiplicative Poisson noise, Electron. J. Probab. 15 (2010), no. 49, 1528--1555.
E. Pardoux, Equations aux derives partielles stochastiques non lineaires monotones: Etude de solutions fortes de type Ito, PhD Thesis, A LUniversite Paris Sud, 1975.
S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations With Levy Noise, Cambridge Univ. Press, 2007.
M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Texts Appl. Math. 13, Springer-Verlag, New York, 2004.
C. Prevot and M. Rockner, A Concise Course on Stochastic Partial Differential Equations, Springer, 2007.
E. Salavati and B.Z. Zangeneh, Semilinear stochastic evolution equations of monotone type with Levy noise. in: Dynamic Systems and Applications. Vol. 6, pp. 380--387, Dynamic, Atlanta, GA, 2012.
S. Zamani and B.Z. Zangeneh, Random motion of strings and related stochastic evolution equations with monotone nonlinearities, Stoch. Anal. Appl. 23 (2005), no. 5, 903--920.
B. Z. Zangeneh, Semilinear Stochastic Evolution Equations, PhD Thesis, University of British Columbia, Vancouver, B.C. Canada, 1990.
B.Z. Zangeneh, Measurability of the Solution of a Semilinear Evolution Equation, in: Seminar on Stochastic Processes, 1990 (Vancouver, BC, 1990), pp. 335--351, Progr. Probab. 24, Birkhauser, Boston, MA, 1991.
B.Z. Zangeneh, Semilinear stochastic evolution equations with monotone nonlinearities, Stochastics Stochastics Rep. 53 (1995), no. 1-2, 129--174.
B.Z. Zangeneh, Stationarity of the solution for the semilinear stochastic integral equation on the whole real line, in: Malliavin Calculus and Stochastic Analysis, A Festschrift in Honor of David Nualart, F.G. Viens, J. Feng, Y. Hu, E. Nualart (eds.), pp. 315--331,Springer Proc. Math. Stat. 34, Springer, New York, 2013.
Bulletin of the Iranian Mathematical Society
Volume 43, Issue 5
October 2017
Pages 1287-1299

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History

  • Receive Date: 10 October 2015
  • Revise Date: 18 May 2016
  • Accept Date: 23 May 2016

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