Document Type : Research Paper
Authors
Department of Mathematics and Computer Sciences, Amirkabir University of Technology, Tehran, Iran.
Abstract
The present paper seeks to prove the existence and uniqueness of solutions to stochastic evolution equations in Hilbert spaces driven by both Poisson random measure and Wiener process with non-Lipschitz drift term. The proof is provided by the theory of measure of noncompactness and condensing operators. Moreover, we give some examples to illustrate the application of our main theorem.
Keywords
[1] R.R. Akhmerov, M.I. Kamenskii, A.S. Patapov, A.E. Rodkina and B.N. Sadovskii, Measures of noncompactness and condensing operators, Birkhäuser--Verlag, Basel, Switzerland, 1992.
[2] 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), pp. 835-863.
[3] A. Anguraj and K. Banupriya, Existence, uniqueness and stability results for impulsive stochastic functional differential equations with infinite delay and Poisson jumps, Malaya J., 5 (2017), pp. 653-659.
[4] D. Barbu, Local and global existence for mild solutions of stochastic differential equations, Port. Math., 55 (1998), pp. 411-424.
[5] D. Barbu and G. Bocsan, Approximations to mild solutions of stochastic semilinear equations with non-Lipschitz coefficients, Czechoslovak Math. J., 52 (2002), pp. 87-95.
[6] D. Barbu and G. Bocsan, Successive approximations to solutions of stochastic semilinear functional differential equations in Hilbert spaces, Technical Report 162, University of Timisoara, Faculty of Mathematics, STPA - Seminar of Probability Theory and Applications, Timișoara, Romania, 2004.
[7] Z. Brzezniak, E. Hausenblas and P.A. Razafimandimby, Stochastic reaction-diffusion equations driven by jump processes, Potential Anal., 49 (2018), pp. 131-201.
[8] G. Cao and K. He, Successive approximation of infinite dimensional semilinear backward stochastic evolution equations with jumps, Stochastic Process. Appl., 117 (2007), pp. 1251-1264.
[9] G. Da Prato and F. Flandoli, Pathwise uniqueness for a class of SDE in Hilbert spaces and applications, J. Funct. Anal., 259 (2010), pp. 243-267.
[10] G. Da Prato, F. Flandoli, M. Rockner and A.Y. Veretennikov, Strong uniqueness for SDEs in Hilbert spaces with nonregular drift, Ann. Probab., 44 (2016), pp. 1985-2023.
[11] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, UK, 2014.
[12] X.L. Fan, Non autonomous semilinear stochastic evolution equations, Comm. Statist. Theory Methods, 44 (2015), pp. 1806-1818.
[13] L. Guedda and P.R. Fitte, Existence and dependence results for semilinear functional stochastic differential equations with infinite delay in a Hilbert space, Mediterr. J. Math., 13 (2016), pp. 4153-4174.
[14] E. Hausenblas, Existence, uniqueness and regularity of parabolic SPDEs driven by Poisson random measure, Electron. J. Probab., 10 (2005), pp. 1496-1546.
[15] E. Hausenblas, SPDEs driven by Poisson random measure with non-Lipschitz coefficients: existence results, Probab. Theory Related Fields, 137 (2007), pp. 161-200.
[16] E. Hausenblas and J. Seidler, Stochastic convolutions driven by martingales: maximal inequalities and exponential integrability, Stoch. Anal. Appl., 26 (2007), pp. 98-119.
[17] A. Jakubowski, M. Kamenskii and P.R. Fitte, Existence of weak solutions to stochastic evolution inclusions, Stoch. Anal. Appl., 23 (2005), pp. 723-749.
[18] E. Pardoux, Stochastic partial differential equations and filtering of diffusion propcesses, Stochastics, 6 (1979), pp. 193-231.
[19] S. Peszat and J. Zabczyk, Stochastic partial differential equations with Levy noise: An evolution equation approach, Cambridge University Press, Cambridge, UK, 2007.
[20] M. Rockner and T. Zhang, Stochastic evolution equations of jump type: existence, uniqueness and large deviation principles, Potential Anal., 26 (2007), pp. 255-279.
[21] T. Taniguchi, Successive approximations to solutions of stochastic differential equations, J. Differ. Equ., 96 (1992), pp. 152-169.
[22] T. Taniguchi, Successive approximations to solutions of stochastic differential equations, J. Differ. Equ., 96 (1992), pp. 152-169.
[23] L. Wang, The existence and uniqueness of mild solutions to stochastic differential equations with Levy noise, Adv. Differ. Equ., 2017 (2017), pp. 1-12.
[24] B. Xie, Stochastic differential equations with non-Lipschitz coefficients in Hilbert spaces, Stoch. Anal. Appl., 26 (2008), pp. 408-433.
[25] T. Yamada, On the successive approximation of solutions of stochastic differential equations, J. Math. Kyoto Univ., 21 (1981), pp. 501-515.
[2] 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), pp. 835-863.
[3] A. Anguraj and K. Banupriya, Existence, uniqueness and stability results for impulsive stochastic functional differential equations with infinite delay and Poisson jumps, Malaya J., 5 (2017), pp. 653-659.
[4] D. Barbu, Local and global existence for mild solutions of stochastic differential equations, Port. Math., 55 (1998), pp. 411-424.
[5] D. Barbu and G. Bocsan, Approximations to mild solutions of stochastic semilinear equations with non-Lipschitz coefficients, Czechoslovak Math. J., 52 (2002), pp. 87-95.
[6] D. Barbu and G. Bocsan, Successive approximations to solutions of stochastic semilinear functional differential equations in Hilbert spaces, Technical Report 162, University of Timisoara, Faculty of Mathematics, STPA - Seminar of Probability Theory and Applications, Timișoara, Romania, 2004.
[7] Z. Brzezniak, E. Hausenblas and P.A. Razafimandimby, Stochastic reaction-diffusion equations driven by jump processes, Potential Anal., 49 (2018), pp. 131-201.
[8] G. Cao and K. He, Successive approximation of infinite dimensional semilinear backward stochastic evolution equations with jumps, Stochastic Process. Appl., 117 (2007), pp. 1251-1264.
[9] G. Da Prato and F. Flandoli, Pathwise uniqueness for a class of SDE in Hilbert spaces and applications, J. Funct. Anal., 259 (2010), pp. 243-267.
[10] G. Da Prato, F. Flandoli, M. Rockner and A.Y. Veretennikov, Strong uniqueness for SDEs in Hilbert spaces with nonregular drift, Ann. Probab., 44 (2016), pp. 1985-2023.
[11] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, UK, 2014.
[12] X.L. Fan, Non autonomous semilinear stochastic evolution equations, Comm. Statist. Theory Methods, 44 (2015), pp. 1806-1818.
[13] L. Guedda and P.R. Fitte, Existence and dependence results for semilinear functional stochastic differential equations with infinite delay in a Hilbert space, Mediterr. J. Math., 13 (2016), pp. 4153-4174.
[14] E. Hausenblas, Existence, uniqueness and regularity of parabolic SPDEs driven by Poisson random measure, Electron. J. Probab., 10 (2005), pp. 1496-1546.
[15] E. Hausenblas, SPDEs driven by Poisson random measure with non-Lipschitz coefficients: existence results, Probab. Theory Related Fields, 137 (2007), pp. 161-200.
[16] E. Hausenblas and J. Seidler, Stochastic convolutions driven by martingales: maximal inequalities and exponential integrability, Stoch. Anal. Appl., 26 (2007), pp. 98-119.
[17] A. Jakubowski, M. Kamenskii and P.R. Fitte, Existence of weak solutions to stochastic evolution inclusions, Stoch. Anal. Appl., 23 (2005), pp. 723-749.
[18] E. Pardoux, Stochastic partial differential equations and filtering of diffusion propcesses, Stochastics, 6 (1979), pp. 193-231.
[19] S. Peszat and J. Zabczyk, Stochastic partial differential equations with Levy noise: An evolution equation approach, Cambridge University Press, Cambridge, UK, 2007.
[20] M. Rockner and T. Zhang, Stochastic evolution equations of jump type: existence, uniqueness and large deviation principles, Potential Anal., 26 (2007), pp. 255-279.
[21] T. Taniguchi, Successive approximations to solutions of stochastic differential equations, J. Differ. Equ., 96 (1992), pp. 152-169.
[22] T. Taniguchi, Successive approximations to solutions of stochastic differential equations, J. Differ. Equ., 96 (1992), pp. 152-169.
[23] L. Wang, The existence and uniqueness of mild solutions to stochastic differential equations with Levy noise, Adv. Differ. Equ., 2017 (2017), pp. 1-12.
[24] B. Xie, Stochastic differential equations with non-Lipschitz coefficients in Hilbert spaces, Stoch. Anal. Appl., 26 (2008), pp. 408-433.
[25] T. Yamada, On the successive approximation of solutions of stochastic differential equations, J. Math. Kyoto Univ., 21 (1981), pp. 501-515.
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