Stability in distribution of mild solutions to stochastic partial differential equations
HTML articles powered by AMS MathViewer
- by Jianhai Bao, Zhenting Hou and Chenggui Yuan PDF
- Proc. Amer. Math. Soc. 138 (2010), 2169-2180 Request permission
Abstract:
In the present paper, we investigate stochastic partial differential equations. By introducing a suitable metric between the transition probability functions of mild solutions, we derive sufficient conditions for stability in distribution of mild solutions. Consequently, we generalize some existing results to infinite dimensional cases. Finally, one example is constructed to demonstrate the applicability of our theory.References
- Gopal K. Basak, Arnab Bisi, and Mrinal K. Ghosh, Stability of a random diffusion with linear drift, J. Math. Anal. Appl. 202 (1996), no. 2, 604–622. MR 1406250, DOI 10.1006/jmaa.1996.0336
- Tomás Caraballo and José Real, On the pathwise exponential stability of non-linear stochastic partial differential equations, Stochastic Anal. Appl. 12 (1994), no. 5, 517–525. MR 1297111, DOI 10.1080/07362999408809370
- Tomás Caraballo and Kai Liu, On exponential stability criteria of stochastic partial differential equations, Stochastic Process. Appl. 83 (1999), no. 2, 289–301. MR 1708210, DOI 10.1016/S0304-4149(99)00045-9
- Mu-Fa Chen, From Markov chains to non-equilibrium particle systems, 2nd ed., World Scientific Publishing Co., Inc., River Edge, NJ, 2004. MR 2091955, DOI 10.1142/9789812562456
- Pao Liu Chow, Stability of nonlinear stochastic-evolution equations, J. Math. Anal. Appl. 89 (1982), no. 2, 400–419. MR 677738, DOI 10.1016/0022-247X(82)90110-X
- Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. MR 1207136, DOI 10.1017/CBO9780511666223
- G. Da Prato and J. Zabczyk, Ergodicity for infinite-dimensional systems, London Mathematical Society Lecture Note Series, vol. 229, Cambridge University Press, Cambridge, 1996. MR 1417491, DOI 10.1017/CBO9780511662829
- T. E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics 77 (2005), no. 2, 139–154. MR 2151664, DOI 10.1080/10451120512331335181
- U. G. Haussmann, Asymptotic stability of the linear Itô equation in infinite dimensions, J. Math. Anal. Appl. 65 (1978), no. 1, 219–235. MR 501750, DOI 10.1016/0022-247X(78)90211-1
- Akira Ichikawa, Stability of semilinear stochastic evolution equations, J. Math. Anal. Appl. 90 (1982), no. 1, 12–44. MR 680861, DOI 10.1016/0022-247X(82)90041-5
- Akira Ichikawa, Absolute stability of a stochastic evolution equation, Stochastics 11 (1983), no. 1-2, 143–158. MR 729494, DOI 10.1080/17442508308833282
- Ruifeng Liu and V. Mandrekar, Stochastic semilinear evolution equations: Lyapunov function, stability, and ultimate boundedness, J. Math. Anal. Appl. 212 (1997), no. 2, 537–553. MR 1464896, DOI 10.1006/jmaa.1997.5534
- Takeshi Taniguchi, The exponential stability for stochastic delay partial differential equations, J. Math. Anal. Appl. 331 (2007), no. 1, 191–205. MR 2305998, DOI 10.1016/j.jmaa.2006.08.055
- Chenggui Yuan and Xuerong Mao, Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stochastic Process. Appl. 103 (2003), no. 2, 277–291. MR 1950767, DOI 10.1016/S0304-4149(02)00230-2
Additional Information
- Jianhai Bao
- Affiliation: School of Mathematics, Central South University, Changsha, Hunan 410075, People’s Republic of China
- Zhenting Hou
- Affiliation: School of Mathematics, Central South University, Changsha, Hunan 410075, People’s Republic of China
- Chenggui Yuan
- Affiliation: Department of Mathematics, Swansea University, Swansea SA2 8PP, United Kingdom
- Email: C.Yuan@Swansea.ac.uk
- Received by editor(s): March 16, 2009
- Received by editor(s) in revised form: September 23, 2009, and September 24, 2009
- Published electronically: January 14, 2010
- Additional Notes: This work was partially supported by the NNSF of China (Grant No. 10671212)
- Communicated by: Richard C. Bradley
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 138 (2010), 2169-2180
- MSC (2010): Primary 60H15, 60H30
- DOI: https://doi.org/10.1090/S0002-9939-10-10230-5
- MathSciNet review: 2596056