A note on stochastic semilinear equations and their associated Fokker-Planck equations

Röckner M, Zhu R, Zhu X (2014)
Journal of Mathematical Analysis and Applications 415(1): 83-109.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
Abstract / Bemerkung
The main purpose of this paper is to prove existence and uniqueness of (probabilistically weak and strong) solutions to stochastic differential equations (SDE) on Hilbert spaces under a new approximation condition on the drift, recently proposed in [6] to solve Fokker-Planck equations (FPE), extended in this paper to a considerably larger class of drifts. As a consequence we prove existence of martingale solutions to the SDE (whose time marginals then solve the corresponding FPE). Applications include stochastic semilinear partial differential equations with white noise and a non-linear drift part which is the sum of a Burgers-type part and a reaction diffusion part. The main novelty is that the latter is no longer assumed to be of at most linear, but of at most polynomial growth. This case so far had not been covered by the existing literature. We also give a direct and more analytic proof for existence of solutions to the corresponding FPE, extending the technique from [6] to our more general framework, which in turn requires to work on a suitable Gelfand triple rather than just the Hilbert state space. (C) 2014 Elsevier Inc. All rights reserved.
Erscheinungsjahr
Zeitschriftentitel
Journal of Mathematical Analysis and Applications
Band
415
Ausgabe
1
Seite(n)
83-109
ISSN
PUB-ID

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Röckner M, Zhu R, Zhu X. A note on stochastic semilinear equations and their associated Fokker-Planck equations. Journal of Mathematical Analysis and Applications. 2014;415(1):83-109.
Röckner, M., Zhu, R., & Zhu, X. (2014). A note on stochastic semilinear equations and their associated Fokker-Planck equations. Journal of Mathematical Analysis and Applications, 415(1), 83-109. doi:10.1016/j.jmaa.2014.01.058
Röckner, M., Zhu, R., and Zhu, X. (2014). A note on stochastic semilinear equations and their associated Fokker-Planck equations. Journal of Mathematical Analysis and Applications 415, 83-109.
Röckner, M., Zhu, R., & Zhu, X., 2014. A note on stochastic semilinear equations and their associated Fokker-Planck equations. Journal of Mathematical Analysis and Applications, 415(1), p 83-109.
M. Röckner, R. Zhu, and X. Zhu, “A note on stochastic semilinear equations and their associated Fokker-Planck equations”, Journal of Mathematical Analysis and Applications, vol. 415, 2014, pp. 83-109.
Röckner, M., Zhu, R., Zhu, X.: A note on stochastic semilinear equations and their associated Fokker-Planck equations. Journal of Mathematical Analysis and Applications. 415, 83-109 (2014).
Röckner, Michael, Zhu, Rongchan, and Zhu, Xiangchan. “A note on stochastic semilinear equations and their associated Fokker-Planck equations”. Journal of Mathematical Analysis and Applications 415.1 (2014): 83-109.