Abstract
This paper includes in an unified way several results about existence and uniqueness of solutions of Fokker–Planck equations from (Bogachev et al., J Funct Anal, 256:1269–1298, 2009) [2], (Bogachev et al., J Evol Equ, 10(3):487–509, 2010) [3], (Bogachev et al., Partial Differ Equ, 36:925–939, 2011) [4] and (Bogachev et al., Bull London Math Soc 39:631–640, 2007) [1], using probabilistic methods. Several applications are provided including Burgers and 2D-Navier–Stokes equations perturbed by noise. Some of these applications were also studied by a different analytic approach in (Bogachev et al., J Differ Equ, 259(8):3854–3873, 2015) [5], (Bogachev et al., Ann Sc Norm Super Pisa Cl Sci 14(3):983–1023, 2015) [6], (Da Prato et al., Commun Math Stat, 1(3):281–304, 2013) [11].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The upper index 1 in the definitions of \(S^{0,1}\) and \(\mathcal K^1_0\) recalls the space \(C_{b,1}(H)\).
- 2.
\(\epsilon _0\) was defined in Hypothesis 1(ii).
- 3.
\(\psi \) is defined in (21).
- 4.
\(\mathbb Q\) denotes the set of rational numbers.
- 5.
We take F independent of t for simplicity.
- 6.
The aforementioned paper concerns Dirichlet boundary conditions but all its results generalise easily to periodic ones.
- 7.
We proceed here as in Sect. 2 (with \(F_\alpha =0\)), but working in \(C_b(H)\) rather than in \(C_{1,b}(H)\).
References
Bogachev, V., Da Prato, G., Röckner, M., Stannat, W.: Uniqueness of solutions to weak parabolic equations for measures. Bull. London Math. Soc. 39, 631–640 (2007)
Bogachev, V., Da Prato, G., Röckner, M.: Fokker-Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces. J. Funct. Anal. 256, 1269–1298 (2009)
Bogachev, V., Da Prato, G., Röckner, M.: Existence and uniqueness of solutions for Fokker-Planck equations on Hilbert spaces. J. Evol. Equ. 10(3), 487–509 (2010)
Bogachev, V., Da Prato, G., Röckner, M.: Uniqueness for solutions of Fokker-Planck equations on infinite dimensional spaces. Commun. Partial Differ. Equ. 36, 925–939 (2011)
Bogachev, V., Da Prato, G., Röckner, M., Shaposhnikov, S.: On the uniqueness of solutions to continuity equations. J. Differ. Equ. 259(8), 3854–3873 (2015)
Bogachev, V., Da Prato, G., Röckner, M., Shaposhnikov, S.: An analytic approach to infinite-dimensional continuity and Fokker-Planck-Kolmogorov equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. 14(3), 983–1023 (2015)
Cerrai, S.: A Hille-Yosida theorem for weakly continuous semigroups. Semigroup Forum 49, 349–367 (1994)
Da Prato, G.: Kolmogorov Equations for Stochastic PDEs. Birkhäuser, Boston (2004)
Da Prato, G., Tubaro, L.: Some results about dissipativity of Kolmogorov operators. Czechoslov. Math. J. 51(126), 685–699 (2001)
Da Prato, G., Debussche, A.: \(m\)-dissipativity of Kolmogorov operators corresponding to Burgers equations with space-time white noise. Potential Anal 26, 31–55 (2007)
Da Prato, G., Flandoli, F., Röckner, M.: Fokker-Planck equations for SPDE with non-trace class noise. Commun. Math. Stat. 1(3), 281–304 (2013)
Eberle, A.: Uniqueness and Non-uniqueness of Semigroups Generated by Singular Diffusion Operators. Lecture Notes in Mathematics, vol. 1718. Springer, Berlin (1999)
Liu, W., Röckner, M.: Stochastic Partial Differential Equations: An Introduction. Universitext. Springer, Berlin (2015)
Priola, E.: On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions. Studia Math. 136, 271–295 (1999)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Da Prato, G. (2018). Fokker–Planck Equations in Hilbert Spaces. In: Eberle, A., Grothaus, M., Hoh, W., Kassmann, M., Stannat, W., Trutnau, G. (eds) Stochastic Partial Differential Equations and Related Fields. SPDERF 2016. Springer Proceedings in Mathematics & Statistics, vol 229. Springer, Cham. https://doi.org/10.1007/978-3-319-74929-7_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-74929-7_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-74928-0
Online ISBN: 978-3-319-74929-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)