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2016 | OriginalPaper | Chapter

7. Transition Semigroup

Authors : Viorel Barbu, Giuseppe Da Prato, Michael Röckner

Published in: Stochastic Porous Media Equations

Publisher: Springer International Publishing

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Abstract

This chapter is devoted to existence of invariant measures for transition semigroups associated with stochastic porous media equations with additive noise studied in previous chapters.

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Footnotes
1
β is called strictly monotone if there exists a > 0 such that (β(r) −β(s)(rs) ≥ a | rs | 2, for all \(r,s \in \mathbb{R}\).
 
2
\(\frac{1} {\frac{1} {t}\int _{0}^{t}hdt} \leq \frac{1} {t} \int _{0}^{t} \frac{1} {h}\;dt.\)
 
Literature
6.
go back to reference V. Barbu, Nonlinear Differential Equations of Monotone Type in Banach Spaces (Springer, New York, 2010)CrossRefMATH V. Barbu, Nonlinear Differential Equations of Monotone Type in Banach Spaces (Springer, New York, 2010)CrossRefMATH
11.
go back to reference V. Barbu, G. Da Prato, Invariant measures and the Kolmogorov equation for the stochastic fast diffusion equation. Stoch. Process. Appl. 120 (7), 1247–1266 (2010)MathSciNetCrossRefMATH V. Barbu, G. Da Prato, Invariant measures and the Kolmogorov equation for the stochastic fast diffusion equation. Stoch. Process. Appl. 120 (7), 1247–1266 (2010)MathSciNetCrossRefMATH
12.
go back to reference V. Barbu, G. Da Prato, Ergodicity for the phase-field equations perturbed by Gaussian noise. Infinite Dimen. Anal. Quantum Probab. Relat. Top. 14 (1), 35–55 (2011)MathSciNetCrossRefMATH V. Barbu, G. Da Prato, Ergodicity for the phase-field equations perturbed by Gaussian noise. Infinite Dimen. Anal. Quantum Probab. Relat. Top. 14 (1), 35–55 (2011)MathSciNetCrossRefMATH
20.
go back to reference V. Barbu, V. Bogachev, G. Da Prato, M. Röckner, Weak solutions to the stochastic porous media equation via Kolmogorov equations: the degenerate case. J. Funct. Anal. 237, 54–75 (2006)MathSciNetCrossRefMATH V. Barbu, V. Bogachev, G. Da Prato, M. Röckner, Weak solutions to the stochastic porous media equation via Kolmogorov equations: the degenerate case. J. Funct. Anal. 237, 54–75 (2006)MathSciNetCrossRefMATH
46.
49.
go back to reference G. Da Prato, M. Röckner, Well posedness of Fokker–Planck equations for generators of time-inhomogeneous Markovian transition probabilities. Rend. Lincei Mat. Appl. 23 (4), 361–376 (2012)MathSciNetMATH G. Da Prato, M. Röckner, Well posedness of Fokker–Planck equations for generators of time-inhomogeneous Markovian transition probabilities. Rend. Lincei Mat. Appl. 23 (4), 361–376 (2012)MathSciNetMATH
50.
go back to reference G. Da Prato, J. Zabczyk, Ergodicity for Infinite Dimensional Systems. London Mathematical Society Lecture Notes, vol. 229 (Cambridge University Press, Cambridge, 1996) G. Da Prato, J. Zabczyk, Ergodicity for Infinite Dimensional Systems. London Mathematical Society Lecture Notes, vol. 229 (Cambridge University Press, Cambridge, 1996)
51.
go back to reference G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications, 2nd edn. (Cambridge University Press, Cambridge, 2014) G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications, 2nd edn. (Cambridge University Press, Cambridge, 2014)
53.
go back to reference G. Da Prato, M. Röckner, B.L. Rozovskii, F. Wang, Strong solutions of stochastic generalized porous media equations: existence, uniqueness, and ergodicity. Commun. Partial Differ. Equ. 31 (1–3), 277–291 (2006)MathSciNetCrossRefMATH G. Da Prato, M. Röckner, B.L. Rozovskii, F. Wang, Strong solutions of stochastic generalized porous media equations: existence, uniqueness, and ergodicity. Commun. Partial Differ. Equ. 31 (1–3), 277–291 (2006)MathSciNetCrossRefMATH
76.
go back to reference W. Liu, M. Röckner, Stochastic Partial Differential Equations: An Introduction. Universitext. (Springer, Cham, 2015) W. Liu, M. Röckner, Stochastic Partial Differential Equations: An Introduction. Universitext. (Springer, Cham, 2015)
77.
go back to reference W. Liu, J. Tölle, Existence and uniqueness of invariant measures for stochastic evolution equations with weakly dissipative drifts. Electronic Commun. Probab. 16, 447–457 (2011)MathSciNetCrossRef W. Liu, J. Tölle, Existence and uniqueness of invariant measures for stochastic evolution equations with weakly dissipative drifts. Electronic Commun. Probab. 16, 447–457 (2011)MathSciNetCrossRef
83.
go back to reference E. Priola, On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions. Stud. Math. 136, 271–295 (1999)MathSciNetMATH E. Priola, On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions. Stud. Math. 136, 271–295 (1999)MathSciNetMATH
Metadata
Title
Transition Semigroup
Authors
Viorel Barbu
Giuseppe Da Prato
Michael Röckner
Copyright Year
2016
DOI
https://doi.org/10.1007/978-3-319-41069-2_7