• Previous Article
    Analysis of a reduced-order approximate deconvolution model and its interpretation as a Navier-Stokes-Voigt regularization
  • DCDS-B Home
  • This Issue
  • Next Article
    On the spectral stability of standing waves of the one-dimensional $M^5$-model
June  2016, 21(4): 1051-1077. doi: 10.3934/dcdsb.2016.21.1051

Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces

1. 

Department of Mathematics, University of York, Heslington Road, York YO10 5DD, United Kingdom

2. 

Department of Mathematics and Information Technology, Montanuniversität Leoben, Franz Josef Straße 18, 8700 Leoben, Austria

Received  January 2015 Revised  October 2015 Published  March 2016

The main goal of this paper is to generalize to Banach spaces the well-known results for diffusions on Hilbert spaces obtained in [Peszat, S. and Zabczyk, J. (1995). Strong Feller property and irreducibility for diffusions on Hilbert spaces. Ann. Probab. 23(1): 157-172.]. More precisely, we are aiming to prove the strong Feller property and irreducibility of the solutions to a stochastic evolution equations (SEEs) in Banach spaces. We give sufficient conditions on the path space and the coefficients of the SEEs for these aforementioned properties to hold. We apply our result to investigate the long-time behavior of a stochastic nonlinear heat equations on $L^p$-space with $p>4$. Our result implies the uniqueness of the invariant measure, if it exists, for the stochastic nonlinear heat equations on $L^p$-space with $p>4$.
Citation: Zdzisław Brzeźniak, Paul André Razafimandimby. Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1051-1077. doi: 10.3934/dcdsb.2016.21.1051
References:
[1]

Z. Brzeźniak, On stochastic convolution in Banach spaces and applications,, Stochastics Stochastics Rep., 61 (1997), 245. doi: 10.1080/17442509708834122. Google Scholar

[2]

Z. Brzeźniak and D. Gatarek, Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces,, Stochastic Process. Appl., 84 (1999), 187. doi: 10.1016/S0304-4149(99)00034-4. Google Scholar

[3]

Z. Brzeźniak and S. Peszat, Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process,, Studia Math., 137 (1999), 261. Google Scholar

[4]

Z. Brzeźniak and J. M. A. M. van Neerven, Space-time regularity for linear stochastic evolution equations driven by spatially homogeneous noise,, J. Math. Kyoto Univ., 43 (2003), 261. Google Scholar

[5]

Z. Brzeźniak, J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Itô's formula in UMD Banach spaces and regularity of solutions of the Zakai equation,, J. Differential Equations, 245 (2008), 30. doi: 10.1016/j.jde.2008.03.026. Google Scholar

[6]

Z. Brzeźniak, H. Long and I. Simão, Invariant measures for stochastic evolution equations in M-type 2 Banach spaces,, J. Evol. Equ., 10 (2010), 785. doi: 10.1007/s00028-010-0070-2. Google Scholar

[7]

H. Cartan, Differential Calculus,, Translated from the French. Hermann, (1971). Google Scholar

[8]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Second Edition,, Encyclopedia of Mathematics and its Applications, 44 (2014). doi: 10.1017/CBO9781107295513. Google Scholar

[9]

A. Chojnowska-Michalik and B. Goldys, Existence, uniqueness and invariant measures for stochastic semilinear equations in Hilbert spaces,, Probab. Th. Rel. Fields, 102 (1995), 331. doi: 10.1007/BF01192465. Google Scholar

[10]

G. Da Prato, K. D. Elworthy and J. Zabczyk, Strong Feller property for stochastic semilinear equations,, Stoch. Anal. Appl., 13 (1995), 35. doi: 10.1080/07362999508809381. Google Scholar

[11]

R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces,, Pitman Monographs and Surveys in Pure and Applied Mathematics,64. Longman Scientific & Technical, 64 (1993). Google Scholar

[12]

D. Gątarek and B. Dariusz, On invariant measures for diffusions on Banach spaces,, Potential Anal., 7 (1997), 539. doi: 10.1023/A:1008663614438. Google Scholar

[13]

M. Hairer and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing,, Ann. of Math. (2), 164 (2006), 993. doi: 10.4007/annals.2006.164.993. Google Scholar

[14]

M. Hairer and J. C. Mattingly, A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs,, Electron. J. Probab., 16 (2011), 658. doi: 10.1214/EJP.v16-875. Google Scholar

[15]

P. Hájek and M. Johanis, Smooth approximations,, J. Funct. Anal., 259 (1020), 561. doi: 10.1016/j.jfa.2010.04.020. Google Scholar

[16]

A. Ichikawa, Semilinear stochastic evolution equations: Boundedness, stability and invariant measures,, Stochastics, 12 (1984), 1. doi: 10.1080/17442508408833293. Google Scholar

[17]

T. Komorowski, S. Peszat and T. Szarek, On ergodicity of some Markov processes,, Ann. Probab., 38 (2010), 1401. doi: 10.1214/09-AOP513. Google Scholar

[18]

S. Kuksin and V. Nersesyan, Stochastic CGL equations without linear dispersion in any space dimension,, Stochastic Partial Differential Equations: Analysis and Computations, 1 (2013), 389. doi: 10.1007/s40072-013-0010-6. Google Scholar

[19]

S. Kuksin and A. Shirikyan, Coupling approach to white-forced nonlinear PDEs,, J. Math. Pures Appl. (9), 81 (2002), 567. doi: 10.1016/S0021-7824(02)01259-X. Google Scholar

[20]

S. Kuksin and A. Shirikyan, Mathematics of Two-Dimensional Turbulence,, Cambridge University Press, (2012). doi: 10.1017/CBO9781139137119. Google Scholar

[21]

G. Leha and G. Ritter, Lyapunov-type conditions for stationary distributions of diffusion processes on Hilbert spaces,, Stoch. Stoch. Rep., 48 (1994), 195. doi: 10.1080/17442509408833906. Google Scholar

[22]

J. Maas, Malliavin calculus and decoupling inequalities in Banach spaces,, J. Math. Anal. Appl., 363 (2010), 383. doi: 10.1016/j.jmaa.2009.08.041. Google Scholar

[23]

J. Maas and J. M. A. M. van Neerven, A Clark-Ocone formula in UMD Banach spaces,, Electron. Commun. Probab., 13 (2008), 151. doi: 10.1214/ECP.v13-1361. Google Scholar

[24]

B. Maslowski, Uniqueness and stability of invariant measures for stochastic differential equations in Hilbert spaces,, Stoch. Stoch. Rep., 28 (1989), 85. doi: 10.1080/17442508908833585. Google Scholar

[25]

B. Maslowski and J. Seidler, Invariant measures for nonlinear SPDEs: Uniqueness and stability,, Arch. Math., 34 (1998), 153. Google Scholar

[26]

J. M. A. M. van Neerven, Uniqueness of invariant measures for the stochastic Cauchy problem in Banach spaces,, Recent advances in operator theory and related topics (Szeged, 127 (2001), 491. Google Scholar

[27]

J. M. A. M. van Neerven and J. Zhu, A maximal inequality for stochastic convolutions in 2-smooth Banach spaces,, Electron. Commun. Probab., 16 (2011), 689. doi: 10.1214/ECP.v16-1677. Google Scholar

[28]

A. L. Neidhardt, Stochastic Integrals in 2-Uniformly Smooth Banach Spaces,, PhD thesis, (1978). Google Scholar

[29]

M. Ondreját, Uniqueness for stochastic evolution equations in Banach spaces,, Dissertationes Math. (Rozprawy Mat.), 426 (2004). doi: 10.4064/dm426-0-1. Google Scholar

[30]

S. Peszat and J. Zabczyk, Strong Feller property and irreducibility for diffusions on Hilbert spaces,, Ann. Probab., 23 (1995), 157. doi: 10.1214/aop/1176988381. Google Scholar

[31]

G. Pisier, Martingales with values in uniformly convex spaces,, Israel J. Math., 20 (1975), 326. doi: 10.1007/BF02760337. Google Scholar

[32]

M. Pronk and M. Veraar, Tools for Malliavin calculus in UMD Banach spaces,, Potential Anal., 40 (2014), 307. doi: 10.1007/s11118-013-9350-0. Google Scholar

[33]

E. Shamarova, A version of the Hörmander-Malliavin theorem in 2-smooth Banach spaces,, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 17 (2014). doi: 10.1142/S0219025714500040. Google Scholar

[34]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, Second edition. Johann Ambrosius Barth, (1995). Google Scholar

[35]

J. Zabczyk, Symmetric solutions of semilinear stochastic equations,, in Stochastic partial differential equations and applications, 1390 (1989), 237. doi: 10.1007/BFb0083952. Google Scholar

[36]

J. Zhu, Z. Brzeźniak and E. Hausenblas, Maximal inequality of Stochastic convolution driven by compensated Poisson random measures in Banach spaces,, To appear in Ann. Inst. Henri Poincar Probab. Stat. preprint, (). Google Scholar

show all references

References:
[1]

Z. Brzeźniak, On stochastic convolution in Banach spaces and applications,, Stochastics Stochastics Rep., 61 (1997), 245. doi: 10.1080/17442509708834122. Google Scholar

[2]

Z. Brzeźniak and D. Gatarek, Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces,, Stochastic Process. Appl., 84 (1999), 187. doi: 10.1016/S0304-4149(99)00034-4. Google Scholar

[3]

Z. Brzeźniak and S. Peszat, Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process,, Studia Math., 137 (1999), 261. Google Scholar

[4]

Z. Brzeźniak and J. M. A. M. van Neerven, Space-time regularity for linear stochastic evolution equations driven by spatially homogeneous noise,, J. Math. Kyoto Univ., 43 (2003), 261. Google Scholar

[5]

Z. Brzeźniak, J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Itô's formula in UMD Banach spaces and regularity of solutions of the Zakai equation,, J. Differential Equations, 245 (2008), 30. doi: 10.1016/j.jde.2008.03.026. Google Scholar

[6]

Z. Brzeźniak, H. Long and I. Simão, Invariant measures for stochastic evolution equations in M-type 2 Banach spaces,, J. Evol. Equ., 10 (2010), 785. doi: 10.1007/s00028-010-0070-2. Google Scholar

[7]

H. Cartan, Differential Calculus,, Translated from the French. Hermann, (1971). Google Scholar

[8]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Second Edition,, Encyclopedia of Mathematics and its Applications, 44 (2014). doi: 10.1017/CBO9781107295513. Google Scholar

[9]

A. Chojnowska-Michalik and B. Goldys, Existence, uniqueness and invariant measures for stochastic semilinear equations in Hilbert spaces,, Probab. Th. Rel. Fields, 102 (1995), 331. doi: 10.1007/BF01192465. Google Scholar

[10]

G. Da Prato, K. D. Elworthy and J. Zabczyk, Strong Feller property for stochastic semilinear equations,, Stoch. Anal. Appl., 13 (1995), 35. doi: 10.1080/07362999508809381. Google Scholar

[11]

R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces,, Pitman Monographs and Surveys in Pure and Applied Mathematics,64. Longman Scientific & Technical, 64 (1993). Google Scholar

[12]

D. Gątarek and B. Dariusz, On invariant measures for diffusions on Banach spaces,, Potential Anal., 7 (1997), 539. doi: 10.1023/A:1008663614438. Google Scholar

[13]

M. Hairer and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing,, Ann. of Math. (2), 164 (2006), 993. doi: 10.4007/annals.2006.164.993. Google Scholar

[14]

M. Hairer and J. C. Mattingly, A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs,, Electron. J. Probab., 16 (2011), 658. doi: 10.1214/EJP.v16-875. Google Scholar

[15]

P. Hájek and M. Johanis, Smooth approximations,, J. Funct. Anal., 259 (1020), 561. doi: 10.1016/j.jfa.2010.04.020. Google Scholar

[16]

A. Ichikawa, Semilinear stochastic evolution equations: Boundedness, stability and invariant measures,, Stochastics, 12 (1984), 1. doi: 10.1080/17442508408833293. Google Scholar

[17]

T. Komorowski, S. Peszat and T. Szarek, On ergodicity of some Markov processes,, Ann. Probab., 38 (2010), 1401. doi: 10.1214/09-AOP513. Google Scholar

[18]

S. Kuksin and V. Nersesyan, Stochastic CGL equations without linear dispersion in any space dimension,, Stochastic Partial Differential Equations: Analysis and Computations, 1 (2013), 389. doi: 10.1007/s40072-013-0010-6. Google Scholar

[19]

S. Kuksin and A. Shirikyan, Coupling approach to white-forced nonlinear PDEs,, J. Math. Pures Appl. (9), 81 (2002), 567. doi: 10.1016/S0021-7824(02)01259-X. Google Scholar

[20]

S. Kuksin and A. Shirikyan, Mathematics of Two-Dimensional Turbulence,, Cambridge University Press, (2012). doi: 10.1017/CBO9781139137119. Google Scholar

[21]

G. Leha and G. Ritter, Lyapunov-type conditions for stationary distributions of diffusion processes on Hilbert spaces,, Stoch. Stoch. Rep., 48 (1994), 195. doi: 10.1080/17442509408833906. Google Scholar

[22]

J. Maas, Malliavin calculus and decoupling inequalities in Banach spaces,, J. Math. Anal. Appl., 363 (2010), 383. doi: 10.1016/j.jmaa.2009.08.041. Google Scholar

[23]

J. Maas and J. M. A. M. van Neerven, A Clark-Ocone formula in UMD Banach spaces,, Electron. Commun. Probab., 13 (2008), 151. doi: 10.1214/ECP.v13-1361. Google Scholar

[24]

B. Maslowski, Uniqueness and stability of invariant measures for stochastic differential equations in Hilbert spaces,, Stoch. Stoch. Rep., 28 (1989), 85. doi: 10.1080/17442508908833585. Google Scholar

[25]

B. Maslowski and J. Seidler, Invariant measures for nonlinear SPDEs: Uniqueness and stability,, Arch. Math., 34 (1998), 153. Google Scholar

[26]

J. M. A. M. van Neerven, Uniqueness of invariant measures for the stochastic Cauchy problem in Banach spaces,, Recent advances in operator theory and related topics (Szeged, 127 (2001), 491. Google Scholar

[27]

J. M. A. M. van Neerven and J. Zhu, A maximal inequality for stochastic convolutions in 2-smooth Banach spaces,, Electron. Commun. Probab., 16 (2011), 689. doi: 10.1214/ECP.v16-1677. Google Scholar

[28]

A. L. Neidhardt, Stochastic Integrals in 2-Uniformly Smooth Banach Spaces,, PhD thesis, (1978). Google Scholar

[29]

M. Ondreját, Uniqueness for stochastic evolution equations in Banach spaces,, Dissertationes Math. (Rozprawy Mat.), 426 (2004). doi: 10.4064/dm426-0-1. Google Scholar

[30]

S. Peszat and J. Zabczyk, Strong Feller property and irreducibility for diffusions on Hilbert spaces,, Ann. Probab., 23 (1995), 157. doi: 10.1214/aop/1176988381. Google Scholar

[31]

G. Pisier, Martingales with values in uniformly convex spaces,, Israel J. Math., 20 (1975), 326. doi: 10.1007/BF02760337. Google Scholar

[32]

M. Pronk and M. Veraar, Tools for Malliavin calculus in UMD Banach spaces,, Potential Anal., 40 (2014), 307. doi: 10.1007/s11118-013-9350-0. Google Scholar

[33]

E. Shamarova, A version of the Hörmander-Malliavin theorem in 2-smooth Banach spaces,, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 17 (2014). doi: 10.1142/S0219025714500040. Google Scholar

[34]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, Second edition. Johann Ambrosius Barth, (1995). Google Scholar

[35]

J. Zabczyk, Symmetric solutions of semilinear stochastic equations,, in Stochastic partial differential equations and applications, 1390 (1989), 237. doi: 10.1007/BFb0083952. Google Scholar

[36]

J. Zhu, Z. Brzeźniak and E. Hausenblas, Maximal inequality of Stochastic convolution driven by compensated Poisson random measures in Banach spaces,, To appear in Ann. Inst. Henri Poincar Probab. Stat. preprint, (). Google Scholar

[1]

Tôn Việt Tạ. Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: Strict solutions and maximal regularity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4507-4542. doi: 10.3934/dcds.2017193

[2]

Goro Akagi, Mitsuharu Ôtani. Evolution equations and subdifferentials in Banach spaces. Conference Publications, 2003, 2003 (Special) : 11-20. doi: 10.3934/proc.2003.2003.11

[3]

Sonja Cox, Arnulf Jentzen, Ryan Kurniawan, Primož Pušnik. On the mild Itô formula in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2217-2243. doi: 10.3934/dcdsb.2018232

[4]

Fatihcan M. Atay, Lavinia Roncoroni. Lumpability of linear evolution Equations in Banach spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 15-34. doi: 10.3934/eect.2017002

[5]

Laura Levaggi. Existence of sliding motions for nonlinear evolution equations in Banach spaces. Conference Publications, 2013, 2013 (special) : 477-487. doi: 10.3934/proc.2013.2013.477

[6]

Aleksander Ćwiszewski, Piotr Kokocki. Krasnosel'skii type formula and translation along trajectories method for evolution equations. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 605-628. doi: 10.3934/dcds.2008.22.605

[7]

Arnulf Jentzen, Felix Lindner, Primož Pušnik. On the Alekseev-Gröbner formula in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4475-4511. doi: 10.3934/dcdsb.2019128

[8]

Wei-guo Wang, Wei-chao Wang, Ren-cang Li. Deflating irreducible singular M-matrix algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 491-518. doi: 10.3934/naco.2013.3.491

[9]

Marina V. Plekhanova. Strong solutions of quasilinear equations in Banach spaces not solvable with respect to the highest-order derivative. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 833-846. doi: 10.3934/dcdss.2016031

[10]

Mustapha Mokhtar-Kharroubi. On permanent regimes for non-autonomous linear evolution equations in Banach spaces with applications to transport theory. Kinetic & Related Models, 2010, 3 (3) : 473-499. doi: 10.3934/krm.2010.3.473

[11]

Pengyu Chen, Yongxiang Li, Xuping Zhang. On the initial value problem of fractional stochastic evolution equations in Hilbert spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1817-1840. doi: 10.3934/cpaa.2015.14.1817

[12]

Alexandre Nolasco de Carvalho, Stefanie Sonner. Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results. Communications on Pure & Applied Analysis, 2013, 12 (6) : 3047-3071. doi: 10.3934/cpaa.2013.12.3047

[13]

Alexandre Nolasco de Carvalho, Stefanie Sonner. Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1141-1165. doi: 10.3934/cpaa.2014.13.1141

[14]

Ugo Bessi. The stochastic value function in metric measure spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076

[15]

Huijie Qiao, Jiang-Lun Wu. On the path-independence of the Girsanov transformation for stochastic evolution equations with jumps in Hilbert spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1449-1467. doi: 10.3934/dcdsb.2018215

[16]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085

[17]

Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140

[18]

Yijing Sun, Yuxin Tan. Kirchhoff type equations with strong singularities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 181-193. doi: 10.3934/cpaa.2019010

[19]

Paola Mannucci, Claudio Marchi, Nicoletta Tchou. Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 119-128. doi: 10.3934/dcdss.2019008

[20]

B. S. Lee, Arif Rafiq. Strong convergence of an implicit iteration process for a finite family of Lipschitz $\phi -$uniformly pseudocontractive mappings in Banach spaces. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 287-293. doi: 10.3934/naco.2014.4.287

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (0)

[Back to Top]