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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

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