Shift Harnack inequality and integration by parts formula for semilinear stochastic partial differential equations

Abstract

Shift Harnack inequality and integration by parts formula are established for semilinear stochastic partial differential equations and stochastic functional partial differential equations by modifying the coupling used by F. -Y. Wang [Ann. Probab., 2012, 42(3): 994–1019]. Log-Harnack inequality is established for a class of stochastic evolution equations with non-Lipschitz coefficients which includes hyperdissipative Navier-Stokes/Burgers equations as examples. The integration by parts formula is extended to the path space of stochastic functional partial differential equations, then a Dirichlet form is defined and the log-Sobolev inequality is established.

Keywords

Shift Harnack inequality / integration by parts formula / stochastic partial differential equation (SPDE) / stochastic functional partial differential equation (SFPDE) / path space / log-Sobolev inequality

References

[1] Bao J, Yuan C, Wang, F-Y. Bismut formulae and applications for functional SPDEs. Bull Sci Math, 2013, 137: 509–522

[2] Bensoussan A, Da Prato G, Delfour M C, Mitter S K. Representation and Control of Infinite Dimensional Systems. 2nd ed. Basel: Birkhäuser, 2007

[3] Capitaine M, Hsu E P, Ledoux M. Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces. Electron Commun Probab, 1997, 2: 71–81

[4] Da Prato G, Zabczyk J. Stochastic Equations in Infinite Dimensions. Cambridge: Cambridge University Press, 1992

[5] Driver B. A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold. J Funct Anal, 1992, 110: 272–376

[6] Driver B. Integration by parts for heat kernel measures revisited. J Math Pures Appl, 1997, 76: 703–737

[7] Driver B. Integration by parts and quasi-invariance for heat kernel measures on loop groups. J Funct Anal, 1997, 149: 470–547

[8] Fan X-L. Derivative formula, integration by parts formula and applications for SDEs driven by fractional Brownian motion. Stoch Anal Appl, 2012, 33: 199–212

[9] Fan X-L. Stochastic Volterra equations driven by fractional Brownian motion. Front Math China, 2015, 10: 595–620

[10] Fang S. Inégalit’e du type de Poincar’e sur l’espace des chemins riemanniens. C R Acad Sci Paris Sér I, 1994, 318: 257–260

[11] Fang S, Li H, Luo D, Heat semi-group and generalized flows on complete Riemannian manifolds. Bull Sci Math, 2011, 135: 565–600

[12] Guillin A, Wang F-Y. Degenerate Fokker-Planck equations: Bismut formula, gradient estimate and Harnack inequality. J Differential Equations, 2012, 253: 20–40

[13] Hsu E P. Stochastic Analysis on Manifolds. Providence: Amer Math Soc, 2002

[14] Ouyang S-X, Röckner M, Wang F-Y. Harnack inequalities and applications for Ornstein-Uhlenbeck semigroups with jump. Potential Anal, 2012, 36: 301–315

[15] Reed M, Simon B. Methods of Modern Mathematical Physics, Vol I. New York: Academic Press, 1980

[16] Röckner M, Liu W. SPDE in Hilbert space with locally monotone coefficients. J Funct Anal, 2010, 259: 2902–2922

[17] Wang F-Y. Analysis on path spaces over Riemannian manifolds with boundary. Comm Math Sci, 2011, 9: 1203–1212

[18] Wang F-Y. Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds. Ann Probab, 2011, 39: 1449–1467

[19] Wang F-Y. Integration by parts formula and shift Harnack inequality for stochastic equations. Ann Probab, 2012, 42: 994–1019

[20] Wang F-Y. Analysis for Diffusion Processes on Riemannian Manifolds. Singapore: World Scientific, 2013

[21] Wang F-Y. Harnack Inequalities for Stochastic Partial Differential Equations. Berlin: Springer, 2013

[22] Wang F-Y, Xu L. Derivative formula and applications for hyperdissipative stochastic Navier-Stokes/Burgers equations. Infin Dimens Anal Quantum Probab Relat Top, 2012, 15: 1250020

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