# zbMATH — the first resource for mathematics

Log-Harnack inequality for stochastic Burgers equations and applications. (English) Zbl 1227.60079
Authors’ abstract: “By proving an $$L^2$$ gradient estimate for the corresponding Galerkin approximation, the log Harnack inequality is established for the semigroup associated to a class of stochastic Burgers equations. As application, we derive the strong Feller property of the semigroups, the irreducibility of the solution, the entropy-cost inequality for the adjoint semigroups, and entropy upper bounds of the transition density.”

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H30 Applications of stochastic analysis (to PDEs, etc.)
##### Keywords:
stochastic Burgers equation; strong Feller property
Full Text:
##### References:
 [1] Arnaudon, M.; Thalmaier, A.; Wang, F.-Y., Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below, Bull. sci. math., 130, 223-233, (2006) · Zbl 1089.58024 [2] Arnaudon, M.; Thalmaier, A.; Wang, F.-Y., Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds, Stochastic process. appl., 119, 3653-3670, (2009) · Zbl 1178.58013 [3] Da Prato, G., Kolmogorov equations for stochastic pdes, Adv. courses math. CRM Barcelona, (2004), Birkhäuser Verlag Basel · Zbl 1066.60061 [4] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press · Zbl 0761.60052 [5] Da Prato, G.; Zabczyk, J., Ergodicity for infinite-dimensional systems, London math. soc. lecture note ser., vol. 229, (1996), Cambridge University Press Cambridge · Zbl 0849.60052 [6] Da Prato, G.; Röckner, M.; Wang, F.-Y., Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups, J. funct. anal., 257, 992-1017, (2009) · Zbl 1193.47047 [7] Es-Sarhir, A.; Renesse, M.-K.v.; Scheutzow, M., Harnack inequality for functional SDEs with bounded memory, Electron. comm. probab., 14, 560-565, (2009) · Zbl 1195.34124 [8] Liu, W.; Wang, F.-Y., Harnack inequality and strong Feller property for stochastic fast-diffusion equations, J. math. anal. appl., 342, 651-662, (2008) · Zbl 1151.60032 [9] Röckner, M.; Wang, F.-Y., Harnack and functional inequalities for generalized mehler semigroups, J. funct. anal., 203, 237-261, (2003) · Zbl 1059.47051 [10] Röckner, M.; Wang, F.-Y., Log-Harnack inequality for stochastic differential equations in Hilbert spaces and its consequences, Infin. dimens. anal. quantum probab. relat. top., 13, 27-37, (2010) · Zbl 1207.60053 [11] Temam, R., Navier-Stokes equations and nonlinear functional analysis, (1995), Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA, xiv+141 pp · Zbl 0833.35110 [12] Wang, F.-Y., Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. theory related fields, 109, 417-424, (1997) · Zbl 0887.35012 [13] Wang, F.-Y., Harnack inequality and applications for stochastic generalized porous media equations, Ann. probab., 35, 1333-1350, (2007) · Zbl 1129.60060 [14] Wang, F.-Y., Harnack inequalities on manifolds with boundary and applications, J. math. pures appl., 94, 304-321, (2010) · Zbl 1207.58028 [15] F.-Y. Wang, Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on non-convex manifolds, Ann. Probab., doi:10.1214/10-aop600, in press, arXiv:0911.1644. [16] Wang, F.-Y.; Xu, L., Bismut type formula and its application to stochastic hyperdissipative Navier-Stokes/Burgers equations
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.