Abstract
By using local and global versions of Bismut type derivative formulas, gradient estimates are derived for the Neumann semigroup on a narrow strip. Applications to functional/cost inequalities and heat kernel estimates are presented. Since the narrow strip we consider is non-convex with zero injectivity radius, and does not satisfy the volume doubling condition, existing results in the literature do not apply.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bakry, D.: $\text{ L }^{\prime }$hypercontractivité et son utilisation en théorie des semigroups. Lectures on Probability Theory, Lecture Notes in Math, vol. 1581, pp. 1-114. Springer, Berlin (1994)
Bismut, J.-M.: Large Deviations and the Malliavin Calculus. Birkhäuser, Boston, MA (1984)
Boutayeb, S., Coulhon, T., Sikora, A.: A new approach to poitwise heat kernel upper bounds on doubling metric measure spaces. Adv. Math. 270, 302-374 (2015)
Elworthy, K.-D., Li, X.-M.: Formulae for the derivatives of heat semigroups. J. Funct. Anal. 125, 252-286 (1994)
Grigor’yan, A., Andras, T.: Two-sided estimates of heat kernels on metric spaces. Ann. Probab. 40, 1212-1284 (2012)
Hsu, E.P.: Multiplicative functional for the heat equation on manifolds with boundary. Mich. Math. J. 20, 351-367 (2002)
Thalmaier, A.: On the differentiation of heat semigroups and Poisson integrals. Stoch. Stoch. Rep. 61, 297-321 (1997)
Thalmaier, A., Wang, F.-Y.: Gradient estimates for harmonic functions on regular domains in Riemannian manifolds. J. Funct. Anal. 155, 109-124 (1998)
Léandre, R.: Long-time behaviour for the Brownian heat kernel on a compact Riemannian manifold and Bismut’s integration-by-parts formula. Seminar on Stochastic Analysis, Random Fields and Applications V, 197C201, Progr. Probab., 59, Birkhäuser, Basel (2008)
Röckner, M., Wang, F.-Y.: Log-Harnack inequality for Stochastic differential equations in Hilbert spaces and its consequences. Infin. Dimens. Anal. Quant. Probab. Relat. Top. 13, 27-37 (2010)
Wang, F.-Y.: Gradient estimates and the first Neumann eigenvalue on manifolds with boundary. Stoch. Proc. Appl. 115, 1475-1486 (2005)
Wang, F.-Y.: Harnack inequalities on manifolds with boundary and applications. J. Math. Pures Appl. 94, 304-321 (2010)
Wang, F.-Y.: Gradient and Harnack inequalities on noncompact manifolds with boundary. Pac. J. Math. 245, 185-200 (2010)
Wang, F.-Y.: Semigroup properties for the second fundamental form. Doc. Math. 15, 543-559 (2010)
Wang, F.-Y.: Analysis for Diffusion Processes on Riemannian Manifolds. World Scientific, Singapore (2013)
Wang, F.-Y., Yuan, C.: Harnack inequalities for functional SDEs with multiplicative noise and applications. Stoch. Process. Appl. 121, 2692-2710 (2011)
Wang, J.: Global heat kernel estimates. Pac. J. Math. 178, 377-398 (1997)
Acknowledgments
The author would like to thank Professor Lixin Yan for stimulating conversations and the referee for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
NNSFC (11131003, 11431014), the 985 project and the Laboratory of Mathematical and Complex Systems.
Rights and permissions
About this article
Cite this article
Wang, FY. Gradient estimates and applications for Neumann semigroup on narrow strip. Math. Z. 282, 43–60 (2016). https://doi.org/10.1007/s00209-015-1531-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-015-1531-7