Abstract
A symmetric Markov process X is said to be in Class (T) if it is irreducible, strong Feller and possesses a tightness property. We give some properties of X in Class (T) and of the semi-group \(p_t\) of X: the uniform large deviation principle of X, \(L^p\)-independence of growth bounds of \(p_t\), compactness of \(p_t\) as an operator in \(L^2\), and boundedness of every eigenfunction of \(p_t\).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Chen, Z.-Q.: Gaugeability and conditional gaugeability. Trans. Am. Math. Soc. 354, 4639–4679 (2002)
Chung, K.L., Zhao, Z.X.: From Brownian Motion to Schrödinger’s Equation. Springer, Berlin (1995)
Deuschel, J.-D., Stroock, D.W.: Large Deviations. American Mathematical Society, Providence (2001)
Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Oxford University Press, Oxford (1987)
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, de Gruyter, 2nd rev. and ext. ed. (2011)
Itô, K.: Essentials of Stochastic Processes. American Mathematical Society, Providence (2006)
Kaleta, K., Kulczycki, T.: Intrinsic ultracontractivity for Schrödinger operators based on fractional Laplacians. Potential Anal. 33, 313–339 (2010)
Lenz, D., Stollmann, P., Wingert, D.: Compactness of Schrödinger semigroups. Math. Nachr. 283, 94–103 (2010)
Schilling, R. L.: Measures, Integrals and Martingales. Cambridge University Press, Cambridge (2005)
Simon, B.: Schrödinger operators with purely discrete spectra. Methods Funct. Anal. Topol. 15, 61–66 (2009)
Simon, B.: Operator Theory, A Comprehensive Course in Analysis, Part 4. American Mathematical Society (2015)
Stollmann, P., Voigt, J.: Perturbation of Dirichlet forms by measures. Potential Anal. 5, 109–138 (1996)
Takeda, M.: Lp-independence of spectral bounds of Schrödinger type semigroups. J. Funct. Anal. 252, 550–565 (2007)
Takeda, M.: A tightness property of a symmetric Markov process and the uniform large deviation principle. Proc. Am. Math. Soc. 141, 4371–4383 (2013)
Takeda, M.: A variational formula for Dirichlet forms and existence of ground states. J. Funct. Anal. 266, 660–675 (2014)
Takeda, M.: Criticality and subcriticality of generalized Schrödinger forms. Illinois J. Math. 58, 251–277 (2014)
Takeda, M., Tawara, Y., Tsuchida, K.: Compactness of Markov and Schrödinger semi-groups: a probabilistic approach. Osaka J. Math. 54, 517–532 (2017)
Wu, L.: Some notes on large deviations of Markov processes, Acta Math. Sin. (Engl. Ser.) 16, 369-394 (2000)
Acknowledgements
The author would like to thank the referee for helpful comments on this paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Takeda, M. (2018). Symmetric Markov Processes with Tightness Property. In: Eberle, A., Grothaus, M., Hoh, W., Kassmann, M., Stannat, W., Trutnau, G. (eds) Stochastic Partial Differential Equations and Related Fields. SPDERF 2016. Springer Proceedings in Mathematics & Statistics, vol 229. Springer, Cham. https://doi.org/10.1007/978-3-319-74929-7_33
Download citation
DOI: https://doi.org/10.1007/978-3-319-74929-7_33
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-74928-0
Online ISBN: 978-3-319-74929-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)