Ultracontractivity for Brownian motion with Markov switching

Abstract

Consider the transition density functions for Brownian motion with two-state Markov switching. The characteristic functions for transition density functions are presented. Then, we show that the semigroup-associated Brownian motion with Markov switching is ultracontractive. And an explicit time-dependent upper bound for heat kernels are presented. Moreover, we prove that the Dirichlet form associated Brownian motion with Markov switching satisfies the Nash inequality.

Keywords:

• Heat kernels
• Nash inequality
• ultracontractivity

Mathematics Subject Classification 2000:

• 60H10
• 37A25

Acknowledgments

The authors are grateful to Prof. Zhenqing Chen and Dr. Guohuang Zhao for his valuable comments and suggestions which led to improvements in this manuscript. Besides, the authors would like to thank the anonymous reviewers for careful reading of the article and for helpful comments that led to improvement of the first version of this article.

Additional information

Funding

The research of Z. Zhang was partially supported by the Humanities and Social Sciences Fund of Ministry of Education of China (No. 17YJA910004). The research of J. Tong was partially supported by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China (Nos. 11401093 and 11571071). The research of L. Hu was partially supported by the National Natural Science Foundation of China (No. 11471071).