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.
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.