Harnack Inequality and Long Time Asymptotics of Unbounded Additive Functionals of Regime-Switching Diffusion Processes

Abstract

This work provides explicit conditions to establish the dimension-free Harnack inequalities respectively for state-independent and state-dependent regime-switching processes. Then such Harnack inequalities are applied to establish the hypercontractivity of the associated semigroup, and further to study the long time asymptotics of unbounded additive functionals of regime-switching processes. Much effort is devoted to overcoming the difficulty caused by the close interaction between the diffusion processes and the Markov chains in the setting of regime-switching processes.