Estimates of deviations of transition probabilities of inhomogeneous Markov processes
Let \(X^{(i)}=(X^{(i)}(t))\), \(i=1,2\), be two Markov processes with values in a measurable space (X,\({\mathfrak B}_ x)\). Let \(p^{(i)}(t,x,s,A)\) \((i=1,2\); \(x\in X\), \(A\in {\mathfrak B}_ x\), \(0\leq t\leq s<\infty)\) denote their transition probabilities. The author introduces the mixing coefficients \[ \rho^{(i)}(t,s)=\sup \{| p^{(i)}(t,x_ 1,s,A)-p^{(i)}(t,x_ 2,s,A)|:\quad x_ 1,x_ 2\in X,\quad A\in {\mathfrak B}_ x\},\quad 0\leq t\leq s<\infty,\text{ and the bounds} \] \[ \psi^{(1,2)}(t,s)=\sup_{x,A}| p^{(1)}(t,x,s,A)-p^{(2)}(t,x,s,A)|,\quad 0\leq t\leq s<\infty \] for the deviations of the transition probabilities of \(X^{(1)}\) and \(X^{(2)}\). Conditions in terms of \(\rho^{(1)}\) and \(\psi^{(1,2)}\) are given which are sufficient for \(\rho^{(2)}(t,s)\) to decrease exponentially fast as \(0\leq t-s\to \infty.\) As a consequence, the author shows in the case when \(X^{(1)}\) and \(X^{(2)}\) are time-homogeneous and satisfy certain conditions that \(X^{(i)}\) has a stationary distribution \(\pi^{(i)}\) \((i=1,2)\), and he obtains an upper bound for the total variation between \(\pi^{(1)}\) and \(\pi^{(2)}\). Finally, large deviation results for certain hitting times are derived.
scientific article; zbMATH DE number 4105994