Hamiltonian Feynman measures, Kolmogorov integral and infinite dimensional pseudo-differential operators

Properties of infinite dimensional pseudo-differential operators (PDO) are discussed; in particular, the connection between two definitions of the PDO is considered: one given in terms of the Hamiltonian Feynman measure, and another introduced in this work in terms of the Kolmogorov integral.

COUNTABLE STATE MARKOV PROCESSES: NON-EXPLOSIVENESS AND MOMENT FUNCTION

The existence of a moment function satisfying a drift function condition is well known to guarantee non-explosiveness of the associated minimal Markov process (cf. [1,9]), under standard technical conditions. Surprisingly, the reverse is true as well for a countable space Markov process. We prove this result by showing that recurrence of an associated jump process, that we call the α-jump process, is equivalent to non-explosiveness. Non-explosiveness corresponds in a natural way to the validity of the Kolmogorov integral relation for the function identically equal to 1. In particular, we show that positive recurrence of the α-jump chain implies that all bounded functions satisfy the Kolmogorov integral relation. We present a drift function criterion characterizing positive recurrence of this α-jump chain.Suppose that to a drift functionVthere corresponds another drift functionW, which is a moment with respect toV. Via a transformation argument, the above relations hold for the transformed process with respect toV. Transferring the results back to the original process, allows to characterize theV-bounded functions that satisfy the Kolmogorov forward equation.