Extending Markov Processes in Weak Duality by Poisson Point Processes of Excursions

Abstract

Summary. Let a be a non-isolated point of a topological space E. Suppose we are given standard processes X 0 and b X 0 on E0 = E \ {a} in weak duality with respect to a σ-finite measure m on E0 which are of no killings inside E0 but approachable to a. We first show that their extensions X and bX to E admitting no sojourn at a and keeping the weak duality are uniquely determined by the approaching probabilities of X 0 , b X 0 and m up to a non-negative constant δ0 representing the killing rate of X at a. We then construct, starting from X 0 , such X by piecing together returning excursions around a and a possible non-returning excursion including the instant killing. This extends a recent result by M. Fukushima and H. Tanaka [16] which treats the case where X 0 , X are m-symmetric diffusions and X admits no sojourn nor killing at a. Typical examples of jump type symmetric Markov processes and non-symmetric diffusions on Euclidean domains are given at the end of the paper.

One-point extensions of Markov processes by darning

“…In this case, a is just the point 0. However recent papers [14] and [8] show that Itô's program works equally well in the construction of X * by conceiving a certain set K as a single point a.…”

“…In fact, this approach of constructions of X * and X * from X 0 and X 0 together with their characterizations has been accomplished by [14] in the symmetric diffusion case and by [8] in the weak dual standard processes case under certain additional conditions on X 0 and X 0 . We have made use of Itô's Poisson point processes of excursions of X 0 and X 0 around the point a whose characteristic measures are governed by the uniquely determined entrance laws {µ t , t > 0} of X 0 and { µ t , t > 0} of X 0 from the one point a.…”

“…We point out that it is assumed in both [8] and [14] that X 0 and X 0 admit no killings inside E 0 . Furthermore, to ensure the finiteness of µ t (E 0 ) for each t > 0, a crucial condition imposed in Sect.…”

“…The first is to present in Sect. 3 general conditions to ensure the one-point extendability of a pair of standard Markov processes in weak duality by developing those methods employed in Fukushima-Tanaka [14] and ChenFukushima-Ying [8]. The second is to characterize in Sect.…”

“…4 of [14] and in the first part of Sect. 5 of [8] is the m 0 -integrability of the α-order approaching probability u α of X 0 defined by u α (x) := E 0 x e −αζ 0 ; X ζ 0 − ∈ K = E x e −ασ K , x ∈ E 0 , α > 0.…”