Extending Markov processes in duality by Poisson point processes of excursions

Masatoshi Fukushima, Kansai University, Osaka

Let {a} be a non-isolated point of a topological space E and X^{0} = {X_{t}^{0}, ζ^{0}, P_{x}^{0}} be a strong Markov process on E_{0} = E \ {a} such that

φ(x) := P_{x}^{0}(ζ^{0} < ∞, X_{ζ^{0}−}^{0} = a) > 0, x ∈ E_{0}.

We are concerned with the uniqueness and construction of a strong Markov extension X of X^{0} from E_{0} to E such that X admits no sojourn at the point {a}. When both X^{0} and X are required to be diffusions which are symmetric with respect to a σ-finite measure m on E_{0}, affirmative answers have been given in [4]: the entrance law and the absorption rate for the absorbed Poisson point processes of excursions attached to X ([1],[2]) is necessarily determined by the function φ, and X can be recovered from X^{0} and these data by piecing together the associated excursions away from {a}. In this talk, we generalize the results of [4] to a standard process X^{0} possessing a weak dual one X̂^{0} with respect to m based on the recent works [5], [6] on the associated exit system. Here φ and its dual object φ̂ play the roles. As an application, we extend the censored stable process studied in [3].

References

[1] K. Itô, Poisson point processes attached to Markov processes, in: Proc. Sixth Berkeley Symp. Math. Stat. Probab. III, 1970, pp225-239

[2] P.A. Meyer, Processus de Poisson ponctuels, d’aprés K. Itô, Séminaire de Probab. V, in: Lecture Notes in Math., Vol.191, Springer,1971, pp.177-190 [3] K. Bogdan, K. Burdzy, Z.Q. Chen, Censored stable processes, Probab. Theory Relat. Fields 127(2003), 89-152

[4] M. Fukushima and H. Tanaka, Poisson point processes attached to symmetric diffusions, Ann. Inst. H. Poincaré Probab.Statist. 41(2005), 419-459 [5] P.J. Fitzsimmons and R.K. Getoor, Excursion theory revisited, Preprint, 2005

[6] Z.Q. Chen, M. Fukushima and J. Ying, Entrance law, exit system and Lévy system of time changed processes, Preprint