Interactive infinite Markov particle systems with jumps

Abstract

In [2] we investigated independent infinite Markov particle systems as measure-valued Markov processes with jumps, and we gave sample path properties and martingale characterizations. In particular, we investigated the exponent of Hölder-right continuity in case that the motion process is absorbing a-stable motion on ð0;yÞ with 0 < a < 2, that is, time-changed absorbing Brownian motions on ð0;yÞ by the increasing a=2-stable Lévy processes. In the present paper we shall extend the results to the case of simple interactive infinite Markov particle systems. We also consider the absorbing stable motion on a half space H 1⁄4 R 1 ð0;yÞ as a motion process. 1. Settings and Previous Results In this section we give the general settings and the known results which are given in [2] in order to describe the main results in § 3 and § 4. Let S be a domain of R . Let ðwðtÞ;PxÞtb0;x AS be a S-valued Markov process having life time zðwÞ A ð0;y such that w A Dð1⁄20; zðwÞÞ ! SÞ, i.e., w : 1⁄20; zðwÞÞ ! S is right continuous and has left-hand limits. For convenience, we fix an extra point D B S and set wðtÞ 1⁄4 D if tb zðwÞ. Moreover we shall extend functions f on S to on fDg by f ðDÞ 1⁄4 0, if necessary. We use the following notations: Let SHR be a domain. If x 1⁄4 ðx1; . . . ; xdÞ A R , then q i1 ik 1⁄4 q k=ðqxi1 qxik Þ, q i 1⁄4 q =ðqxk i Þ and qi 1⁄4 q i for each k 1⁄4 0; 1; . . . , i 1⁄4 1; . . . ; d. Moreover qt 1⁄4 q=qt for time tb 0. 2000 Mathematics Subject Classification: Primary 60G57; Secondary 60G75.