Abstract.
We say that n independent trajectories ξ1(t),…,ξ n (t) of a stochastic process ξ(t)on a metric space are asymptotically separated if, for some ɛ > 0, the distance between ξ i (t i ) and ξ j (t j ) is at least ɛ, for some indices i, j and for all large enough t 1,…,t n , with probability 1. We prove sufficient conitions for asymptotic separationin terms of the Green function and the transition function, for a wide class of Markov processes. In particular,if ξ is the diffusion on a Riemannian manifold generated by the Laplace operator Δ, and the heat kernel p(t, x, y) satisfies the inequality p(t, x, x) ≤ Ct −ν/2 then n trajectories of ξ are asymptotically separated provided . Moreover, if for some α∈(0, 2)then n trajectories of ξ(α) are asymptotically separated, where ξ(α) is the α-process generated by −(−Δ)α/2.
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Received: 10 June 1999 / Revised version: 20 April 2000 / Published online: 14 December 2000
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ID="*" Supported by the EPSRC Research Fellowship B/94/AF/1782
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ID="**" Partially supported by the EPSRC Visiting Fellowship GR/M61573
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Grigor'yan, A., Kelbert, M. Asymptotic separation for independent trajectories of Markov processes. Probab Theory Relat Fields 119, 31–69 (2001). https://doi.org/10.1007/PL00012738
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DOI: https://doi.org/10.1007/PL00012738
- Mathematics Subject Classification (2000): 58J65, 60G17, 60G52, 60J45