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Asymptotic separation for independent trajectories of Markov processes
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  • Published: January 2001

Asymptotic separation for independent trajectories of Markov processes

  • Alexander Grigor'yan1 &
  • Mark Kelbert2 

Probability Theory and Related Fields volume 119, pages 31–69 (2001)Cite this article

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  • 2 Citations

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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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Authors and Affiliations

  1. Imperial College, 180 Queens Gate, London SW7 2BZ, United Kingdom. e-mail: a.grigoryan@ic.ac.uk, , , , , , GB

    Alexander Grigor'yan

  2. University of Wales, Swansea, Singleton Park, Swansea SA2 8PP, United Kingdom. e-mail: m.kelbert@swansea.ac.uk, , , , , , GB

    Mark Kelbert

Authors
  1. Alexander Grigor'yan
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  2. Mark Kelbert
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Additional information

Received: 10 June 1999 / Revised version: 20 April 2000 / Published online: 14 December 2000

RID="*"

ID="*" Supported by the EPSRC Research Fellowship B/94/AF/1782

RID="**"

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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  • Issue Date: January 2001

  • DOI: https://doi.org/10.1007/PL00012738

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  • Mathematics Subject Classification (2000): 58J65, 60G17, 60G52, 60J45
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