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Self-avoiding random walk: A Brownian motion model with local time drift
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  • Published: June 1987

Self-avoiding random walk: A Brownian motion model with local time drift

  • J. R. Norris1,
  • L. C. G. Rogers1 &
  • David Williams1 

Probability Theory and Related Fields volume 74, pages 271–287 (1987)Cite this article

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  • 51 Citations

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Summary

A natural model for a ‘self-avoiding’ Brownian motion inR d, when specialised and simplified tod=1, becomes the stochastic differential equation\(X_t = B_t - \int\limits_0^t g (X_s ,L(s,X_s ))ds\), where {L(t, x):t≧0,x∈R} is the local time process ofX. ThoughX is not Markovian, an analogue of the Ray-Knight theorem holds for {L(∞,x):x∈R}, which allows one to prove in many cases of interest that\(\mathop {\lim }\limits_{t \to \infty } X_t /t\) exists almost surely, and to identify the limit.

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Authors and Affiliations

  1. Statistical Laboratory, 16 Mill Lane, CB2 1SB, Cambridge, Great Britain

    J. R. Norris, L. C. G. Rogers & David Williams

Authors
  1. J. R. Norris
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  2. L. C. G. Rogers
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  3. David Williams
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Norris, J.R., Rogers, L.C.G. & Williams, D. Self-avoiding random walk: A Brownian motion model with local time drift. Probab. Th. Rel. Fields 74, 271–287 (1987). https://doi.org/10.1007/BF00569993

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  • Received: 17 July 1985

  • Revised: 05 September 1986

  • Issue Date: June 1987

  • DOI: https://doi.org/10.1007/BF00569993

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Keywords

  • Differential Equation
  • Time Process
  • Stochastic Process
  • Brownian Motion
  • Random Walk
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