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Limit laws for local times of the Brownian sheet
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  • Published: March 1990

Limit laws for local times of the Brownian sheet

  • Michael T. Lacey1 

Probability Theory and Related Fields volume 86, pages 63–85 (1990)Cite this article

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Summary

LetB(s, t), s, t>0 be a Brownian sheet. Then for alls>0, the processB s (t)≔B(s, t), t>0 is a (scaled) Brownian motion which admits a local timeL s (x; t), which is jointly continuous inx∈R ands, t>0.s 1/2 L s is a standard Brownian local time. We prove that

$$\mathop {\lim }\limits_{\lambda \to + \infty } - \lambda ^{ - 2} \log P(\mathop {\sup }\limits_{1< s< 2} \mathop {\sup }\limits_x s^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} L_s (x;1) > \lambda ) = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}.$$

This result has several corollaries, both forL s and the local time ofB(s, t), most of them new. The new ingredient in the proof has applications to other questions concerning local times. In particular, we give a new proof of the well known large deviations result for Brownian local time,

$$\mathop {\lim }\limits_{\lambda \to + \infty } - \lambda ^{ - 2} \log P(\mathop {\sup }\limits_x L_1 (x;1) > \lambda ) = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}.$$

Previous proofs of this, unlike the present one, have relied on the techniques specific to Brownian motion.

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Authors and Affiliations

  1. Department of Mathematics, Indiana University, 47405, Bloomington, IN, USA

    Michael T. Lacey

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  1. Michael T. Lacey
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Lacey, M.T. Limit laws for local times of the Brownian sheet. Probab. Th. Rel. Fields 86, 63–85 (1990). https://doi.org/10.1007/BF01207514

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  • Received: 18 July 1988

  • Revised: 16 February 1990

  • Issue Date: March 1990

  • DOI: https://doi.org/10.1007/BF01207514

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Keywords

  • Stochastic Process
  • Brownian Motion
  • Probability Theory
  • Local Time
  • Mathematical Biology
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