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
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,
Previous proofs of this, unlike the present one, have relied on the techniques specific to Brownian motion.
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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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DOI: https://doi.org/10.1007/BF01207514
Keywords
- Stochastic Process
- Brownian Motion
- Probability Theory
- Local Time
- Mathematical Biology