A CLT for the L² modulus of continuity of Brownian local time

2009/01/08 by Xia Chen, Wenbo Li, Chen, Xia +5
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.0901.1102

arxiv created 2009/01/08 · arxiv updated 2009/12/01

Abstract

Let \L^x_t ; (x,t)∈ R¹× R¹₊\ denote the local time of Brownian motion and α_t:=∫_-∞^∞ (L^x_t)² dx . Let η=N(0,1) be independent of α_t. For each fixed t ∫_-∞^∞ (L^x+h_t- L^x_t)² dx- 4ht\over h^3/2 \stackrelL→(64 \over 3)^1/2√α_t η, as h\rar 0. Equivalently ∫_-∞^∞ (L^x+1_t- L^x_t)² dx- 4t\over t^3/4 \stackrelL→(64 \over 3 )^1/2√α₁ η, as t\rar∞.

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