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February 2015 Brownian motion and thermal capacity
Davar Khoshnevisan, Yimin Xiao
Ann. Probab. 43(1): 405-434 (February 2015). DOI: 10.1214/14-AOP910


Let $W$ denote $d$-dimensional Brownian motion. We find an explicit formula for the essential supremum of Hausdorff dimension of $W(E)\cap F$, where $E\subset(0,\infty)$ and $F\subset\mathbf{R} ^{d}$ are arbitrary nonrandom compact sets. Our formula is related intimately to the thermal capacity of Watson [Proc. Lond. Math. Soc. (3) 37 (1978) 342–362]. We prove also that when $d\ge2$, our formula can be described in terms of the Hausdorff dimension of $E\times F$, where $E\times F$ is viewed as a subspace of space time.


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Davar Khoshnevisan. Yimin Xiao. "Brownian motion and thermal capacity." Ann. Probab. 43 (1) 405 - 434, February 2015.


Published: February 2015
First available in Project Euclid: 12 November 2014

zbMATH: 1305.60077
MathSciNet: MR3298476
Digital Object Identifier: 10.1214/14-AOP910

Primary: 60G17 , 60J65
Secondary: 28A78 , 28A80 , 60G15 , 60J45

Keywords: Brownian motion , Euclidean and space–time Hausdorff dimension , thermal capacity

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.43 • No. 1 • February 2015
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