Brownian motion and thermal capacity

Details Brownian motion and thermal capacity Brownian motion and thermal capacity

2011-04-19

arxiv.org/abs/1104.3768v1

Mathematics

Probability

26 pages

Scientific paper

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 $F\subset\R^d$ are arbitrary nonrandom compact sets. Our formula is related intimately to the thermal capacity of Watson (1978). We prove also that when $d\ge 2$, 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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