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Almost sure asymptotic behaviour of the $r$ -neighbourhood surface area of Brownian paths

Ondřej Honzl; Jan Rataj — 2012

Czechoslovak Mathematical Journal

We show that whenever the $q$ -dimensional Minkowski content of a subset $A \subset ℝ^{d}$ exists and is finite and positive, then the “S-content” defined analogously as the Minkowski content, but with volume replaced by surface area, exists as well and equals the Minkowski content. As a corollary, we obtain the almost sure asymptotic behaviour of the surface area of the Wiener sausage in $ℝ^{d}$ , $d \geq 3$ .

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Almost sure asymptotic behaviour of the r -neighbourhood surface area of Brownian paths

Almost sure asymptotic behaviour of the $r$ -neighbourhood surface area of Brownian paths