Geodesic
Almost sure asymptotic behaviour of the $r$-neighbourhood surface area of Brownian paths
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 1, pp. 67-75
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We show that whenever the $q$-dimensional Minkowski content of a subset $A\subset \mathbb R^d$ 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 $\mathbb R^d$, $d\geq 3$.
We show that whenever the $q$-dimensional Minkowski content of a subset $A\subset \mathbb R^d$ 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 $\mathbb R^d$, $d\geq 3$.
DOI : 10.1007/s10587-012-0017-6
Classification : 28A75, 52A20, 52A38, 60D05, 60J65
Keywords: Minkowski content; Kneser function; Brownian motion; Wiener sausage 
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Honzl, Ondřej; Rataj, Jan. Almost sure asymptotic behaviour of the $r$-neighbourhood surface area of Brownian paths. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 1, pp. 67-75. doi: 10.1007/s10587-012-0017-6

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