| Title:
|
Almost sure asymptotic behaviour of the $r$-neighbourhood surface area of Brownian paths (English) |
| Author:
|
Honzl, Ondřej |
| Author:
|
Rataj, Jan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
1 |
| Year:
|
2012 |
| Pages:
|
67-75 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $\mathbb R^d$, $d\geq 3$. (English) |
| Keyword:
|
Minkowski content |
| Keyword:
|
Kneser function |
| Keyword:
|
Brownian motion |
| Keyword:
|
Wiener sausage |
| MSC:
|
28A75 |
| MSC:
|
52A20 |
| MSC:
|
52A38 |
| MSC:
|
60D05 |
| MSC:
|
60J65 |
| idZBL:
|
Zbl 1249.28003 |
| idMR:
|
MR2899735 |
| DOI:
|
10.1007/s10587-012-0017-6 |
| . |
| Date available:
|
2012-03-05T07:11:56Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142041 |
| . |
| Reference:
|
[1] Berezhkovskii, A. M., Makhnovskii, Yu. A., Suris, R. A.: Wiener sausage volume moments.J. Stat. Phys. 57 (1989), 333-346. MR 1031416, 10.1007/BF01023647 |
| Reference:
|
[2] Gall, J.-F. Le: Fluctuation results for the Wiener sausage.Ann. Probab. 16 (1988), 991-1018. Zbl 0665.60080, MR 0942751, 10.1214/aop/1176991673 |
| Reference:
|
[3] Kneser, M.: Über den Rand von Parallelkörpern.Math. Nachr. 5 (1951), 241-251 German. Zbl 0042.40802, MR 0042729, 10.1002/mana.19510050309 |
| Reference:
|
[4] Rataj, J., Schmidt, V., Spodarev, E.: On the expected surface area of the Wiener sausage.Math. Nachr. 282 (2009), 591-603. Zbl 1166.60049, MR 2504619, 10.1002/mana.200610757 |
| Reference:
|
[5] Rataj, J., Winter, S.: On volume and surface area of parallel sets.Indiana Univ. Math. J. 59 (2010), 1661-1685. MR 2865426, 10.1512/iumj.2010.59.4165 |
| Reference:
|
[6] Spitzer, F.: Electrostatic capacity, heat-flow, and Brownian motion.Z. Wahrscheinlichkeitstheor. Verw. Geb. 3 (1964), 110-121. Zbl 0126.33505, MR 0172343, 10.1007/BF00535970 |
| Reference:
|
[7] Stachó, L. L.: On the volume function of parallel sets.Acta Sci. Math. 38 (1976), 365-374. Zbl 0342.52014, MR 0442202 |
| . |
