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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 65 open journal systems 


Moderate deviations and laws of the iterated logarithm for the volume of the intersections of Wiener sausages

Fuqing Gao, Wuhan University
Yanqing Wang, Wuhan University


Abstract
Using the high moment method and the Feynman-Kac semigroup technique, we obtain moderate deviations and laws of the iterated logarithm for the volume of the intersections of two and three dimensional Wiener sausages.


Full text: PDF

Pages: 1900-1935

Published on: September 9, 2009


Bibliography
  1. Bass R. F., Chen. X and Rosen. J, Moderate deviations and laws of the iterated logarithm for the renormalized self-intersection local times of planar random walks. Electron. J. Probab., 11(2006), 993-1030. Math. Review 2007m:60064 MR2261059
  2. Bass, Richard F.; Chen, Xia; Rosen, Jay. Moderate deviations for the range of planar random walks. Mem. Amer. Math. Soc. 198 (2009), no. 929, viii+82 pp. ISBN: 978-0-8218-4287-4 MR2493313
  3. Bass, Richard F.; Kumagai, Takashi. Laws of the iterated logarithm for the range of random walks in two and three dimensions. Ann. Probab. 30 (2002), no. 3, 1369--1396. MR1920111 (2003d:60086)
  4. Bass, Richard F.; Rosen, Jay. An almost sure invariance principle for the range of planar random walks. Ann. Probab. 33 (2005), no. 5, 1856--1885. MR2165582 (2006h:60076)
  5. van den Berg, M. On the expected volume of intersection of independent Wiener sausages and the asymptotic behaviour of some related integrals. J. Funct. Anal. 222 (2005), no. 1, 114--128. MR2129767 (2006e:60117)
  6. van den Berg, Michiel; Bolthausen, Erwin. Asymptotics of the generating function for the volume of the Wiener sausage. Probab. Theory Related Fields 99 (1994), no. 3, 389--397. MR1283118 (95h:60121)
  7. van den Berg, M.; Bolthausen, E.; den Hollander, F. Moderate deviations for the volume of the Wiener sausage. Ann. of Math. (2) 153 (2001), no. 2, 355--406. MR1829754 (2002f:60041)
  8. van den Berg, M.; Bolthausen, E.; den Hollander, F. On the volume of the intersection of two Wiener sausages. Ann. of Math. (2) 159 (2004), no. 2, 741--782. MR2081439 (2005j:60050)
  9. van den Berg M. and Toth. B. Exponential estimates for the Wiener sausage. Probab. Theory Relat. Fields, 88 (1991), 249-259. Math. Review 92b:60075 MR1096482
  10. Bolthausen, E. On the volume of the Wiener sausage. Ann. Probab. 18 (1990), no. 4, 1576--1582. MR1071810 (92e:60151)
  11. Chen, Xia. Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks. Ann. Probab. 32 (2004), no. 4, 3248--3300. MR2094445 (2005m:60174)
  12. Chen, Xia. Moderate deviations and law of the iterated logarithm for intersections of the ranges of random walks. Ann. Probab. 33 (2005), no. 3, 1014--1059. MR2135311 (2006d:60050)
  13. Chen. X, Random walk intersections: large deviations and some related topics. Preprint, 2008. Math. Review number not available.
  14. Chen, Xia; Li, Wenbo V. Large and moderate deviations for intersection local times. Probab. Theory Related Fields 128 (2004), no. 2, 213--254. MR2031226 (2005m:60175)
  15. Csáki, E.; Hu, Y. Strong approximations of three-dimensional Wiener sausages. Acta Math. Hungar. 114 (2007), no. 3, 205--226. MR2296543 (2007k:60085)
  16. Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications.Second edition.Applications of Mathematics (New York), 38. Springer-Verlag, New York, 1998. xvi+396 pp. ISBN: 0-387-98406-2 MR1619036 (99d:60030)
  17. Donsker M. D. and S. R. S. Varadhan, Asymptotics for the Wiener sausage. Comm. Pure Appl. Math., 28 (1975), 525-565. Math. Review 53#1757a MR0397901
  18. Le Gall, J.-F. Propriétés d'intersection des marches aléatoires. I. Convergence vers le temps local d'intersection.(French) [Intersection properties of random walks. I. Convergence to local time of intersection] Comm. Math. Phys. 104 (1986), no. 3, 471--507. MR0840748 (88d:60182)
  19. Le Gall, J.-F. Propriétés d'intersection des marches aléatoires. II. Étude des cas critiques.(French) [Intersection properties of random walks. II. Critical cases] Comm. Math. Phys. 104 (1986), no. 3, 509--528. MR0840749 (88d:60183)
  20. Le Gall, Jean-François. Sur la saucisse de Wiener et les points multiples du mouvement brownien.(French) [Wiener sausages and multiple points in Brownian motion] Ann. Probab. 14 (1986), no. 4, 1219--1244. MR0866344 (88e:60097)
  21. Le Gall, Jean-François. Fluctuation results for the Wiener sausage. Ann. Probab. 16 (1988), no. 3, 991--1018. MR0942751 (90a:60080)
  22. Le Gall, J.-F. Sur une conjecture de M. Kac.(French) [On a conjecture of M. Kac] Probab. Theory Related Fields 78 (1988), no. 3, 389--402. MR0949180 (89m:60195)
  23. Le Gall, Jean-François. Some properties of planar Brownian motion. École d'Été de Probabilités de Saint-Flour XX---1990, 111--235, Lecture Notes in Math., 1527, Springer, Berlin, 1992. MR1229519 (94g:60156)
  24. Le Gall, Jean-François; Rosen, Jay. The range of stable random walks. Ann. Probab. 19 (1991), no. 2, 650--705. MR1106281 (92j:60083)
  25. Hamana, Yuji; Kesten, Harry. A large-deviation result for the range of random walk and for the Wiener sausage. Probab. Theory Related Fields 120 (2001), no. 2, 183--208. MR1841327 (2002e:60161)
  26. König, Wolfgang; Mörters, Peter. Brownian intersection local times: upper tail asymptotics and thick points. Ann. Probab. 30 (2002), no. 4, 1605--1656. MR1944002 (2003m:60230)
  27. Spitzer, Frank. Electrostatic capacity, heat flow, and Brownian motion. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 1964 110--121. MR0172343 (30 #2562)
  28. Sznitman, Alain-Sol. Long time asymptotics for the shrinking Wiener sausage. Comm. Pure Appl. Math. 43 (1990), no. 6, 809--820. MR1059329 (92e:60152)
  29. Taylor, S. J. Multiple points for the sample paths of the symmetric stable process. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 5 1966 247--264. MR0202193 (34 #2066)
















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Electronic Journal of Probability. ISSN: 1083-6489