2004/10/01 by Xia Chen · 1 citation
Decision Sciences · Mathematics · #Probability and Risk Models #Random Matrices and Applications #Stochastic processes and statistical mechanics #math.PR #msc:60B12 #msc:60F10 #msc:60F15 #msc:60G50 #msc:60J55 #msc:60J65.
paper · pdf · doi:10.1214/009117904000000513
published as Annals of Probability 2004, Vol. 32, No. 4, 3248-3300 · Published at http://dx.doi.org/10.1214/009117904000000513 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2004/10/01 · arxiv created 2005/03/25 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Let α([0,1]p) denote the intersection local time of p independent d-dimensional Brownian motions running up to the time 1. Under the conditions p(d−2)<d and d≥2, we prove limt→∞t-1log ℙ\α([0,1]p)≥ t(d(p-1))/2\=-γα(d,p) with the right-hand side being identified in terms of the the best constant of the Gagliardo–Nirenberg inequality. Within the scale of moderate deviations, we also establish the precise tail asymptotics for the intersection local time In=#\(k1,…,kp)∈[1,n]p ; S1(k1)=⋯=Sp(kp)\ run by the independent, symmetric, ℤd-valued random walks S1(n), …,Sp(n). Our results apply to the law of the iterated logarithm. Our approach is based on Feynman–Kac type large deviation, time exponentiation, moment computation and some technologies along the lines of probability in Banach space. As an interesting coproduct, we obtain the inequality (𝔼I_n1+⋯ +nam)1/p≤ ∑_\mathopk1+⋯ +ka=m_k1,…,ka≥ 0\fracm!k1!⋯ ka!(𝔼I_n1^k1)1/p⋯ (𝔼I_na^ka)1/p in the case of random walks.