Let \alpha ([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\ge 2, we prove lim_{ t\to\infty}t^{ -1}\log P\bigl{ \alpha([0,1]^p)\ge t^{ (d(p-1))/2}\bigr}=-\gamma_{ \alpha}(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 I_n=#{ (k_1,...,k_p)\in [1,n]^p;S_1(k_1)=... =S_p(k_p)} run by the independent, symmetric, Z^d-valued random walks S_1(n),...,S_p(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 \bigl(EI_{ n_1+... +n_a}^m\bigr)^{ 1/p}\le \sum_{ k_1+... +k_a=m\limits_{ k_1,...,k_a\ge 0}}\frac{ m!}{ k_1!... k_a!}\bigl(EI_{ n_1}^{ k_1}\bigr)^{ 1/p}... \bigl(EI_{ n_a}^{ k_a}\bigr)^{ 1/p} in the case of random walks.
Categories: math.PR