Laws of the iterated logarithm for the Brownian snake

Abstract

We consider the path-valued process (W _s , ζ _s ) called the Brownian snake, with lifetime process (ζ _s ) a reflected Brownian motion. We first give an estimate of the probability that this process exits a “big” ball. Then we show the following laws of the iterated logarithm for the euclidean norm of the “terminal point” of the Brownian snake:

$$\mathop {\lim \sup }\limits_{s \uparrow + \infty } \frac{{\left| {W_s (\zeta _s )} \right|}}{{s^{1/4} (\log \log s)^{3/4} }} = c,\mathop {\lim \sup }\limits_{s \downarrow 0} \frac{{\left| {W_s (\zeta _s )} \right|}}{{s^{1/4} (\log \log (1/s))^{3/4} }} = c$$

where c = 2.3^−3/4.