Brownian Motion in a Truncated Weyl Chamber

#### W. Konig, P. Schmid

2011, v.17, Issue 4, 499-522

ABSTRACT

We examine the non-exit probability of a multidimensional Brownian motion from a growing truncated Weyl chamber. More precisely, we compute, for fixed time $t$, the probability that the motion does not leave by time $t$ the intersection of a Weyl chamber and a $t$-dependent centred box, and we identify its asymptotics for $t\to\infty$. Different regimes are identified according to the growth speed, ranging from polynomial decay over stretched exponential (that is, exponential of a power function, here with exponent in $(0,1)$) to exponential decay. Furthermore we derive associated large deviation principles for the empirical measure of the properly rescaled and transformed Brownian motion as the dimension grows to infinity. Our main tool is an explicit eigenvalue expansion for the transition probabilities before exiting the truncated Weyl chamber.

Keywords: Weyl chamber,non-colliding Brownian motions,Karlin - McGregorformula,non-colliding probability,non-exit probability,eigenvalue expansion,reduite