Limiting Behaviors for Brownian Motion Reflected on Brownian Motion

Abstract

Suppose that g(t) and W_t are independent Brownian motions starting from g(0) = W₀ = 0. Consider the Brownian motion Y_t reflected on g(t), obtained from W_t by the means of the Skorohod lemma. The upper and lower limiting behaviors of Y_t are presented. The upper tail estimate on exit time is computed via principal eigenvalue.