Suppose that g(t) and Wt are independent Brownian motions starting from g(0) = W0 = 0. Consider the Brownian motion Yt reflected on g(t), obtained from Wt by the means of the Skorohod lemma. The upper and lower limiting behaviors of Yt are presented. The upper tail estimate on exit time is computed via principal eigenvalue.
"Limiting Behaviors for Brownian Motion Reflected on Brownian Motion." Methods Appl. Anal. 9 (3) 377 - 392, September 2002.