Numerical bounds for critical exponents of crossing Brownian motion (Q2750908)

Consider \(d\)-dimensional crossing Brownian motion in a truncated Poissonian potential conditional to reach a fixed hyperplane at distance \(L\) from the starting point. The transverse fluctuation of the path is expected to be of order \(L^\xi\). It is proved that \(\xi\leq 3/4\) for \(d\geq 2\). The author also studies a second critical exponent \(\chi^{(2)}\), which describes the fluctuations of naturally defined distance functions for crossing Brownian motion. An improvement of the author's earlier result is presented: \(\chi^{(2)}\geq 1/5\) if \(d=2\) and if the killing rate \(\lambda\) is strictly positive. It is noticeable that both upper and lower bounds for the same definition of the exponents are obtained.