Statistics of first-passage Brownian functionals

We study the distribution of first-passage functionals of the type A = integral(tf)(0) x(n)(t) dt where x(t) represents a Brownian motion ( with or without drift) with di.usion constant D, starting at x(0) > 0, and tf is the first-passage time to the origin. In the driftless case, we compute exactly, for all n > -2, the probability density P-n(A vertical bar x(0)) = Prob.(A = A). We show that P-n(A vertical bar x(0)) has an essential singular tail as A -> 0 and a power-law tail similar to A(-(n+3)/(n+2)) as A -> infinity. The leading essential singular behavior for small A can be obtained using the optimal fluctuation method (OFM), which also predicts the optimal paths of the conditioned process in this limit. For the case with a drift toward the origin, where no exact solution is known for general n > -1, we show that the OFM successfully predicts the tails of the distribution. For A -> 0 it predicts the same essential singular tail as in the driftless case. For A -> infinity it predicts a stretched exponential tail - ln P-n(A vertical bar x(0)) similar to A(1/(n+1)) for all n > 0. In the limit of large Peclet number Pe = mu x(0)/(2D), where mu is the drift velocity toward the origin, the OFM predicts an exact large-deviation scaling behavior, valid for all A: - ln P-n(A vertical bar x(0)) similar or equal to Pe Phi(n) (z = A/(A) over bar, where (A) over bar = x(0)(n+1)/mu(n + 1) is the mean value of A in this limit. We compute the rate function Phi(n)(z) analytically for all n > -1. We show that, while for n > 0 the rate function Phi(n)(z) is analytic for all z, it has a non-analytic behavior at z = 1 for -1 < n < 0 which can be interpreted as a dynamical phase transition. The order of this transition is 2 for -1/2 < n < 0, while for -1 < n < -1/2 the order of transition is 1/(n + 1); it changes continuously with n. We also provide an illuminating alternative derivation of the OFM result by using a WKB-type asymptotic perturbation theory for large Pe. Finally, we employ the OFM to study the case of mu < 0 (drift away from the origin). We show that, when the process is conditioned on reaching the origin, the distribution of A coincides with the distribution of A for mu > 0 with the same vertical bar mu vertical bar.