Corrigendum: Statistics of first-passage Brownian functionals (2020 J. Stat. Mech. 023202)
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Abstract
. We study the distribution of first-passage functionals of the type x n ( t ) d t where x ( t ) represents a Brownian motion ( with or without drift ) with diusion constant D , starting at x 0 > 0, and t f 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 | x 0 ) = Prob.( A = A ) . We show that P n ( A | x 0 ) has an essential singular tail as A → 0 and a power-law tail ∼ A − ( n +3) / ( n +2) as A → ∞ . 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 → ∞ it predicts a stretched exponential tail − ln P n ( A | x 0 ) ∼ A 1 / ( n +1) for all n > 0. In the , We analytically We while for n > 0 the rate function Φ n ( ) is analytic all , 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 while for 1 < 1/2 the order of transition is it We also provide an of theory for large Pe . Finally, we employ the OFM to study the case of µ < 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 µ > 0 with the same | µ | .