Statistics of first-passage Brownian functionals

Satya N. Majumdar, Baruch Meerson

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18 Scopus citations


We study the distribution of first-passage functionals of the type where represents a Brownian motion (with or without drift) with diffusion 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. We show that has an essential singular tail as and a power-law tail as. 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 it predicts the same essential singular tail as in the driftless case. For it predicts a stretched exponential tail for all n > 0. In the limit of large Péclet number, where is the drift velocity toward the origin, the OFM predicts an exact large-deviation scaling behavior, valid for all A: Where is the mean value of in this limit. We compute the rate function analytically for all n >-1. We show that, while for n > 0 the rate function 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 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. Finally, we employ the OFM to study the case of (drift away from the origin). We show that, when the process is conditioned on reaching the origin, the distribution of coincides with the distribution of for with the same.

Original languageAmerican English
Article number023202
JournalJournal of Statistical Mechanics: Theory and Experiment
Issue number2
StatePublished - 1 Feb 2020

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© 2020 IOP Publishing Ltd and SISSA Medialab srl.


  • Brownian motion
  • classical phase transitions
  • fluctuation phenomena
  • large deviations in non-equilibrium systems


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