On singularity as a function of time of a conditional distribution of an exit time

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Abstract

We establish the singularity with respect to Lebesgue measure as a function of time of the conditional probability distribution that the sum of two one-dimensional Brownian motions will exit from the unit interval before time t, given the trajectory of the second Brownian motion up to the same time. On the way of doing so we show that if one solves the one-dimensional heat equation with zero condition on a trajectory of a one-dimensional Brownian motion, which is the lateral boundary, then for each moment of time with probability one the normal derivative of the solution is zero, provided that the diffusion of the Brownian motion is sufficiently large.

Original languageEnglish (US)
Pages (from-to)541-557
Number of pages17
JournalProbability Theory and Related Fields
Volume165
Issue number3-4
DOIs
StatePublished - Aug 1 2016

Keywords

  • Filtering of partially observable diffusion processes
  • Heat equation in domains with irregular lateral boundaries
  • Stochastic partial differential equations

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