Brownian trajectory is a regular lateral boundary for the heat equation

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Abstract

The one-dimensional heat equation in the domain x > wt, t > 0, is considered. Here wt is a trajectory of Brownian motion. For almost any trajectory, it is proved that if the boundary data are continuous, then the solution is continuous in the closure of the domain. The proof is based on Davis's law of square root for Brownian motion or on its weaker version, which is obtained by using the theory of stochastic partial differential equations.

Original languageEnglish (US)
Pages (from-to)1167-1182
Number of pages16
JournalSIAM Journal on Mathematical Analysis
Volume34
Issue number5
DOIs
StatePublished - Nov 20 2003

Keywords

  • Fine properties of Brownian motion
  • Heat equation in irregular cylinders
  • Stochastic partial differential equations

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