TY - JOUR

T1 - Brownian trajectory is a regular lateral boundary for the heat equation

AU - Krylov, Nicolai V

PY - 2003/11/20

Y1 - 2003/11/20

N2 - 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.

AB - 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.

KW - Fine properties of Brownian motion

KW - Heat equation in irregular cylinders

KW - Stochastic partial differential equations

UR - http://www.scopus.com/inward/record.url?scp=0242596346&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0242596346&partnerID=8YFLogxK

U2 - 10.1137/S0036141002402980

DO - 10.1137/S0036141002402980

M3 - Article

AN - SCOPUS:0242596346

VL - 34

SP - 1167

EP - 1182

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 5

ER -