Reflected Brownian motion: selection, approximation and linearization
File(s)
Published version
Accepted version
Author(s)
Arnaudon, M
Li, X-M
Type
Journal Article
Abstract
We construct a family of SDEs with smooth coefficients whose solutions select a reflected Brownian flow as well as a corresponding stochastic damped transport process (Wt)(Wt), the limiting pair gives a probabilistic representation for solutions of the heat equations on differential 1-forms with the absolute boundary conditions. The transport process evolves pathwise by the Ricci curvature in the interior, by the shape operator on the boundary where it is driven by the boundary local time, and with its normal part erased at the end of the excursions to the boundary of the reflected Brownian motion. On the half line, this construction selects the Skorohod solution (and its derivative with respect to initial points), not the Tanaka solution; on the half space it agrees with the construction of N. Ikeda and S. Watanabe [29] by Poisson point processes. The construction leads also to an approximation for the boundary local time, in the topology of uniform convergence but not in the semi-martingale topology, indicating the difficulty in proving convergence of solutions of a family of random ODE’s to the solution of a stochastic equation driven by the local time and with jumps. In addition, we obtain a differentiation formula for the heat semi-group with Neumann boundary condition and prove also that (Wt)(Wt) is the weak derivative of a family of reflected Brownian motions with respect to the initial point.
Date Issued
2017-05-25
Date Acceptance
2017-02-27
Citation
Electronic Journal of Probability, 2017, 22, pp.1-55
ISSN
1083-6489
Publisher
Institute of Mathematical Statistics
Start Page
1
End Page
55
Journal / Book Title
Electronic Journal of Probability
Volume
22
Copyright Statement
Creative Commons Attribution 4.0 International License.
License URL
Identifier
http://dx.doi.org/10.1214/17-EJP41
Subjects
Science & Technology
Physical Sciences
Statistics & Probability
Mathematics
Brownian motion
reflection
local time
boundary
stochastic flow
heat equation
STOCHASTIC DIFFERENTIAL-EQUATIONS
PARTS FORMULAS
HARMONIC-FUNCTIONS
MANIFOLDS
BOUNDARY
INTEGRATION
DOMAINS
THEOREM
SDES
math.PR
0104 Statistics
Notes
mrclass: 60H10 (58J65 60J65) mrnumber: 3629875
Publication Status
Published
Article Number
31