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Brownian Motion on a Hypersurface

Brownian Motion on a Hypersurface M. VAN DEN BERG AND J. T. LEWIS §1. Introduction This work arose from a study of a note by Price and Williams [7] in which they give the following novel description of a BM (S ), a Brownian motion on the unit 2 3 sphere S in U : 3 3 3 Let B be a BM ((R ), a Brownian motion in 1R. Let X be a process on U with \X \ = 1 and dX = XxdB-Xdt, (1.1) 3 2 where x is the vector product in U ; then X is a BM(5 ) in the sense that X is a diffusion and, for any g in C u(x, t) = E[g{X )l X = x, (1.2) t 0 solves the initial value problem du, = {An , t > 0,) (1-3) u(x,O) = 0(x), j where A is the Laplace-Beltrami operator on S . It is surprising that this elegant description of BM(S ) was not pointed out before 1981. Ito [4] discusses the equation dX = (l-XX^dB-Xdt, (1.4) which he attributes to Stroock [8]. But (1.4) is quadratic in X whereas (1.1) is linear; it follows that in many instances there is http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Brownian Motion on a Hypersurface

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/17.2.144
Publisher site
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Abstract

M. VAN DEN BERG AND J. T. LEWIS §1. Introduction This work arose from a study of a note by Price and Williams [7] in which they give the following novel description of a BM (S ), a Brownian motion on the unit 2 3 sphere S in U : 3 3 3 Let B be a BM ((R ), a Brownian motion in 1R. Let X be a process on U with \X \ = 1 and dX = XxdB-Xdt, (1.1) 3 2 where x is the vector product in U ; then X is a BM(5 ) in the sense that X is a diffusion and, for any g in C u(x, t) = E[g{X )l X = x, (1.2) t 0 solves the initial value problem du, = {An , t > 0,) (1-3) u(x,O) = 0(x), j where A is the Laplace-Beltrami operator on S . It is surprising that this elegant description of BM(S ) was not pointed out before 1981. Ito [4] discusses the equation dX = (l-XX^dB-Xdt, (1.4) which he attributes to Stroock [8]. But (1.4) is quadratic in X whereas (1.1) is linear; it follows that in many instances there is

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Mar 1, 1985

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