The Brownian fan
File(s)
Accepted version
Author(s)
Hairer, Martin
Weare, Jonathan
Type
Journal Article
Abstract
We provide a mathematical study of the modified diffusion Monte Carlo (DMC) algorithm introduced in the companion article [3]. DMC is a simulation technique that uses branching particle systems to represent expectations associated with Feynman-Kac formulae. We provide a detailed heuristic explanation of why, in cases in which a stochastic integral appears in the Feynman-Kac formula (e.g., in rare event simulation, continuous time filtering, and other settings), the new algorithm is expected to converge in a suitable sense to a limiting process as the time interval between branching steps goes to 0. The situation studied here stands in stark contrast to the “naï ve” generalization of the DMC algorithm, which would lead to an exponential explosion of the number of particles, thus precluding the existence of any finite limiting object. Convergence is shown rigorously in the simplest possible situation of a random walk, biased by a linear potential. The resulting limiting object, which we call the “Brownian fan,” is a very natural new mathematical object of independent interest. © 2014 Wiley Periodicals, Inc.
Date Issued
2014-10-06
Date Acceptance
2014-10-01
Citation
Communications on Pure and Applied Mathematics, 2014, 68 (1), pp.1-60
ISSN
0010-3640
Publisher
Wiley
Start Page
1
End Page
60
Journal / Book Title
Communications on Pure and Applied Mathematics
Volume
68
Issue
1
Copyright Statement
© 2014 Wiley Periodicals, Inc. This is the accepted version of the following article, which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/cpa.21544/abstract
Identifier
http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000344804800001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=1ba7043ffcc86c417c072aa74d649202
Subjects
Science & Technology
Physical Sciences
Mathematics, Applied
Mathematics
LARGE PLANAR MAPS
CONVERGENCE
Publication Status
Published