NDLI: Measurability of optimal transportation and convergence rate for Landau type interacting particle systems

Content Provider	Springer Nature Link
Author	Guérin, Hélène Méléard, Sylvie Fontbona, Joaquin
Copyright Year	2007
Abstract	In this paper, we consider nonlinear diffusion processes driven by space-time white noises, which have an interpretation in terms of partial differential equations. For a specific choice of coefficients, they correspond to the Landau equation arising in kinetic theory. The main goal of the paper is to construct an easily simulable diffusive interacting particle system, converging towards this nonlinear process and to obtain an explicit pathwise rate. This requires to find a significant coupling between finitely many Brownian motions and the infinite dimensional white noise process. The key idea will be to construct the right Brownian motions by pushing forward the white noise processes, through the Brenier map realizing the optimal transport between the law of the nonlinear process, and the empirical measure of independent copies of it. A crucial problem will then be to establish the joint measurability of this optimal transport map, with respect to the space variable and the parameter (time-randomness) that makes the marginals vary. To overcome this point, we shall prove a general measurability result for the mass transportation problem and for the supports of the optimal transfer plans, in the sense of set-valued mappings. This will allow us to construct the coupling and to obtain explicit convergence rates.
Ending Page	351
Page Count	23
Starting Page	329
File Format	PDF
ISSN	01788051
e-ISSN	14322064
Journal	Probability Theory and Related Fields
Issue Number	3
Volume Number	143
Language	English
Publisher	Springer-Verlag
Publisher Date	2008-01-04
Publisher Place	Berlin/Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Nonlinear white noise driven SDE Pathwise coupling Predictable transport process Variational problems in a geometric measure-theoretic setting General theory of processes Landau type interacting particle systems Kinetic theory of gases Mathematical and Computational Physics Interacting random processes; statistical mechanics type models; percolation theory Probability Theory and Stochastic Processes Quantitative Finance Numerical methods (Monte Carlo, series resummation, etc.) Statistics for Business/Economics/Mathematical Finance/Insurance Operations Research/Decision Theory Mathematical Biology in General Measurability of optimal transport
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Analysis Statistics, Probability and Uncertainty

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