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

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

2007-03-14

arxiv.org/abs/math/0703432v1

Mathematics

Probability

Scientific paper

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. A particular feature is that the diffusion matrix of this process is a linear function the law of the process, and not a quadratic one, as in the McKean-Vlasov model. 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 striking problem then is to establish the joint measurability of this optimal transport map with respect to the space variable and the parameters (time and randomness) making the marginals vary. We shall prove a general measurability result for the mass transportation problem in terms of the support of the transfert plans, in the sense of set-valued mappings. This will allow us to construct the coupling and to obtain explicit convergence rates.

Affiliated with

Also associated with

No associations

Rating

Measurability of optimal transportation and convergence rate for Landau type interacting particle systems does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Measurability of optimal transportation and convergence rate for Landau type interacting particle systems, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Measurability of optimal transportation and convergence rate for Landau type interacting particle systems will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-715886