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.
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The first author was supported by Fondecyt Projects 1040689 and 1070743, ECOS-Conicyt C05E02, Millennium Nucleus Information and Randomness ICM P04-069-F and FONDAP Applied Mathematics.
The second and third authors were supported by ECOS-Conicyt C05E02 and Millennium Nucleus Information and Randomness ICM P04-069-F.
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Fontbona, J., Guérin, H. & Méléard, S. Measurability of optimal transportation and convergence rate for Landau type interacting particle systems. Probab. Theory Relat. Fields 143, 329–351 (2009). https://doi.org/10.1007/s00440-007-0128-4
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DOI: https://doi.org/10.1007/s00440-007-0128-4
Keywords
- Landau type interacting particle systems
- Nonlinear white noise driven SDE
- Pathwise coupling
- Measurability of optimal transport
- Predictable transport process
Mathematics Subject Classification (2000)
- 60K35
- 49Q20
- 82C40
- 82C80
- 60G07