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August  2021, 14(8): 2751-2803. doi: 10.3934/dcdss.2021039

Diffusion-approximation for a kinetic spray-like system with random forcing

Arnaud Debussche 1,, , Angelo Rosello 2, and  Julien Vovelle 3,

1. 

Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France, Institut Universitaire de France (IUF)
2. 

Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
3. 

Université de Lyon, CNRS, ENS de Lyon, UMR5669, Unité de Mathématiques Pures et Appliquées, 46, allée d'Italie, 69364 Lyon Cedex 07, France

* Corresponding author: Arnaud Debussche

Received  February 2020 Revised  September 2020 Published  August 2021 Early access  April 2021

Fund Project: A. Debussche and A. Rosello are partially supported by the French government thanks to the "Investissements d'Avenir" program ANR-11-LABX-0020-01. This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX- 0007) operated by the French National Research Agency (ANR). A. Debussche and J. Vovelle are partially supported by the ANR project ANR-19-CE40-0019

We study a kinetic toy model for a spray of particles immersed in an ambient fluid, subject to some additional random forcing given by a mixing, space-dependent Markov process. Using the perturbed test function method, we derive the hydrodynamic limit of the kinetic system. The law of the limiting density satisfies a stochastic conservation equation in Stratonovich form, whose drift and diffusion coefficients are completely determined by the law of the stationary process associated with the Markovian perturbation.

Keywords: Kinetic equations, stochastic partial differential equations, diffusion-approximation, hydrodynamic limit.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Arnaud Debussche, Angelo Rosello, Julien Vovelle. Diffusion-approximation for a kinetic spray-like system with random forcing. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2751-2803. doi: 10.3934/dcdss.2021039
References:
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J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc., 63 (1977), 370-373.  doi: 10.2307/2041821.

[2]

P. Billingsley, Convergence of Probability Measures, 2nd edition, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons Inc., New York, 1999. doi: 10.1002/9780470316962.

[3]

D. L. Burkholder, B. J. Davis and R. F. Gundy, Integral inequalities for convex functions of operators on martingales, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Probability Theory, University of California Press, Berkeley, Calif., 1972,223-240.

[4]

N. Caillerie and J. Vovelle, Diffusion-approximation of a kinetic equation with stochastically perturbed velocity redistribution process., preprint, arXiv: 1712.10173v2.

[5] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.
[6]

A. Debussche, S. De Moor and J. Vovelle, Diffusion limit for the radiative transfer equation perturbed by a Markovian process, Asymptotic Analysis, 98 (2016), 31-58.  doi: 10.3233/ASY-161360.

[7]

A. Debussche and J. Vovelle, Diffusion limit for a stochastic kinetic problem, Communications on Pure and Applied Mathematics, 11 (2012), 2305-2326.  doi: 10.3934/cpaa.2012.11.2305.

[8]

A. Debussche and J. Vovelle, Diffusion-approximation in stochastically forced kinetic equations, Tunis. J. Math., 3 (2021), 1-53.  doi: 10.2140/tunis.2021.3.1.

[9]

K. Du and Q. Meng, A revisit to $W^2_n$-theory of super-parabolic backward stochastic partial differential equations in $\mathbb{R}^d$, Stochastic Processes and their Applications, 120 (2010), 1996-2015.  doi: 10.1016/j.spa.2010.06.001.

[10]

J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, Wave Propagation and Time Reversal in Randomly Layered Media, vol. 56 of Stochastic Modelling and Applied Probability, 1st edition, Springer-Verlag New York, 2007.

[11]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, vol. 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edition, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-05265-5.

[12]

G. Papanicolaou, D. Stroock and S. Varadhan, Martingale approach to some limit theorems, Duke University Series, 3.

[13]

S. Roch, Modern discrete probability, an essential toolkit, Available from: https://www.math.wisc.edu/ roch/mdp/roch-mdp-chap3.pdf, 2015.

show all references

References:
[1]

J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc., 63 (1977), 370-373.  doi: 10.2307/2041821.

[2]

P. Billingsley, Convergence of Probability Measures, 2nd edition, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons Inc., New York, 1999. doi: 10.1002/9780470316962.

[3]

D. L. Burkholder, B. J. Davis and R. F. Gundy, Integral inequalities for convex functions of operators on martingales, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Probability Theory, University of California Press, Berkeley, Calif., 1972,223-240.

[4]

N. Caillerie and J. Vovelle, Diffusion-approximation of a kinetic equation with stochastically perturbed velocity redistribution process., preprint, arXiv: 1712.10173v2.

[5] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.
[6]

A. Debussche, S. De Moor and J. Vovelle, Diffusion limit for the radiative transfer equation perturbed by a Markovian process, Asymptotic Analysis, 98 (2016), 31-58.  doi: 10.3233/ASY-161360.

[7]

A. Debussche and J. Vovelle, Diffusion limit for a stochastic kinetic problem, Communications on Pure and Applied Mathematics, 11 (2012), 2305-2326.  doi: 10.3934/cpaa.2012.11.2305.

[8]

A. Debussche and J. Vovelle, Diffusion-approximation in stochastically forced kinetic equations, Tunis. J. Math., 3 (2021), 1-53.  doi: 10.2140/tunis.2021.3.1.

[9]

K. Du and Q. Meng, A revisit to $W^2_n$-theory of super-parabolic backward stochastic partial differential equations in $\mathbb{R}^d$, Stochastic Processes and their Applications, 120 (2010), 1996-2015.  doi: 10.1016/j.spa.2010.06.001.

[10]

J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, Wave Propagation and Time Reversal in Randomly Layered Media, vol. 56 of Stochastic Modelling and Applied Probability, 1st edition, Springer-Verlag New York, 2007.

[11]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, vol. 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edition, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-05265-5.

[12]

G. Papanicolaou, D. Stroock and S. Varadhan, Martingale approach to some limit theorems, Duke University Series, 3.

[13]

S. Roch, Modern discrete probability, an essential toolkit, Available from: https://www.math.wisc.edu/ roch/mdp/roch-mdp-chap3.pdf, 2015.

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