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