# American Institute of Mathematical Sciences

November  2012, 11(6): 2305-2326. doi: 10.3934/cpaa.2012.11.2305

## Diffusion limit for a stochastic kinetic problem

 1 IRMAR and ENS de Cachan, antenne de Bretagne, Campus de Ker Lann, 35170 BRUZ Cedex 2 Université de Lyon, CNRS UMR 5208 & Université Lyon 1, Institut Camille Jordan, 43 bd du 11 novembre 1918, F-69622 Villeurbanne cedex

Received  February 2011 Revised  July 2011 Published  April 2011

We study the limit of a kinetic evolution equation involving a small parameter and perturbed by a smooth random term which also involves the small parameter. Generalizing the classical method of perturbed test functions, we show the convergence to the solution of a stochastic diffusion equation.
Citation: Arnaud Debussche, Julien Vovelle. Diffusion limit for a stochastic kinetic problem. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2305-2326. doi: 10.3934/cpaa.2012.11.2305
##### References:
 [1] P. Billingsley, "Convergence of Probability Measures," second ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons Inc., New York, 1999, A Wiley-Interscience Publication. [2] T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations," Oxford Lecture Series in Mathematics and its Applications, vol. 13, The Clarendon Press Oxford University Press, New York, 1998. [3] A. de Bouard and A. Debussche, The nonlinear Schrödinger equation with white noise dispersion, J. Funct. Anal., 259 (2010), 1300-1321. doi: 10.1016/j.jfa.2010.04.002. [4] A. de Bouard and M. Gazeau, A diffusion approximation theorem for a nonlinear pde with application to random birefringent optical fibers, preprint, arXiv:1105.4048. [5] P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversible processes, Indiana Univ. Math. J., 49 (2000), 1175-1198. [6] G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. [7] A. Debussche and Y. Tsutsumi, 1d quintic nonlinear schrodinger equation with white noise dispersion, Journal de Math. Pures et Appl., (2010), To appear. arXiv:1010.4011. [8] S. N. Ethier and T. G. Kurtz, "Markov Processes," Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1986, Characterization and convergence. [9] L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375. doi: 10.1017/S0308210500018631. [10] J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, "Wave Propagation and Time Reversal in Randomly Layered Media," Stochastic Modelling and Applied Probability, vol. 56, Springer, New York, 2007. [11] R. Z. Has'minskiĭ, A limit theorem for solutions of differential equations with a random right hand part, Teor. Verojatnost. i Primenen, 11 (1966), 444-462. [12] R. Z. Has'minskiĭ, Stochastic processes defined by differential equations with a small parameter, Teor. Verojatnost. i Primenen, 11 (1966), 240-259. [13] J. Jacod and A. N. Shiryaev, "Limit Theorems for Stochastic Processes," second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, Springer-Verlag, Berlin, 2003. [14] H. J. Kushner, Approximation and weak convergence methods for random processes, with applications to stochastic systems theory, MIT Press Series in Signal Processing, Optimization, and Control, 6, MIT Press, Cambridge, MA, 1984. [15] R. Marty, On a splitting scheme for the nonlinear Schrödinger equation in a random medium, Commun. Math. Sci., 4 (2006), 679-705. [16] E. Pardoux and A. L. Piatnitski, Homogenization of a nonlinear random parabolic partial differential equation, Stochastic Process. Appl., 104 (2003), 1-27. doi: 10.1016/S0304-4149(02)00221-1. [17] G. C. Papanicolaou, D. Stroock and S. R. S. Varadhan, Martingale approach to some limit theorems, Papers from the Duke Turbulence Conference (Duke Univ., Durham, N.C., 1976), Paper No. 6, Duke Univ., Durham, N.C., 1977, pp. ii+120 pp. Duke Univ. Math. Ser., Vol. III.

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##### References:
 [1] P. Billingsley, "Convergence of Probability Measures," second ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons Inc., New York, 1999, A Wiley-Interscience Publication. [2] T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations," Oxford Lecture Series in Mathematics and its Applications, vol. 13, The Clarendon Press Oxford University Press, New York, 1998. [3] A. de Bouard and A. Debussche, The nonlinear Schrödinger equation with white noise dispersion, J. Funct. Anal., 259 (2010), 1300-1321. doi: 10.1016/j.jfa.2010.04.002. [4] A. de Bouard and M. Gazeau, A diffusion approximation theorem for a nonlinear pde with application to random birefringent optical fibers, preprint, arXiv:1105.4048. [5] P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversible processes, Indiana Univ. Math. J., 49 (2000), 1175-1198. [6] G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. [7] A. Debussche and Y. Tsutsumi, 1d quintic nonlinear schrodinger equation with white noise dispersion, Journal de Math. Pures et Appl., (2010), To appear. arXiv:1010.4011. [8] S. N. Ethier and T. G. Kurtz, "Markov Processes," Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1986, Characterization and convergence. [9] L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375. doi: 10.1017/S0308210500018631. [10] J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, "Wave Propagation and Time Reversal in Randomly Layered Media," Stochastic Modelling and Applied Probability, vol. 56, Springer, New York, 2007. [11] R. Z. Has'minskiĭ, A limit theorem for solutions of differential equations with a random right hand part, Teor. Verojatnost. i Primenen, 11 (1966), 444-462. [12] R. Z. Has'minskiĭ, Stochastic processes defined by differential equations with a small parameter, Teor. Verojatnost. i Primenen, 11 (1966), 240-259. [13] J. Jacod and A. N. Shiryaev, "Limit Theorems for Stochastic Processes," second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, Springer-Verlag, Berlin, 2003. [14] H. J. Kushner, Approximation and weak convergence methods for random processes, with applications to stochastic systems theory, MIT Press Series in Signal Processing, Optimization, and Control, 6, MIT Press, Cambridge, MA, 1984. [15] R. Marty, On a splitting scheme for the nonlinear Schrödinger equation in a random medium, Commun. Math. Sci., 4 (2006), 679-705. [16] E. Pardoux and A. L. Piatnitski, Homogenization of a nonlinear random parabolic partial differential equation, Stochastic Process. Appl., 104 (2003), 1-27. doi: 10.1016/S0304-4149(02)00221-1. [17] G. C. Papanicolaou, D. Stroock and S. R. S. Varadhan, Martingale approach to some limit theorems, Papers from the Duke Turbulence Conference (Duke Univ., Durham, N.C., 1976), Paper No. 6, Duke Univ., Durham, N.C., 1977, pp. ii+120 pp. Duke Univ. Math. Ser., Vol. III.
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