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 & Applied Analysis, 2012, 11 (6) : 2305-2326. doi: 10.3934/cpaa.2012.11.2305
References:
[1]

P. Billingsley, "Convergence of Probability Measures,", second ed., (1999). Google Scholar

[2]

T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations,", Oxford Lecture Series in Mathematics and its Applications, (1998). Google Scholar

[3]

A. de Bouard and A. Debussche, The nonlinear Schrödinger equation with white noise dispersion,, J. Funct. Anal., 259 (2010), 1300. doi: 10.1016/j.jfa.2010.04.002. Google Scholar

[4]

A. de Bouard and M. Gazeau, A diffusion approximation theorem for a nonlinear pde with application to random birefringent optical fibers,, preprint, (). Google Scholar

[5]

P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversible processes,, Indiana Univ. Math. J., 49 (2000), 1175. Google Scholar

[6]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions,", Encyclopedia of Mathematics and its Applications, (1992). Google Scholar

[7]

A. Debussche and Y. Tsutsumi, 1d quintic nonlinear schrodinger equation with white noise dispersion,, Journal de Math. Pures et Appl., (2010). Google Scholar

[8]

S. N. Ethier and T. G. Kurtz, "Markov Processes,", Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, (1986). Google Scholar

[9]

L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE,, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359. doi: 10.1017/S0308210500018631. Google Scholar

[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, (2007). Google Scholar

[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. Google Scholar

[12]

R. Z. Has'minskiĭ, Stochastic processes defined by differential equations with a small parameter,, Teor. Verojatnost. i Primenen, 11 (1966), 240. Google Scholar

[13]

J. Jacod and A. N. Shiryaev, "Limit Theorems for Stochastic Processes,", second ed., (2003). Google Scholar

[14]

H. J. Kushner, Approximation and weak convergence methods for random processes, with applications to stochastic systems theory,, MIT Press Series in Signal Processing, (1984). Google Scholar

[15]

R. Marty, On a splitting scheme for the nonlinear Schrödinger equation in a random medium,, Commun. Math. Sci., 4 (2006), 679. Google Scholar

[16]

E. Pardoux and A. L. Piatnitski, Homogenization of a nonlinear random parabolic partial differential equation,, Stochastic Process. Appl., 104 (2003), 1. doi: 10.1016/S0304-4149(02)00221-1. Google Scholar

[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., (1976). Google Scholar

show all references

References:
[1]

P. Billingsley, "Convergence of Probability Measures,", second ed., (1999). Google Scholar

[2]

T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations,", Oxford Lecture Series in Mathematics and its Applications, (1998). Google Scholar

[3]

A. de Bouard and A. Debussche, The nonlinear Schrödinger equation with white noise dispersion,, J. Funct. Anal., 259 (2010), 1300. doi: 10.1016/j.jfa.2010.04.002. Google Scholar

[4]

A. de Bouard and M. Gazeau, A diffusion approximation theorem for a nonlinear pde with application to random birefringent optical fibers,, preprint, (). Google Scholar

[5]

P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversible processes,, Indiana Univ. Math. J., 49 (2000), 1175. Google Scholar

[6]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions,", Encyclopedia of Mathematics and its Applications, (1992). Google Scholar

[7]

A. Debussche and Y. Tsutsumi, 1d quintic nonlinear schrodinger equation with white noise dispersion,, Journal de Math. Pures et Appl., (2010). Google Scholar

[8]

S. N. Ethier and T. G. Kurtz, "Markov Processes,", Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, (1986). Google Scholar

[9]

L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE,, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359. doi: 10.1017/S0308210500018631. Google Scholar

[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, (2007). Google Scholar

[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. Google Scholar

[12]

R. Z. Has'minskiĭ, Stochastic processes defined by differential equations with a small parameter,, Teor. Verojatnost. i Primenen, 11 (1966), 240. Google Scholar

[13]

J. Jacod and A. N. Shiryaev, "Limit Theorems for Stochastic Processes,", second ed., (2003). Google Scholar

[14]

H. J. Kushner, Approximation and weak convergence methods for random processes, with applications to stochastic systems theory,, MIT Press Series in Signal Processing, (1984). Google Scholar

[15]

R. Marty, On a splitting scheme for the nonlinear Schrödinger equation in a random medium,, Commun. Math. Sci., 4 (2006), 679. Google Scholar

[16]

E. Pardoux and A. L. Piatnitski, Homogenization of a nonlinear random parabolic partial differential equation,, Stochastic Process. Appl., 104 (2003), 1. doi: 10.1016/S0304-4149(02)00221-1. Google Scholar

[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., (1976). Google Scholar

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