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.

show all references

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.

[1]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[2]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515

[3]

Naoufel Ben Abdallah, Antoine Mellet, Marjolaine Puel. Fractional diffusion limit for collisional kinetic equations: A Hilbert expansion approach. Kinetic and Related Models, 2011, 4 (4) : 873-900. doi: 10.3934/krm.2011.4.873

[4]

Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control and Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231

[5]

Min Yang, Guanggan Chen. Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1565-1581. doi: 10.3934/dcdsb.2019240

[6]

Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021020

[7]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295

[8]

Sergio Albeverio, Sonia Mazzucchi. Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena. Journal of Geometric Mechanics, 2019, 11 (2) : 123-137. doi: 10.3934/jgm.2019006

[9]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209

[10]

Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203

[11]

Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks and Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014

[12]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221

[13]

Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061

[14]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control and Related Fields, 2021, 11 (4) : 797-828. doi: 10.3934/mcrf.2020047

[15]

Emmanuel N. Barron, Rafal Goebel, Robert R. Jensen. The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1693-1706. doi: 10.3934/dcdsb.2012.17.1693

[16]

Dariusz Idczak. A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional order. Mathematical Control and Related Fields, 2022, 12 (1) : 225-243. doi: 10.3934/mcrf.2021019

[17]

Ishak Alia. Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach. Mathematical Control and Related Fields, 2020, 10 (4) : 785-826. doi: 10.3934/mcrf.2020020

[18]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[19]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473

[20]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (84)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]