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January  2017, 37(1): 33-76. doi: 10.3934/dcds.2017003

On the Smoluchowski-Kramers approximation for SPDEs and its interplay with large deviations and long time behavior

1. 

University of Maryland, College Park, United States

2. 

Boston University, Boston, United States

* Corresponding author

* Partially supported by the NSF grant DMS 1407615

Received  February 2016 Revised  September 2016 Published  November 2016

Fund Project: Partially supported by the NSF grant DMS 1407615

We discuss here the validity of the small mass limit (the so-called Smoluchowski-Kramers approximation) on a fixed time interval for a class of semi-linear stochastic wave equations, both in the case of the presence of a constant friction term and in the case of the presence of a constant magnetic field. We also consider the small mass limit in an infinite time interval and we see how the approximation is stable in terms of the invariant measure and of the large deviation estimates and the exit problem from a bounded domain of the space of square integrable functions.

Citation: Sandra Cerrai, Mark Freidlin, Michael Salins. On the Smoluchowski-Kramers approximation for SPDEs and its interplay with large deviations and long time behavior. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 33-76. doi: 10.3934/dcds.2017003
References:
[1]

R. Carmona and D. Nualart, Random non-linear wave equation: Smoothness of solutions, Probability Theory and Related Fields, 79 (1988), 469-508. doi: 10.1007/BF00318783. Google Scholar

[2]

S. Cerrai, Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach Lecture Notes in Mathematics 1762, Springer Verlag, 2001. doi: 10.1007/b80743. Google Scholar

[3]

S. Cerrai and M. Freidlin, On the Smoluchowski-Kramers approximation for a system with an infinite number of degrees of freedom, Probability Theory and Related Fields, 135 (2006), 363-394. doi: 10.1007/s00440-005-0465-0. Google Scholar

[4]

S. Cerrai and M. Freidlin, Smoluchowski-Kramers approximation for a general class of SPDE's, Journal of Evolution Equations, 6 (2006), 657-689. doi: 10.1007/s00028-006-0281-8. Google Scholar

[5]

S. Cerrai and M. Freidlin, Approximation of quasi-potentials and exit problems for multidimensional RDE's with noise, Transactions of the AMS, 363 (2011), 3853-3892. doi: 10.1090/S0002-9947-2011-05352-3. Google Scholar

[6]

S. Cerrai and M. Freidlin, Small mass asymptotics for a charged particle in magnetic field and long-time influence of small perturbations, Journal of Statistical Physics, 144 (2011), 101-123. doi: 10.1007/s10955-011-0238-3. Google Scholar

[7]

S. Cerrai and M. Röckner, Large deviations for invariant measures of stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term, Annales de l'Institut Henri Poincaré (Probabilités et Statistiques), 41 (2005), 69-105. doi: 10.1016/j.anihpb.2004.03.001. Google Scholar

[8]

S. Cerrai and M. Salins, On the Smoluchowski-Kramers approximation for a system with an infinite number of degrees of freedom subject to a magnetic field, arXiv: 1409.0803.Google Scholar

[9]

S. Cerrai and M. Salins, Smoluchowski-Kramers approximation and large deviations for infinite dimensional non-gradient systems with applications to the exit problem, Ann. Probab., 44 (2016), 2591-2642. doi: 10.1214/15-AOP1029. Google Scholar

[10]

S. Cerrai and M. Salins, Smoluchowski-Kramers approximation and large deviations for infinite dimensional gradient systems, Asymptotics Analysis, 88 (2014), 201-215. Google Scholar

[11]

Z. Chen and M. Freidlin, Smoluchowski-Kramers approximation and exit problems, Stochastics and Dynamics, 5 (2005), 569-585. doi: 10.1142/S0219493705001560. Google Scholar

[12]

R. Dalang and N. E. Frangos, The stochastic wave equation in two spatial dimensions, The Annals of Probability, 26 (1998), 187-212. doi: 10.1214/aop/1022855416. Google Scholar

[13]

G. Da Prato and V. Barbu, The stochastic non-linear damped wave equation, Applied Mathematics and Optimization, 46 (2002), 125-141. doi: 10.1007/s00245-002-0744-4. Google Scholar

[14]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223. Google Scholar

[15]

G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems London Mathematical Society, Lecture Notes Series 229, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829. Google Scholar

[16]

E. B. Davies, Heat Kernels and Spectral Theory Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158. Google Scholar

[17]

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications Springer-Verlag, 1998. doi: 10.1007/978-1-4612-5320-4. Google Scholar

[18]

M. Freidlin, Random perturbations of reaction-diffusion equations: The quasi deterministic approximation, Transactions of the AMS, 305 (1988), 665-697. doi: 10.2307/2000884. Google Scholar

[19]

M. Freidlin, Some remarks on the Smoluchowski-Kramers approximation, Journal of Statistical Physics, 117 (2004), 617-634. doi: 10.1007/s10955-004-2273-9. Google Scholar

[20]

M. Freidlin and W. Hu, Smoluchowski-Kramers approximation in the case of variable friction, J. Math. Sci., 179 (2011), 184-207. doi: 10.1007/s10958-011-0589-y. Google Scholar

[21]

M. FreidlinW. Hu and A. Wentzell, Small mass asymptotic for the motion with vanishing friction, Stochastic Processes and Applications, 123 (2013), 45-75. doi: 10.1016/j.spa.2012.08.013. Google Scholar

[22]

M. Freidlin and A. Wentzell, Random Perturbations of Dynamical Systems Springer-Verlag, 1998. doi: 10.1007/978-1-4612-0611-8. Google Scholar

[23]

I. Gyöngy and N. V. Krylov, Existence of strong solutions for Itô's stochastic equations via approximations, Probability Theory and Related Fields, 105 (1996), 143-158. doi: 10.1007/BF01203833. Google Scholar

[24]

S. HottovyG. Volpe and J. Wehr, Noise-induced drift in stochastic differential equations with arbitrary friction and diffusion in the Smoluchowski-Kramers limit, Journal of Statistical Physics, 146 (2012), 762-773. doi: 10.1007/s10955-012-0418-9. Google Scholar

[25]

S. HottovyA. McDanielG. VolpeGiovanni and J. Wehr, The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction, Communications in Mathematical Physics, 336 (2015), 1259-1283. doi: 10.1007/s00220-014-2233-4. Google Scholar

[26]

A. Karczewska and J. Zabczyk, A note on stochastic wave equations, Evolution Equations and their Applications in Physical and Life Sciences (Bad Herrenalb, 1998), Lecture Notes in Pure and Appl. Math. 215, Dekker (2001), 501-511. Google Scholar

[27]

H. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica, 7 (1940), 284-304. doi: 10.1016/S0031-8914(40)90098-2. Google Scholar

[28]

J.J. Lee, Small mass asymptotics of a charged particle in a variable magnetic field, Asymptotic Analysis, 86 (2014), 99-121. Google Scholar

[29]

A. Millet and M. Sanz-Solé, A stochastic wave equation in two space dimension: Smoothness of the law, The Annals of Probability, 27 (1999), 803-844. doi: 10.1214/aop/1022677387. Google Scholar

[30]

A. Millet and P. Morien, On a non-linear stochastic wave equation in the plane: Existence and uniqueness of the solution, The Annals of Applied Probability, 11 (2001), 922-951. doi: 10.1214/aoap/1015345353. Google Scholar

[31]

M. Ondreját, Existence of global mild and strong solutions to stochastic hyperbolic evolution equations driven by a spatially homogeneous Wiener process, Journal of Evolution Equations, 4 (2004), 169-191. doi: 10.1007/s00028-003-0130-y. Google Scholar

[32]

M. Ondreját, Existence of global martingale solutions to stochastic hyperbolic evolution equations driven by a spatially homogeneous Wiener process, Stochastics and Dynamics, 6 (2006), 23-52. doi: 10.1142/S0219493706001633. Google Scholar

[33]

G. A. Pavliotis and A. Stuart, White noise limits for inertial particles in a random field, Multiscale Modeling and Simulation, 1 (2003), 527-533. doi: 10.1137/S1540345903421076. Google Scholar

[34]

G. A. Pavliotis and A. Stuart, Periodic homogenization for hypoelliptic diffusions, Journal of Statistical Physics, 117 (2004), 261-279. doi: 10.1023/B:JOSS.0000044055.59822.20. Google Scholar

[35]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar

[36]

S. Peszat and J. Zabczyk, Nonlinear stochastic wave and heat equation, Probability Theory and related Fields, 116 (2000), 421-443. doi: 10.1007/s004400050257. Google Scholar

[37]

S. Peszat, The Cauchy problem for a nonlinear stochastic wave equation in any dimension, Journal of Evolution Equations, 2 (2002), 383-394. doi: 10.1007/PL00013197. Google Scholar

[38]

M. Smoluchowski, Drei vortage über diffusion brownsche molekularbewegung und koagulation von kolloidteilchen, Physik Zeitschrift, 17 (1916), 557-571 and 587-599.Google Scholar

show all references

References:
[1]

R. Carmona and D. Nualart, Random non-linear wave equation: Smoothness of solutions, Probability Theory and Related Fields, 79 (1988), 469-508. doi: 10.1007/BF00318783. Google Scholar

[2]

S. Cerrai, Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach Lecture Notes in Mathematics 1762, Springer Verlag, 2001. doi: 10.1007/b80743. Google Scholar

[3]

S. Cerrai and M. Freidlin, On the Smoluchowski-Kramers approximation for a system with an infinite number of degrees of freedom, Probability Theory and Related Fields, 135 (2006), 363-394. doi: 10.1007/s00440-005-0465-0. Google Scholar

[4]

S. Cerrai and M. Freidlin, Smoluchowski-Kramers approximation for a general class of SPDE's, Journal of Evolution Equations, 6 (2006), 657-689. doi: 10.1007/s00028-006-0281-8. Google Scholar

[5]

S. Cerrai and M. Freidlin, Approximation of quasi-potentials and exit problems for multidimensional RDE's with noise, Transactions of the AMS, 363 (2011), 3853-3892. doi: 10.1090/S0002-9947-2011-05352-3. Google Scholar

[6]

S. Cerrai and M. Freidlin, Small mass asymptotics for a charged particle in magnetic field and long-time influence of small perturbations, Journal of Statistical Physics, 144 (2011), 101-123. doi: 10.1007/s10955-011-0238-3. Google Scholar

[7]

S. Cerrai and M. Röckner, Large deviations for invariant measures of stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term, Annales de l'Institut Henri Poincaré (Probabilités et Statistiques), 41 (2005), 69-105. doi: 10.1016/j.anihpb.2004.03.001. Google Scholar

[8]

S. Cerrai and M. Salins, On the Smoluchowski-Kramers approximation for a system with an infinite number of degrees of freedom subject to a magnetic field, arXiv: 1409.0803.Google Scholar

[9]

S. Cerrai and M. Salins, Smoluchowski-Kramers approximation and large deviations for infinite dimensional non-gradient systems with applications to the exit problem, Ann. Probab., 44 (2016), 2591-2642. doi: 10.1214/15-AOP1029. Google Scholar

[10]

S. Cerrai and M. Salins, Smoluchowski-Kramers approximation and large deviations for infinite dimensional gradient systems, Asymptotics Analysis, 88 (2014), 201-215. Google Scholar

[11]

Z. Chen and M. Freidlin, Smoluchowski-Kramers approximation and exit problems, Stochastics and Dynamics, 5 (2005), 569-585. doi: 10.1142/S0219493705001560. Google Scholar

[12]

R. Dalang and N. E. Frangos, The stochastic wave equation in two spatial dimensions, The Annals of Probability, 26 (1998), 187-212. doi: 10.1214/aop/1022855416. Google Scholar

[13]

G. Da Prato and V. Barbu, The stochastic non-linear damped wave equation, Applied Mathematics and Optimization, 46 (2002), 125-141. doi: 10.1007/s00245-002-0744-4. Google Scholar

[14]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223. Google Scholar

[15]

G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems London Mathematical Society, Lecture Notes Series 229, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829. Google Scholar

[16]

E. B. Davies, Heat Kernels and Spectral Theory Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158. Google Scholar

[17]

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications Springer-Verlag, 1998. doi: 10.1007/978-1-4612-5320-4. Google Scholar

[18]

M. Freidlin, Random perturbations of reaction-diffusion equations: The quasi deterministic approximation, Transactions of the AMS, 305 (1988), 665-697. doi: 10.2307/2000884. Google Scholar

[19]

M. Freidlin, Some remarks on the Smoluchowski-Kramers approximation, Journal of Statistical Physics, 117 (2004), 617-634. doi: 10.1007/s10955-004-2273-9. Google Scholar

[20]

M. Freidlin and W. Hu, Smoluchowski-Kramers approximation in the case of variable friction, J. Math. Sci., 179 (2011), 184-207. doi: 10.1007/s10958-011-0589-y. Google Scholar

[21]

M. FreidlinW. Hu and A. Wentzell, Small mass asymptotic for the motion with vanishing friction, Stochastic Processes and Applications, 123 (2013), 45-75. doi: 10.1016/j.spa.2012.08.013. Google Scholar

[22]

M. Freidlin and A. Wentzell, Random Perturbations of Dynamical Systems Springer-Verlag, 1998. doi: 10.1007/978-1-4612-0611-8. Google Scholar

[23]

I. Gyöngy and N. V. Krylov, Existence of strong solutions for Itô's stochastic equations via approximations, Probability Theory and Related Fields, 105 (1996), 143-158. doi: 10.1007/BF01203833. Google Scholar

[24]

S. HottovyG. Volpe and J. Wehr, Noise-induced drift in stochastic differential equations with arbitrary friction and diffusion in the Smoluchowski-Kramers limit, Journal of Statistical Physics, 146 (2012), 762-773. doi: 10.1007/s10955-012-0418-9. Google Scholar

[25]

S. HottovyA. McDanielG. VolpeGiovanni and J. Wehr, The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction, Communications in Mathematical Physics, 336 (2015), 1259-1283. doi: 10.1007/s00220-014-2233-4. Google Scholar

[26]

A. Karczewska and J. Zabczyk, A note on stochastic wave equations, Evolution Equations and their Applications in Physical and Life Sciences (Bad Herrenalb, 1998), Lecture Notes in Pure and Appl. Math. 215, Dekker (2001), 501-511. Google Scholar

[27]

H. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica, 7 (1940), 284-304. doi: 10.1016/S0031-8914(40)90098-2. Google Scholar

[28]

J.J. Lee, Small mass asymptotics of a charged particle in a variable magnetic field, Asymptotic Analysis, 86 (2014), 99-121. Google Scholar

[29]

A. Millet and M. Sanz-Solé, A stochastic wave equation in two space dimension: Smoothness of the law, The Annals of Probability, 27 (1999), 803-844. doi: 10.1214/aop/1022677387. Google Scholar

[30]

A. Millet and P. Morien, On a non-linear stochastic wave equation in the plane: Existence and uniqueness of the solution, The Annals of Applied Probability, 11 (2001), 922-951. doi: 10.1214/aoap/1015345353. Google Scholar

[31]

M. Ondreját, Existence of global mild and strong solutions to stochastic hyperbolic evolution equations driven by a spatially homogeneous Wiener process, Journal of Evolution Equations, 4 (2004), 169-191. doi: 10.1007/s00028-003-0130-y. Google Scholar

[32]

M. Ondreját, Existence of global martingale solutions to stochastic hyperbolic evolution equations driven by a spatially homogeneous Wiener process, Stochastics and Dynamics, 6 (2006), 23-52. doi: 10.1142/S0219493706001633. Google Scholar

[33]

G. A. Pavliotis and A. Stuart, White noise limits for inertial particles in a random field, Multiscale Modeling and Simulation, 1 (2003), 527-533. doi: 10.1137/S1540345903421076. Google Scholar

[34]

G. A. Pavliotis and A. Stuart, Periodic homogenization for hypoelliptic diffusions, Journal of Statistical Physics, 117 (2004), 261-279. doi: 10.1023/B:JOSS.0000044055.59822.20. Google Scholar

[35]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar

[36]

S. Peszat and J. Zabczyk, Nonlinear stochastic wave and heat equation, Probability Theory and related Fields, 116 (2000), 421-443. doi: 10.1007/s004400050257. Google Scholar

[37]

S. Peszat, The Cauchy problem for a nonlinear stochastic wave equation in any dimension, Journal of Evolution Equations, 2 (2002), 383-394. doi: 10.1007/PL00013197. Google Scholar

[38]

M. Smoluchowski, Drei vortage über diffusion brownsche molekularbewegung und koagulation von kolloidteilchen, Physik Zeitschrift, 17 (1916), 557-571 and 587-599.Google Scholar

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