July 2018, 38(7): 3407-3438. doi: 10.3934/dcds.2018146

On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type

1. 

Department of Mathematics, Imperial College London, London SW7 2AZ, UK

2. 

Data Analytics Department, Esmart Systems, Håkon Melbergs vei 16, 1783 Halden, Norway

* Corresponding author: Manh Hong Duong

Received  July 2017 Revised  November 2017 Published  April 2018

In this paper, we construct the fundamental solution to a degenerate diffusion of Kolmogorov type and develop a time-discrete variational scheme for its adjoint equation. The so-called mean squared derivative cost function plays a crucial role occurring in both the fundamental solution and the variational scheme. The latter is implemented by minimizing a free energy functional with respect to the Kantorovich optimal transport cost functional associated with the mean squared derivative cost. We establish the convergence of the scheme to the solution of the adjoint equation generalizing previously known results for the Fokker-Planck equation and the Kramers equation.

Citation: Manh Hong Duong, Hoang Minh Tran. On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3407-3438. doi: 10.3934/dcds.2018146
References:
[1]

S. AdamsN. DirrM. A. Peletier and J. Zimmer, From a large-deviations principle to the Wasserstein gradient flow: A new micro-macro passage, Communications in Mathematical Physics, 307 (2011), 791-815. doi: 10.1007/s00220-011-1328-4.

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics. ETH Zürich. Birkhauser, Basel, 2nd edition, 2008.

[3]

T. Bodineau and R. Lefevere, Large deviations of lattice hamiltonian dynamics coupled to stochastic thermostats, Journal of Statistical Physics, 133 (2008), 1-27. doi: 10.1007/s10955-008-9601-4.

[4]

M. Bramanti, An Invitation to Hypoelliptic Operators and Hörmander's Vector Fields, Springer, Berlin, 2014. doi: 10.1007/978-3-319-02087-7.

[5]

F. Cavalletti, M. Sedjro and M. Westdickenberg, A variational time discretization for the compressible euler equations, http://arXiv.org/abs/1411.1012, 2014.

[6]

F. Delarue and S. Menozzi, Density estimates for a random noise propagating through a chain of differential equations, J. Funct. Anal., 259 (2010), 1577-1630. doi: 10.1016/j.jfa.2010.05.002.

[7]

M. H. DuongV. Laschos and M. Renger, Wasserstein gradient flows from large deviations of many-particle limits, ESAIM Control Optim. Calc. Var., 19 (2013), 1166-1188. doi: 10.1051/cocv/2013049.

[8]

M. H. DuongM. A. Peletier and J. Zimmer, GENERIC formalism of a Vlasov-Fokker-Planck equation and connection to large-deviation principles, Nonlinearity, 26 (2013), 2951-2971. doi: 10.1088/0951-7715/26/11/2951.

[9]

M. H. DuongM. A. Peletier and J. Zimmer, Conservative-dissipative approximation schemes for a generalized Kramers equation, Mathematical Methods in the Applied Sciences, 37 (2014), 2517-2540. doi: 10.1002/mma.2994.

[10]

M. H. Duong and H. M. Tran, Analysis of the mean squared derivative cost functions, Mathematical Methods in the Applied Sciences, 40 (2017), 5222-5240. doi: 10.1002/mma.4382.

[11]

M. H. Duong, Long time behaviour and particle approximation of a generalised Vlasov dynamic, Nonlinear Anal., 127 (2015), 1-16. doi: 10.1016/j.na.2015.06.018.

[12]

J.-P. Eckmann and M. Hairer, Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators, Commun. Math. Phys., 212 (2000), 105-164. doi: 10.1007/s002200000216.

[13]

M. Erbar, J. Maas and M. Renger, From large deviations to wasserstein gradient flows in multiple dimensions, Electron. Commun. Probab. , 20 (2015), 12 pp. doi: 10.1214/ECP.v20-4315.

[14]

M. D. Francesco and A. Pascucci, On a class of degenerate parabolic equations of kolmogorov type, Applied Mathematics Research eXpress, 2005 (2005), 77-116.

[15]

W. Gangbo and M. Westdickenberg, Optimal transport for the system of isentropic Euler equations, Comm. Partial Differential Equations, 34 (2009), 1041-1073. doi: 10.1080/03605300902892345.

[16]

P. HänggiP. Talkner and M. Borkovec, Reaction-rate theory: Fifty years after Kramers, Rev. Modern Phys., 62 (1990), 251-341. doi: 10.1103/RevModPhys.62.251.

[17]

L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. doi: 10.1007/BF02392081.

[18]

C. Huang, A variational principle for the Kramers equation with unbounded external forces, J. Math. Anal. Appl., 250 (2000), 333-367. doi: 10.1006/jmaa.2000.7109.

[19]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the fokker-planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17. doi: 10.1137/S0036141096303359.

[20]

A. Kolmogoroff, Zufallige bewegungen (zur theorie der brownschen bewegung), Annals of Mathematics, 35 (1934), 116-117. doi: 10.2307/1968123.

[21]

L. P. Kuptsov, The fundamental solutions of a certain class of elliptic-parabolic second order equations, Differential Equations, 8 (1972), 1649-1660.

[22]

E. Lanconelli, A. Pascucci and S. Polidoro, Linear and nonlinear ultraparabolic equations of kolmogorov type arising in diffusion theory and in finance, In Nonlinear Problems in Mathematical Physics and Related Topics Vol. Ⅱ In Honor of Professor O. A. Ladyzhenskaya. International Mathematical Series, pages 243-265. Kluwer Ed, 2002. doi: 10.1007/978-1-4615-0701-7_14.

[23]

H. P. Malytska and I. V. Burtnyak, Structure of the fundamenal solution of cauhcy problem for kolmogorov systems of second-order, Journal of Vasyl Stefanyk Precarpathian National University, 2 (2015), 9-22.

[24]

M. Ottobre and G. A. Pavliotis, Asymptotic analysis for the generalized Langevin equation, Nonlinearity, 24 (2011), 1629-1653. doi: 10.1088/0951-7715/24/5/013.

[25]

A. Pascucci, Kolmogorov equations in physics and in finance, Elliptic and Parabolic Problems, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 63 (2005), 353-364. doi: 10.1007/3-7643-7384-9_35.

[26]

S. Polidoro, On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type, Matematiche (Catania), 49 (1994), 53-105(1995).

[27]

H. Risken, The Fokker-Planck Equation, volume 18 of Springer Series in Synergetics, Springer-Verlag, Berlin, 1984. Methods of solution and applications. doi: 10.1007/978-3-642-96807-5.

[28]

C. Villani, Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

[29]

C. Villani, Optimal Transport: Old and New, Grundlehren der mathematischen Wissenschaften. Springer, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

[30]

M. Weber, The fundamental solution of a degenerate partial differential equation of parabolic type, Trans. Amer. Math. Soc., 71 (1951), 24-37. doi: 10.1090/S0002-9947-1951-0042035-0.

[31]

M. Westdickenberg, Projections onto the cone of optimal transport maps and compressible fluid flows, J. Hyperbolic Differ. Equ., 7 (2010), 605-649. doi: 10.1142/S0219891610002244.

show all references

References:
[1]

S. AdamsN. DirrM. A. Peletier and J. Zimmer, From a large-deviations principle to the Wasserstein gradient flow: A new micro-macro passage, Communications in Mathematical Physics, 307 (2011), 791-815. doi: 10.1007/s00220-011-1328-4.

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics. ETH Zürich. Birkhauser, Basel, 2nd edition, 2008.

[3]

T. Bodineau and R. Lefevere, Large deviations of lattice hamiltonian dynamics coupled to stochastic thermostats, Journal of Statistical Physics, 133 (2008), 1-27. doi: 10.1007/s10955-008-9601-4.

[4]

M. Bramanti, An Invitation to Hypoelliptic Operators and Hörmander's Vector Fields, Springer, Berlin, 2014. doi: 10.1007/978-3-319-02087-7.

[5]

F. Cavalletti, M. Sedjro and M. Westdickenberg, A variational time discretization for the compressible euler equations, http://arXiv.org/abs/1411.1012, 2014.

[6]

F. Delarue and S. Menozzi, Density estimates for a random noise propagating through a chain of differential equations, J. Funct. Anal., 259 (2010), 1577-1630. doi: 10.1016/j.jfa.2010.05.002.

[7]

M. H. DuongV. Laschos and M. Renger, Wasserstein gradient flows from large deviations of many-particle limits, ESAIM Control Optim. Calc. Var., 19 (2013), 1166-1188. doi: 10.1051/cocv/2013049.

[8]

M. H. DuongM. A. Peletier and J. Zimmer, GENERIC formalism of a Vlasov-Fokker-Planck equation and connection to large-deviation principles, Nonlinearity, 26 (2013), 2951-2971. doi: 10.1088/0951-7715/26/11/2951.

[9]

M. H. DuongM. A. Peletier and J. Zimmer, Conservative-dissipative approximation schemes for a generalized Kramers equation, Mathematical Methods in the Applied Sciences, 37 (2014), 2517-2540. doi: 10.1002/mma.2994.

[10]

M. H. Duong and H. M. Tran, Analysis of the mean squared derivative cost functions, Mathematical Methods in the Applied Sciences, 40 (2017), 5222-5240. doi: 10.1002/mma.4382.

[11]

M. H. Duong, Long time behaviour and particle approximation of a generalised Vlasov dynamic, Nonlinear Anal., 127 (2015), 1-16. doi: 10.1016/j.na.2015.06.018.

[12]

J.-P. Eckmann and M. Hairer, Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators, Commun. Math. Phys., 212 (2000), 105-164. doi: 10.1007/s002200000216.

[13]

M. Erbar, J. Maas and M. Renger, From large deviations to wasserstein gradient flows in multiple dimensions, Electron. Commun. Probab. , 20 (2015), 12 pp. doi: 10.1214/ECP.v20-4315.

[14]

M. D. Francesco and A. Pascucci, On a class of degenerate parabolic equations of kolmogorov type, Applied Mathematics Research eXpress, 2005 (2005), 77-116.

[15]

W. Gangbo and M. Westdickenberg, Optimal transport for the system of isentropic Euler equations, Comm. Partial Differential Equations, 34 (2009), 1041-1073. doi: 10.1080/03605300902892345.

[16]

P. HänggiP. Talkner and M. Borkovec, Reaction-rate theory: Fifty years after Kramers, Rev. Modern Phys., 62 (1990), 251-341. doi: 10.1103/RevModPhys.62.251.

[17]

L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. doi: 10.1007/BF02392081.

[18]

C. Huang, A variational principle for the Kramers equation with unbounded external forces, J. Math. Anal. Appl., 250 (2000), 333-367. doi: 10.1006/jmaa.2000.7109.

[19]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the fokker-planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17. doi: 10.1137/S0036141096303359.

[20]

A. Kolmogoroff, Zufallige bewegungen (zur theorie der brownschen bewegung), Annals of Mathematics, 35 (1934), 116-117. doi: 10.2307/1968123.

[21]

L. P. Kuptsov, The fundamental solutions of a certain class of elliptic-parabolic second order equations, Differential Equations, 8 (1972), 1649-1660.

[22]

E. Lanconelli, A. Pascucci and S. Polidoro, Linear and nonlinear ultraparabolic equations of kolmogorov type arising in diffusion theory and in finance, In Nonlinear Problems in Mathematical Physics and Related Topics Vol. Ⅱ In Honor of Professor O. A. Ladyzhenskaya. International Mathematical Series, pages 243-265. Kluwer Ed, 2002. doi: 10.1007/978-1-4615-0701-7_14.

[23]

H. P. Malytska and I. V. Burtnyak, Structure of the fundamenal solution of cauhcy problem for kolmogorov systems of second-order, Journal of Vasyl Stefanyk Precarpathian National University, 2 (2015), 9-22.

[24]

M. Ottobre and G. A. Pavliotis, Asymptotic analysis for the generalized Langevin equation, Nonlinearity, 24 (2011), 1629-1653. doi: 10.1088/0951-7715/24/5/013.

[25]

A. Pascucci, Kolmogorov equations in physics and in finance, Elliptic and Parabolic Problems, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 63 (2005), 353-364. doi: 10.1007/3-7643-7384-9_35.

[26]

S. Polidoro, On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type, Matematiche (Catania), 49 (1994), 53-105(1995).

[27]

H. Risken, The Fokker-Planck Equation, volume 18 of Springer Series in Synergetics, Springer-Verlag, Berlin, 1984. Methods of solution and applications. doi: 10.1007/978-3-642-96807-5.

[28]

C. Villani, Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

[29]

C. Villani, Optimal Transport: Old and New, Grundlehren der mathematischen Wissenschaften. Springer, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

[30]

M. Weber, The fundamental solution of a degenerate partial differential equation of parabolic type, Trans. Amer. Math. Soc., 71 (1951), 24-37. doi: 10.1090/S0002-9947-1951-0042035-0.

[31]

M. Westdickenberg, Projections onto the cone of optimal transport maps and compressible fluid flows, J. Hyperbolic Differ. Equ., 7 (2010), 605-649. doi: 10.1142/S0219891610002244.

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