메뉴 건너뛰기




Volumn 259, Issue 6, 2010, Pages 1577-1630

Density estimates for a random noise propagating through a chain of differential equations

Author keywords

Aronson estimates; Gaussian bounds; H rmander conditions; Hypoellipticity; Stochastic control

Indexed keywords


EID: 77954212974     PISSN: 00221236     EISSN: 10960783     Source Type: Journal    
DOI: 10.1016/j.jfa.2010.05.002     Document Type: Article
Times cited : (128)

References (47)
  • 1
    • 84968488966 scopus 로고
    • Bounds for the fundamental solution of a parabolic equation
    • Aronson D.G. Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 1967, 73:890-896.
    • (1967) Bull. Amer. Math. Soc. , vol.73 , pp. 890-896
    • Aronson, D.G.1
  • 2
    • 0013252168 scopus 로고
    • On the connection between the Malliavin covariance matrix and Hörmander's condition
    • Bally V. On the connection between the Malliavin covariance matrix and Hörmander's condition. J. Funct. Anal. 1990, 96(2):219-255.
    • (1990) J. Funct. Anal. , vol.96 , Issue.2 , pp. 219-255
    • Bally, V.1
  • 3
    • 76849114316 scopus 로고    scopus 로고
    • Lower bounds for densities of Asian type stochastic differential equations
    • Bally V., Kohatsu-Higa A. Lower bounds for densities of Asian type stochastic differential equations. J. Funct. Anal. 2010, 258(9):3134-3164.
    • (2010) J. Funct. Anal. , vol.258 , Issue.9 , pp. 3134-3164
    • Bally, V.1    Kohatsu-Higa, A.2
  • 4
  • 6
    • 33847299320 scopus 로고
    • Décroissance exponentielle du noyau de la chaleur sur la diagonale, II
    • Ben Arous G., Léandre R. Décroissance exponentielle du noyau de la chaleur sur la diagonale, II. Probab. Theory Related Fields 1991, 90(3):377-402.
    • (1991) Probab. Theory Related Fields , vol.90 , Issue.3 , pp. 377-402
    • Ben Arous, G.1    Léandre, R.2
  • 7
    • 49249092850 scopus 로고    scopus 로고
    • Large deviations of lattice Hamiltonian dynamics coupled to stochastic thermostats
    • Bodineau T., Lefevere L. Large deviations of lattice Hamiltonian dynamics coupled to stochastic thermostats. J. Stat. Phys. 2008, 133:1-27.
    • (2008) J. Stat. Phys. , vol.133 , pp. 1-27
    • Bodineau, T.1    Lefevere, L.2
  • 9
    • 0000257576 scopus 로고
    • Calcul stochastique et opérateurs dégénérés du second ordre, I. Résolvantes, théorème de Hörmander et applications
    • Cattiaux P. Calcul stochastique et opérateurs dégénérés du second ordre, I. Résolvantes, théorème de Hörmander et applications. Bull. Sci. Math. Sér. 2 1990, 114:421-462.
    • (1990) Bull. Sci. Math. Sér. 2 , vol.114 , pp. 421-462
    • Cattiaux, P.1
  • 11
    • 49249127518 scopus 로고    scopus 로고
    • Control and Nonlinearity
    • Amer. Math. Soc.
    • Coron J.-M. Control and Nonlinearity. Math. Surveys Monogr. 2007, vol. 136. Amer. Math. Soc.
    • (2007) Math. Surveys Monogr. , vol.136
    • Coron, J.-M.1
  • 12
    • 33845277179 scopus 로고    scopus 로고
    • On a class of degenerate parabolic equations of Kolmogorov type
    • Di Francesco M., Pascucci A. On a class of degenerate parabolic equations of Kolmogorov type. Appl. Math. Res. Express. AMRX 2005, 3:77-116.
    • (2005) Appl. Math. Res. Express. AMRX , vol.3 , pp. 77-116
    • Di Francesco, M.1    Pascucci, A.2
  • 13
    • 84881102143 scopus 로고    scopus 로고
    • Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form
    • Di Francesco M., Polidoro S. Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form. Adv. Differential Equations 2006, 11:1261-1320.
    • (2006) Adv. Differential Equations , vol.11 , pp. 1261-1320
    • Di Francesco, M.1    Polidoro, S.2
  • 14
    • 0033243772 scopus 로고    scopus 로고
    • Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures
    • Eckmann J.-P., Pillet C.-A., Rey-Bellet L. Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures. Comm. Math. Phys. 1999, 201(3):657-697.
    • (1999) Comm. Math. Phys. , vol.201 , Issue.3 , pp. 657-697
    • Eckmann, J.-P.1    Pillet, C.-A.2    Rey-Bellet, L.3
  • 16
    • 34250111516 scopus 로고
    • Stochastic variational formula for fundamental solutions of parabolic pde
    • Fleming W.H., Sheu S.J. Stochastic variational formula for fundamental solutions of parabolic pde. Appl. Math. Optim. 1985, 13:193-204.
    • (1985) Appl. Math. Optim. , vol.13 , pp. 193-204
    • Fleming, W.H.1    Sheu, S.J.2
  • 20
    • 1142264370 scopus 로고    scopus 로고
    • Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential
    • Hérau F., Nier F. Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential. Arch. Ration. Mech. Anal. 2004, 171(2):151-218.
    • (2004) Arch. Ration. Mech. Anal. , vol.171 , Issue.2 , pp. 151-218
    • Hérau, F.1    Nier, F.2
  • 21
    • 15244346736 scopus 로고
    • Hypoelliptic second order differential operators
    • Hörmander L. Hypoelliptic second order differential operators. Acta Math. 1967, 119:147-171.
    • (1967) Acta Math. , vol.119 , pp. 147-171
    • Hörmander, L.1
  • 22
    • 0002035256 scopus 로고
    • Second-order linear equations of parabolic type
    • 105
    • Il'in A.M., Kalashnikov A.S., Oleinik O.A. Second-order linear equations of parabolic type. Uspekhi Mat. Nauk 1962, 17(3(105)):3-146.
    • (1962) Uspekhi Mat. Nauk , vol.17 , Issue.3 , pp. 3-146
    • Il'in, A.M.1    Kalashnikov, A.S.2    Oleinik, O.A.3
  • 23
    • 0002824461 scopus 로고
    • Viscosity solutions of fully nonlinear second-order elliptic partial differential equations
    • Ishii H., Lions P.-L. Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differential Equations 1990, 83:26-78.
    • (1990) J. Differential Equations , vol.83 , pp. 26-78
    • Ishii, H.1    Lions, P.-L.2
  • 24
    • 0001334972 scopus 로고
    • Zufällige Bewegungen (zur Theorie der Brownschen Bewegung)
    • Kolmogorov A.N. Zufällige Bewegungen (zur Theorie der Brownschen Bewegung). Ann. of Math. 1934, 2(35):116-117.
    • (1934) Ann. of Math. , vol.2 , Issue.35 , pp. 116-117
    • Kolmogorov, A.N.1
  • 25
    • 0034389016 scopus 로고    scopus 로고
    • Local limit theorems for transition densities of Markov chains converging to diffusions
    • Konakov V., Mammen E. Local limit theorems for transition densities of Markov chains converging to diffusions. Probab. Theory Related Fields 2000, 117:551-587.
    • (2000) Probab. Theory Related Fields , vol.117 , pp. 551-587
    • Konakov, V.1    Mammen, E.2
  • 26
    • 77954213128 scopus 로고    scopus 로고
    • Explicit parametrix and local limit theorems for some degenerate diffusion processes, Ann. Inst. H. Poincaré, Sér. B , in press
    • V. Konakov, S. Menozzi, S. Molchanov, Explicit parametrix and local limit theorems for some degenerate diffusion processes, Ann. Inst. H. Poincaré, Sér. B (2009), in press.
    • (2009)
    • Konakov, S.1    Menozzi, S.2    Molchanov3
  • 27
    • 77956845256 scopus 로고
    • Applications of the Malliavin calculus, I
    • Stochastic Analysis
    • Kusuoka S., Stroock D. Applications of the Malliavin calculus, I. North-Holland Math. Library 1984, vol. 32:271-306.
    • (1984) North-Holland Math. Library , vol.32 , pp. 271-306
    • Kusuoka, S.1    Stroock, D.2
  • 30
    • 61849083018 scopus 로고    scopus 로고
    • Weak solutions for forward-backward SDEs-a martingale problem approach
    • Ma J., Zhang J. Weak solutions for forward-backward SDEs-a martingale problem approach. Ann. Probab. 2008, 36:2092-2125.
    • (2008) Ann. Probab. , vol.36 , pp. 2092-2125
    • Ma, J.1    Zhang, J.2
  • 31
    • 0001694213 scopus 로고
    • Ck-hypoellipticity with degeneracy
    • Academic Press, New York, London
    • Malliavin P. Ck-hypoellipticity with degeneracy. Stochastic Analysis, Proc. Internat. Conf. 1978, 199-214. Academic Press, New York, London.
    • (1978) Stochastic Analysis, Proc. Internat. Conf. , pp. 199-214
    • Malliavin, P.1
  • 32
    • 0002684408 scopus 로고
    • Stochastic calculus of variation and hypoelliptic operators
    • Wiley, New York, Chichester, Brisbane, Proceedings of the International Symposium on Stochastic Differential Equations
    • Malliavin P. Stochastic calculus of variation and hypoelliptic operators. Publ. Res. Inst. Math. Sci. 1978, 195-263. Wiley, New York, Chichester, Brisbane.
    • (1978) Publ. Res. Inst. Math. Sci. , pp. 195-263
    • Malliavin, P.1
  • 34
    • 84972530647 scopus 로고
    • Curvature and the eigenvalues of the Laplacian
    • McKean H.P., Singer I.M. Curvature and the eigenvalues of the Laplacian. J. Differential Geom. 1967, 1:43-69.
    • (1967) J. Differential Geom. , vol.1 , pp. 43-69
    • McKean, H.P.1    Singer, I.M.2
  • 35
    • 0003326139 scopus 로고
    • The Malliavin Calculus and Related Topics
    • Springer, New York
    • Nualart D. The Malliavin Calculus and Related Topics. Probab. Appl. 1995, Springer, New York.
    • (1995) Probab. Appl.
    • Nualart, D.1
  • 36
    • 33750177062 scopus 로고    scopus 로고
    • Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators
    • Pascucci A., Polidoro S. Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators. Trans. Amer. Math. Soc. 2006, 358(11):4873-4893.
    • (2006) Trans. Amer. Math. Soc. , vol.358 , Issue.11 , pp. 4873-4893
    • Pascucci, A.1    Polidoro, S.2
  • 37
    • 0003203092 scopus 로고
    • On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type
    • Polidoro S. On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type. Matematiche (Catania) 1994, 49:53-105.
    • (1994) Matematiche (Catania) , vol.49 , pp. 53-105
    • Polidoro, S.1
  • 38
    • 0344939351 scopus 로고    scopus 로고
    • Continuous Martingales and Brownian Motion
    • Springer, Berlin
    • Revuz D., Yor M. Continuous Martingales and Brownian Motion. Grundlehren Math. Wiss. 1999, vol. 293. Springer, Berlin. third ed.
    • (1999) Grundlehren Math. Wiss. , vol.293
    • Revuz, D.1    Yor, M.2
  • 39
    • 0034336318 scopus 로고    scopus 로고
    • Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators
    • Rey-Bellet L., Lawrence L.E. Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators. Comm. Math. Phys. 2000, 215(1):1-24.
    • (2000) Comm. Math. Phys. , vol.215 , Issue.1 , pp. 1-24
    • Rey-Bellet, L.1    Lawrence, L.E.2
  • 40
    • 0000189968 scopus 로고
    • Some estimates of the transition density of a nondegenerate diffusion Markov process
    • Sheu S.J. Some estimates of the transition density of a nondegenerate diffusion Markov process. Ann. Probab. 1991, 19(2):538-561.
    • (1991) Ann. Probab. , vol.19 , Issue.2 , pp. 538-561
    • Sheu, S.J.1
  • 41
    • 0342904825 scopus 로고
    • The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions
    • World Scientific Publishing Co., Inc., River Edge, NJ
    • Soize C. The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions. Ser. Adv. Math. Appl. Sci. 1994, vol. 17. World Scientific Publishing Co., Inc., River Edge, NJ.
    • (1994) Ser. Adv. Math. Appl. Sci. , vol.17
    • Soize, C.1
  • 42
    • 0038629828 scopus 로고
    • A class of degenerate diffusion processes
    • (in Russian)
    • Sonin I.M. A class of degenerate diffusion processes. Teor. Veroyatn. Primen. 1967, 12:540-547. (in Russian).
    • (1967) Teor. Veroyatn. Primen. , vol.12 , pp. 540-547
    • Sonin, I.M.1
  • 44
    • 0001198512 scopus 로고
    • Some applications of stochastic calculus to partial differential equations
    • Springer, Berlin, Eleventh Saint Flour Probability Summer School-1981
    • Stroock D.W. Some applications of stochastic calculus to partial differential equations. Lecture Notes in Math. 1983, vol. 976:267-382. Springer, Berlin.
    • (1983) Lecture Notes in Math. , vol.976 , pp. 267-382
    • Stroock, D.W.1
  • 46
    • 0345275329 scopus 로고    scopus 로고
    • Stochastic Hamiltonian dissipative systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme
    • Talay D. Stochastic Hamiltonian dissipative systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Markov Process. Related Fields 2002, 8(2):163-198.
    • (2002) Markov Process. Related Fields , vol.8 , Issue.2 , pp. 163-198
    • Talay, D.1
  • 47
    • 0002952245 scopus 로고
    • The fundamental solution of a degenerate partial differential equation of parabolic type
    • Weber M. The fundamental solution of a degenerate partial differential equation of parabolic type. Trans. Amer. Math. Soc. 1951, 71:24-37.
    • (1951) Trans. Amer. Math. Soc. , vol.71 , pp. 24-37
    • Weber, M.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.