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Viscous Aubry-Mather theory and the Vlasov equation
1. | Dip. di Matematica, Università di Roma Tre, L.go S. Leonardo Murialdo 1, 00146, Roma, Italy |
References:
[1] |
L. Ambrosio, Lecture notes on optimal transport problems, in "Mathematical Aspects of Evolving Interfaces" (Funchal, 2000), Lecture Notes in Math., 1812, Springer, Berlin, (2003), 1-52.
doi: 10.1007/978-3-540-39189-0_1. |
[2] |
N. Anantharaman, On the zero-temperature vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics, J. Eur. Math. Soc. (JEMS), 6 (2004), 207-276. |
[3] |
G. Birkhoff, "Lattice Theory," Third edition, AMS Colloquium Publ., Vol. XXV, AMS, Providence, R. I., 1967. |
[4] |
P. Cardialiaguet, "Notes on Mean Field Games,", from P.-L. Lions' lectures at the Collège de France, ().
|
[5] |
E. Carlen, "Lectures on Optimal Mass Transportation and Certain of its Applications," mimeographed notes, 2009. |
[6] |
G. Da Prato, "Introduction to Stochastic Analysis and Malliavin Calculus," Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 6, Edizioni della Normale, Pisa, 2007. |
[7] |
R. L. Dobrušin, Vlasov equations, Functional Analysis and its Applications, 13 (1979), 45-58. |
[8] |
A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics," Fourth preliminary version, mimeographed notes, Lyon, 2003. |
[9] |
W. H. Fleming, The Cauchy problem for a nonlinear first order PDE, Journal of Differential Equations, 5 (1969), 515-530.
doi: 10.1016/0022-0396(69)90091-6. |
[10] |
A. Friedman, "Partial Differential Equations of Parabolic Type," Dover, New York, 1992. |
[11] |
W. Gangbo and A. Tudorascu, Lagrangian dynamics on an infinite-dimensional torus; a weak KAM theorem, Adv. Math., 224 (2010), 260-292.
doi: 10.1016/j.aim.2009.11.005. |
[12] |
W. Gangbo, T. Nguyen and A. Tudorascu, Hamilton-Jacobi equations in the Wasserstein space, Methods Appl. Anal., 15 (2008), 155-183. |
[13] |
D. Gomes, A stochastic analog of Aubry-Mather theory, Nonlinearity, 15 (2002), 581-603.
doi: 10.1088/0951-7715/15/3/304. |
[14] |
T. Hida, "Brownian Motion," Applications of Mathematics, 11, Springer-Verlag, New York-Berlin, 1980. |
[15] |
J. N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.
doi: 10.5802/aif.1377. |
[16] |
M. Viana, "Stochastic Dynamics of Deterministic Systems," mimeographed notes, 2000. |
[17] |
C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003.
doi: 10.1007/b12016. |
show all references
References:
[1] |
L. Ambrosio, Lecture notes on optimal transport problems, in "Mathematical Aspects of Evolving Interfaces" (Funchal, 2000), Lecture Notes in Math., 1812, Springer, Berlin, (2003), 1-52.
doi: 10.1007/978-3-540-39189-0_1. |
[2] |
N. Anantharaman, On the zero-temperature vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics, J. Eur. Math. Soc. (JEMS), 6 (2004), 207-276. |
[3] |
G. Birkhoff, "Lattice Theory," Third edition, AMS Colloquium Publ., Vol. XXV, AMS, Providence, R. I., 1967. |
[4] |
P. Cardialiaguet, "Notes on Mean Field Games,", from P.-L. Lions' lectures at the Collège de France, ().
|
[5] |
E. Carlen, "Lectures on Optimal Mass Transportation and Certain of its Applications," mimeographed notes, 2009. |
[6] |
G. Da Prato, "Introduction to Stochastic Analysis and Malliavin Calculus," Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 6, Edizioni della Normale, Pisa, 2007. |
[7] |
R. L. Dobrušin, Vlasov equations, Functional Analysis and its Applications, 13 (1979), 45-58. |
[8] |
A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics," Fourth preliminary version, mimeographed notes, Lyon, 2003. |
[9] |
W. H. Fleming, The Cauchy problem for a nonlinear first order PDE, Journal of Differential Equations, 5 (1969), 515-530.
doi: 10.1016/0022-0396(69)90091-6. |
[10] |
A. Friedman, "Partial Differential Equations of Parabolic Type," Dover, New York, 1992. |
[11] |
W. Gangbo and A. Tudorascu, Lagrangian dynamics on an infinite-dimensional torus; a weak KAM theorem, Adv. Math., 224 (2010), 260-292.
doi: 10.1016/j.aim.2009.11.005. |
[12] |
W. Gangbo, T. Nguyen and A. Tudorascu, Hamilton-Jacobi equations in the Wasserstein space, Methods Appl. Anal., 15 (2008), 155-183. |
[13] |
D. Gomes, A stochastic analog of Aubry-Mather theory, Nonlinearity, 15 (2002), 581-603.
doi: 10.1088/0951-7715/15/3/304. |
[14] |
T. Hida, "Brownian Motion," Applications of Mathematics, 11, Springer-Verlag, New York-Berlin, 1980. |
[15] |
J. N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.
doi: 10.5802/aif.1377. |
[16] |
M. Viana, "Stochastic Dynamics of Deterministic Systems," mimeographed notes, 2000. |
[17] |
C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003.
doi: 10.1007/b12016. |
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