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On a resonant mean field type equation: A "critical point at Infinity" approach
The stochastic value function in metric measure spaces
Dipartimento di Matematica, Università di Roma Tre, Largo S. Leonardo Murialdo 1,00146 Roma, Italy |
Let $(S,d)$ be a compact metric space and let $m$ be a Borel probability measure on $(S,d)$. We shall prove that, if $(S,d,m)$ is a $RCD(K,\infty)$ space, then the stochastic value function satisfies the viscous Hamilton-Jacobi equation, exactly as in Fleming's theorem on ${\bf{R}}^d$.
References:
[1] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows, Birkhäuser, Basel, 2005. |
[2] |
L. Ambrosio, N. Gigli and G. Savaré, Heat flow and calculus on metric measure spaces with Ricci curvature bounded below -the compact case, in Analysis and Numerics of Partial Differential Equations, Springer, Milano, 2013, 63-115, .
doi: 10.1007/978-88-470-2592-9_8. |
[3] |
L. Ambrosio, N. Gigli and G. Savaré,
Calculus and heat flows in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math., 195 (2014), 289-391.
doi: 10.1007/s00222-013-0456-1. |
[4] |
L. Ambrosio, N. Gigli and G. Savaré,
Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J., 163 (2014), 1405-1490.
doi: 10.1215/00127094-2681605. |
[5] |
L. Ambrosio, N. Gigli and G. Savaré,
Bakry-Emery curvature-dimension condition and Rie-Émannian Ricci curvature bounds, Ann. Probab, 43 (2015), 339-404.
doi: 10.1214/14-AOP907. |
[6] |
N. Anantharaman,
On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics, J. Eur. Math. Soc. (JEMS), 6 (2004), 207-276.
|
[7] |
M. T. Barlow and R. F. Bass,
The construction of Brownian motion on the Sierpinski carpet, Ann. IHP, 25 (1989), 225-257.
|
[8] |
M. T. Barlow and E. A. Perkins,
Brownian motion on the Sierpiski gasket, Probab. Th. Rel. Fields, 79 (1988), 543-623.
doi: 10.1007/BF00318785. |
[9] |
N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Spaces, Berlin, 1991.
doi: 10.1515/9783110858389. |
[10] | |
[11] |
G. Da Prato, Introduction to Stochastic Differential Equations, SNS, Pisa, 1995. |
[12] |
J. Feng and T. Nguyen,
Hamilton-Jacobi equations in space of measures associated with a system of conservation laws, Journal de Mathématiques pures et Appliquées, 97 (2012), 318-390.
doi: 10.1016/j.matpur.2011.11.004. |
[13] |
W. H. Fleming,
The Cauchy problem for a Nonlinear first order Partial Differential Equation, JDE, 5 (1969), 515-530.
doi: 10.1016/0022-0396(69)90091-6. |
[14] |
M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, De Gruyter, Göttingen, 2011. |
[15] |
R. Jordan, D. Kinderleher and F. Otto,
The variational formulation of the Fokker-Planck equation, SIAM Journal on Mathematical Analysis,, 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
[16] |
T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1980. |
[17] |
U. Mosco,
Composite media and asymptotic Dirichlet forms, J. Functional Analysis, 123 (1994), 368-421.
doi: 10.1006/jfan.1994.1093. |
[18] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[19] | |
[20] |
T. K.-Sturm,
Metric measure spaces with variable Ricci bounds and couplings of Brownian motions, in Festschrift Masatoshi Fukushima, Hackensack, NJ, (2015), 553-575.
doi: 10.1142/9789814596534_0027. |
[21] |
C. Villani, Topics in Optimal Transportation, Providence, R. I. , 2003.
doi: 10.1007/b12016. |
show all references
References:
[1] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows, Birkhäuser, Basel, 2005. |
[2] |
L. Ambrosio, N. Gigli and G. Savaré, Heat flow and calculus on metric measure spaces with Ricci curvature bounded below -the compact case, in Analysis and Numerics of Partial Differential Equations, Springer, Milano, 2013, 63-115, .
doi: 10.1007/978-88-470-2592-9_8. |
[3] |
L. Ambrosio, N. Gigli and G. Savaré,
Calculus and heat flows in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math., 195 (2014), 289-391.
doi: 10.1007/s00222-013-0456-1. |
[4] |
L. Ambrosio, N. Gigli and G. Savaré,
Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J., 163 (2014), 1405-1490.
doi: 10.1215/00127094-2681605. |
[5] |
L. Ambrosio, N. Gigli and G. Savaré,
Bakry-Emery curvature-dimension condition and Rie-Émannian Ricci curvature bounds, Ann. Probab, 43 (2015), 339-404.
doi: 10.1214/14-AOP907. |
[6] |
N. Anantharaman,
On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics, J. Eur. Math. Soc. (JEMS), 6 (2004), 207-276.
|
[7] |
M. T. Barlow and R. F. Bass,
The construction of Brownian motion on the Sierpinski carpet, Ann. IHP, 25 (1989), 225-257.
|
[8] |
M. T. Barlow and E. A. Perkins,
Brownian motion on the Sierpiski gasket, Probab. Th. Rel. Fields, 79 (1988), 543-623.
doi: 10.1007/BF00318785. |
[9] |
N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Spaces, Berlin, 1991.
doi: 10.1515/9783110858389. |
[10] | |
[11] |
G. Da Prato, Introduction to Stochastic Differential Equations, SNS, Pisa, 1995. |
[12] |
J. Feng and T. Nguyen,
Hamilton-Jacobi equations in space of measures associated with a system of conservation laws, Journal de Mathématiques pures et Appliquées, 97 (2012), 318-390.
doi: 10.1016/j.matpur.2011.11.004. |
[13] |
W. H. Fleming,
The Cauchy problem for a Nonlinear first order Partial Differential Equation, JDE, 5 (1969), 515-530.
doi: 10.1016/0022-0396(69)90091-6. |
[14] |
M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, De Gruyter, Göttingen, 2011. |
[15] |
R. Jordan, D. Kinderleher and F. Otto,
The variational formulation of the Fokker-Planck equation, SIAM Journal on Mathematical Analysis,, 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
[16] |
T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1980. |
[17] |
U. Mosco,
Composite media and asymptotic Dirichlet forms, J. Functional Analysis, 123 (1994), 368-421.
doi: 10.1006/jfan.1994.1093. |
[18] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[19] | |
[20] |
T. K.-Sturm,
Metric measure spaces with variable Ricci bounds and couplings of Brownian motions, in Festschrift Masatoshi Fukushima, Hackensack, NJ, (2015), 553-575.
doi: 10.1142/9789814596534_0027. |
[21] |
C. Villani, Topics in Optimal Transportation, Providence, R. I. , 2003.
doi: 10.1007/b12016. |
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