The stochastic value function in metric measure spaces

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April 2017, 37(4): 1819-1839. doi: 10.3934/dcds.2017076

The stochastic value function in metric measure spaces

Dipartimento di Matematica, Università di Roma Tre, Largo S. Leonardo Murialdo 1,00146 Roma, Italy

Received April 2016 Revised November 2016 Published December 2016

Fund Project: Work partially supported by the PRIN2009 grant "Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations.

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 ${\bf{R}}^d$.

Keywords: Viscous Hamilton-Jacobi equation, metric measure spaces.

Mathematics Subject Classification: Primary:49L20, 49L25.

Citation: Ugo Bessi. The stochastic value function in metric measure spaces. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076

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]	H. Brezis, Analisi Funzionale, Liguori, Napoli, 1986.
[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]	W. Rudin, Real and Complex Analysis, New Delhi, 1983
[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]	H. Brezis, Analisi Funzionale, Liguori, Napoli, 1986.
[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]	W. Rudin, Real and Complex Analysis, New Delhi, 1983
[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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