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

The stochastic value function in metric measure spaces

Ugo Bessi

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

[7]

M. T. Barlow and R. F. Bass, The construction of Brownian motion on the Sierpinski carpet, Ann. IHP, 25 (1989), 225-257.   Google Scholar

[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.  Google Scholar

[9]

N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Spaces, Berlin, 1991. doi: 10.1515/9783110858389.  Google Scholar

[10]

H. Brezis, Analisi Funzionale, Liguori, Napoli, 1986. Google Scholar

[11]

G. Da Prato, Introduction to Stochastic Differential Equations, SNS, Pisa, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, De Gruyter, Göttingen, 2011.  Google Scholar

[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.  Google Scholar

[16]

T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1980.  Google Scholar

[17]

U. Mosco, Composite media and asymptotic Dirichlet forms, J. Functional Analysis, 123 (1994), 368-421.  doi: 10.1006/jfan.1994.1093.  Google Scholar

[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.  Google Scholar

[19]

W. Rudin, Real and Complex Analysis, New Delhi, 1983 Google Scholar

[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.  Google Scholar

[21]

C. Villani, Topics in Optimal Transportation, Providence, R. I. , 2003. doi: 10.1007/b12016.  Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows, Birkhäuser, Basel, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[7]

M. T. Barlow and R. F. Bass, The construction of Brownian motion on the Sierpinski carpet, Ann. IHP, 25 (1989), 225-257.   Google Scholar

[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.  Google Scholar

[9]

N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Spaces, Berlin, 1991. doi: 10.1515/9783110858389.  Google Scholar

[10]

H. Brezis, Analisi Funzionale, Liguori, Napoli, 1986. Google Scholar

[11]

G. Da Prato, Introduction to Stochastic Differential Equations, SNS, Pisa, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, De Gruyter, Göttingen, 2011.  Google Scholar

[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.  Google Scholar

[16]

T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1980.  Google Scholar

[17]

U. Mosco, Composite media and asymptotic Dirichlet forms, J. Functional Analysis, 123 (1994), 368-421.  doi: 10.1006/jfan.1994.1093.  Google Scholar

[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.  Google Scholar

[19]

W. Rudin, Real and Complex Analysis, New Delhi, 1983 Google Scholar

[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.  Google Scholar

[21]

C. Villani, Topics in Optimal Transportation, Providence, R. I. , 2003. doi: 10.1007/b12016.  Google Scholar

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