Stochastic optimal transport revisited

Tatsuya Mikami1
1Department of Mathematics, Tsuda University, 2-1-1 Tsuda-machi, Kodaira, Tokyo, 187-8577, Japan

Tóm tắt

Từ khóa


Tài liệu tham khảo

Ambrosio, L.: Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158, 227–260 (2004)

Ambrosio, L., Trevisan, D.: Well-posedness of Lagrangian flows and continuity equations in metric measure spaces. Anal. PDE 7(5), 1179–1234 (2014)

Aronson, D.G.: Bounds on the fundamental solution of a parabolic equation. Bull. Am. Math. Soc. 73, 890–896 (1967)

Bogachev, V.I., Krylov, N.V., Röckner, M.: Elliptic and parabolic equations for measures. Russ. Math. Surv. 64(6), 973–1078 (2009)

Bogachev, V. I., Röckner, M., Shaposhnikov, S. V.: On the Ambrosio–Figalli–Trevisan superposition principle for probability solutions to Fokker–Planck–Kolmogorov equations. J. Dyn. Differ. Equ. (2020)

Cacoullos, T., Papathanasiou, V., Utev, S.A.: Another characterization of the normal law and a proof of the central limit theorem connected with it. Theory Probab. Appl. 37, 581–588 (1992)

Cacoullos, T., Papathanasiou, V., Utev, S.A.: Variational inequalities with examples and an application to the central limit theorem. Ann. Probab. 22, 1607–1618 (1994)

Carlen, E.A.: Conservative diffusions. Commun. Math. Phys. 94, 293–315 (1984)

Carlen, E. A.: Existence and sample path properties of the diffusions in Nelson’s stochastic mechanics. In: Albeverio, S., Blanchard, Ph., Streit, L. (eds.) Stochastic processes-Mathematics and Physics, Bielefeld 1984, Lecture Notes in Math., Vol. 1158, pp. 25-51. Springer, Heidelberg (1986)

Carmona, R.: Probabilistic construction of Nelson processes. In: Itô, K., Ikeda, N. (eds.) Proc. Probabilistic Methods in Mathematical Physics, Katata 1985, pp. 55–81. Kinokuniya, Tokyo (1987)

Cattiaux, P., Léonard, C.: Minimization of the Kullback information of diffusion processes. Ann. Inst. H Poincaré Probab. Stat. 30, 83–132 (1994)

Cattiaux, P., Léonard, C.: Correction to: Minimization of the Kullback information of diffusion processes [Ann. Inst. H. Poincaré Probab. Statist. 30 (1994), no. 1, 83–132]. Ann Inst H Poincaré Probab Statist 31, 705–707 (1995)

Cattiaux, P., Léonard, C.: Large deviations and Nelson processes. Forum Math. 7, 95–115 (1995)

Cattiaux, P., Léonard, C.: Minimization of the Kullback information for some Markov processes. In: Azema, J. et al. (eds.) Séminaire de Probabilités, XXX, Lecture Notes in Math., Vol. 1626, pp. 288–311. Springer, Heidelberg (1996)

Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992)

Dai Pra, P.: A stochastic control approach to reciprocal diffusion processes. Appl. Math. Optim. 23, 313–329 (1991)

Dall’Aglio, G.: Sugli estremi dei momenti delle funzioni di ripartizione doppie. Ann. Scuola Normale Superiore Di Pisa, Cl. Sci. 3(1), 33–74 (1956)

Dupuis, P., Ellis, R.S.: A Weak Convergence Approach to the Theory of Large Deviations. John Wiley & Sons, New York (1997)

Figalli, A.: Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254, 109–153 (2008)

Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Springer, New York (1993)

Föllmer, H.: Random fields and diffusion processes. In: Hennequin, PL (ed.) École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87, Lecture Notes in Math., Vol. 1362, pp. 101–203. Springer, Heidelberg (1988)

Friedman, A.: Partial Differential Equations of Parabolic Type. Dover Publications, New York (2013)

Gomes, D.A.: A stochastic analogue of Aubry-Mather theory. Nonlinearity 15, 581–603 (2002)

Gomes, D. A., Mitake, H, Tran, H. V.: The large time profile for Hamilton–Jacobi–Bellman equations. arXiv:2006.04785

Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland/Kodansha, Tokyo (1981)

Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland, Amsterdam (1979)

Jamison, B.: Reciprocal processes. Z. Wahrsch. Verw. Gebiete 30, 65–86 (1974)

Jamison, B.: The Markov process of Schrödinger. Z. Wahrsch. Verw. Gebiete 32, 323–331 (1975)

Koike, S.: A beginner’s guide to the theory of viscosity solutions. MSJ Memoirs, Vol. 13. Math. Soc. Japan., Tokyo (2004)

Léonard, C. : A survey of the Schrödinger problem and some of its connections with optimal transport. Special Issue on Optimal Transport and Applications. Discr. Contin. Dyn. Syst. 34, 1533–1574 (2014)

Liptser, R.S., Shiryaev, A.N.: Statistics of Random Processes I. Springer, Heidelberg (1977)

McCann, R.J.: A convexity principle for interacting gases. Adv. Math. 128, 153–179 (1997)

Mikami, T.: Variational processes from the weak forward equation. Commun. Math. Phys. 135, 19–40 (1990)

Mikami, T.: Equivalent conditions on the central limit theorem for a sequence of probability measures on $$\mathbb{R}$$. Stat. Probab. Lett. 37, 237–242 (1998)

Mikami, T.: Markov marginal problems and their applications to Markov optimal control. In: McEneaney, W. M. etal. (eds.) Stochastic Analysis, Control, Optimization and Applications, A Volume in Honor of W. H. Fleming, pp. 457-476. Birkhäuser, Boston (1999)

Mikami, T.: Dynamical systems in the variational formulation of the Fokker–Planck equation by the Wasserstein metric. Appl. Math. Optim. 42, 203–227 (2000)

Mikami, T.: Optimal control for absolutely continuous stochastic processes and the mass transportation problem. Elect. Commun. Probab. 7, 199–213 (2002)

Mikami, T.: Monge’s problem with a quadratic cost by the zero-noise limit of $$h$$-path processes. Probab. Theory Related Fields 129, 245–260 (2004)

Mikami, T.: Covariance kernel and the central limit theorem in the total variation distance. J. Multivar. Anal. 90, 257–268 (2004)

Mikami, T.: Semimartingales from the Fokker–Planck equation. Appl. Math. Optim. 53, 209–219 (2006)

Mikami, T.: Marginal problem for semimartingales via duality. In: Giga, Y., Ishii, K., Koike, S. et al. (eds) International Conference for the 25th Anniversary of Viscosity Solutions, Gakuto International Series. Mathematical Sciences and Applications 30, pp. 133–152. Gakkotosho, Tokyo (2008)

Mikami, T.: Regularity of Schrödinger’s functional equation and mean field PDEs for h-path processes. Osaka J. Math. 56, 831–842 (2019)

Mikami, T.: Regularity of Schrödinger’s functional equation in the weak topology and moment measures. J. Math. Soc. Jpn. 73, 99–123 (2021)

Mikami, T.: Stochastic optimal transportation. A book in preparation

Mikami, T., Thieullen, M.: Duality theorem for stochastic optimal control problem. Stoc. Proc. Appl. 116, 1815–1835 (2006)

Nagasawa, M.: Transformations of diffusion and Schrödinger process. Probab. Theory Related Fields 82, 109–136 (1989)

Nagasawa, M.: Stochastic Processes in Quantum Physics (Monographs in Mathematics 94). Birkhaüser, Basel (2000)

Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Springer, Heidelberg (2006)

Nelson, E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton (1967)

Nelson, E.: Quantum Fluctuations. Princeton University Press, Princeton (1984)

Rachev, S. T., Rüschendorf, L.: Mass transportation problems, Vol. I: Theory, Vol. II: Application. Springer, Heidelberg (1998)

Röckner, M., Xie, L., Zhang, X.: Superposition principle for non-local Fokker-Planck operators. Probab. Theory Related Fields 178, 699–733 (2020)

Rüschendorf, L., Thomsen, W.: Note on the Schrödinger equation and $$I$$-projections. Statist. Probab. Lett. 17, 369–375 (1993)

Santambrogio, F.: Dealing with moment measures via entropy and optimal transport. J. Funct. Anal. 271, 418–436 (2016)

Schrödinger, E.: Ueber die Umkehrung der Naturgesetze. Sitz. Ber. der Preuss. Akad. Wissen., Berlin, Phys. Math. pp. 144–153 (1931)

Schrödinger, E.: Théorie relativiste de l’electron et l’interprétation de la mécanique quantique. Ann. Inst. H. Poincaré 2, 269–310 (1932)

Schweizer, B., Sklar, A.: Probabilistic Metric Space. Dover Publications, New York (2005)

Sheu, S.J.: Some estimates of the transition density of a nondegenerate diffusion Markov processes. Ann. Probab. 19, 538–561 (1991)

Tan, X., Touzi, N.: Optimal transportation under controlled stochastic dynamics. Ann. Probab. 41, 3201–3240 (2013)

Trevisan, D.: Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients. Electron J. Probab. 21, 1–41 (2016)

Villani, C.: Topics in Optimal Transportation. American Mathematics Society, Providence, RI (2003)

Zambrini, J. C.: Variational processes. In: Albeverio, S. etal. (eds.) Stochastic processes in classical and quantum systems, Ascona 1985, Lecture Notes in Phys., Vol. 262., pp. 517–529. Springer, Heidelberg (1986)

Zheng, W.A.: Tightness results for laws of diffusion processes application to stochastic mechanics. Ann. Inst. Henri Poincaré 21, 103–124 (1985)