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May 2018, 23(3): 1073-1090. doi: 10.3934/dcdsb.2018142

## On the time evolution of Bernstein processes associated with a class of parabolic equations

 1 Inst. Élie Cartan de Lorraine, UMR-CNRS 7502, Nancy, France 2 Dept. de Matemática, Universidade de Lisboa, Lisboa, Portugal

Received  January 2017 Revised  May 2017 Published  February 2018

In this article dedicated to the memory of Igor D. Chueshov, I first summarize in a few words the joint results that we obtained over a period of six years regarding the long-time behavior of solutions to a class of semilinear stochastic parabolic partial differential equations. Then, as the beautiful interplay between partial differential equations and probability theory always was close to Igor's heart, I present some new results concerning the time evolution of certain Markovian Bernstein processes naturally associated with a class of deterministic linear parabolic partial differential equations. Particular instances of such processes are certain conditioned Ornstein-Uhlenbeck processes, generalizations of Bernstein bridges and Bernstein loops, whose laws may evolve in space in a non trivial way. Specifically, I examine in detail the time development of the probability of finding such processes within two-dimensional geometric shapes exhibiting spherical symmetry. I also define a Faedo-Galerkin scheme whose ultimate goal is to allow approximate computations with controlled error terms of the various probability distributions involved.

Citation: Pierre-A. Vuillermot. On the time evolution of Bernstein processes associated with a class of parabolic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1073-1090. doi: 10.3934/dcdsb.2018142
##### References:
 [1] D. G. Aronson, Non-negative solutions of linear parabolic equations, Annali della Scuola Normale Superiore di Pisa, 22 (1968), 607-694. [2] B. Bergé, I. D. Chueshov and P. A. Vuillermot, On the behavior of solutions to certain parabolic SPDEs driven by Wiener processes, Stochastic Processes and their Applications, 92 (2001), 237-263. doi: 10.1016/S0304-4149(00)00082-X. [3] S. Bernfeld, Y. Y. Hu and P. A. Vuillermot, Large-time asymptotic equivalence for a class of non-autonomous semilinear parabolic equations, Bulletin des Sciences Mathématiques, 122 (1998), 337-368. doi: 10.1016/S0007-4497(98)80341-2. [4] S. Bernstein, Sur les liaisons entre les grandeurs aléatoires, Verhandlungen des Internationalen Mathematikerkongress, 1 (1932), 288-309. [5] I. D. Chueshov, Monotone Random Systems -Theory and Applications, Lecture Notes in Mathematics, 1779, Springer Verlag, New York, 2002. [6] I. D. Chueshov and P. A. Vuillermot, Long-time behavior of solutions to a class of quasilinear parabolic equations with random coefficients, Annales de l'Institut Henri-Poincaré C, Analyse Non Linéaire, 15 (1998), 191-232. doi: 10.1016/S0294-1449(97)89299-2. [7] I. D. Chueshov and P. A. Vuillermot, Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Stratonovitch's case, Probability Theory and Related Fields, 112 (1998), 149-202. doi: 10.1007/s004400050186. [8] I. D. Chueshov and P. A. Vuillermot, Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Itô's case, Stochastic Analysis and Applications, 18 (2000), 581-615. doi: 10.1080/07362990008809687. [9] I. D. Chueshov and P. A. Vuillermot, Non-random invariant sets for some systems of parabolic stochastic partial differential equations, Stochastic Analysis and Applications, 22 (2004), 1421-1486. doi: 10.1081/SAP-200029487. [10] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1990. [11] A. Erdélyi, W. Magnus, F. Oberhettinger and G. Tricomi, Higher Transcendental Functions, II, McGraw-Hill, Inc., New York, 1953. [12] A. Galichon, Optimal Transport Methods in Economics, Princeton University Press, Princeton, 2016. [13] B. Jamison, Reciprocal processes, Zeitschrift f ür Wahrscheinlichkeitstheorie und Verwandte Gebiete, 30 (1974), 65-86. doi: 10.1007/BF00532864. [14] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, 113, Springer Verlag, New York, 1991. [15] A. Messiah, Quantum Mechanics, Dover Books on Physics, Dover, 2014. [16] M. Reed and B. Simon, Methods of Modern Mathematical Physics Ⅳ: Analysis of Operators, Academic Press, New York, 1978. [17] S. Roelly and M. Thieullen, A characterisation of reciprocal processes via an integration by parts formula on the path space, Probability Theory and Related Fields, 123 (2002), 97-120. doi: 10.1007/s004400100184. [18] E. Schrödinger, Sur la théorie relativiste de l'électron et l'interprétation de la mécanique quantique, Annales de l'Institut Henri Poincaré, 2 (1932), 269-310. [19] C. Villani, Optimal Transport: Old and New, Grundlehren der Mathematischen Wissenschaften, 338, Springer Verlag, New York, 2009. [20] P. A. Vuillermot, Global exponential attractors for a class of almost-periodic parabolic equations in $\mathbb{R}^{N}$, Proceedings of the American Mathematical Society, 116 (1992), 775-782. [21] P. A. Vuillermot and J. C. Zambrini, Bernstein diffusions for a class of linear parabolic partial differential equations, Journal of Theoretical Probability, 27 (2014), 449-492. doi: 10.1007/s10959-012-0426-3. [22] P. A. Vuillermot and J. C. Zambrini, On some Gaussian Bernstein processes in ${{\mathbb{R}}^{N}}$ and the periodic Ornstein-Uhlenbeck process, Stochastic Analysis and Applications, 34 (2016), 573-597. doi: 10.1080/07362994.2016.1156547.

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##### References:
 [1] D. G. Aronson, Non-negative solutions of linear parabolic equations, Annali della Scuola Normale Superiore di Pisa, 22 (1968), 607-694. [2] B. Bergé, I. D. Chueshov and P. A. Vuillermot, On the behavior of solutions to certain parabolic SPDEs driven by Wiener processes, Stochastic Processes and their Applications, 92 (2001), 237-263. doi: 10.1016/S0304-4149(00)00082-X. [3] S. Bernfeld, Y. Y. Hu and P. A. Vuillermot, Large-time asymptotic equivalence for a class of non-autonomous semilinear parabolic equations, Bulletin des Sciences Mathématiques, 122 (1998), 337-368. doi: 10.1016/S0007-4497(98)80341-2. [4] S. Bernstein, Sur les liaisons entre les grandeurs aléatoires, Verhandlungen des Internationalen Mathematikerkongress, 1 (1932), 288-309. [5] I. D. Chueshov, Monotone Random Systems -Theory and Applications, Lecture Notes in Mathematics, 1779, Springer Verlag, New York, 2002. [6] I. D. Chueshov and P. A. Vuillermot, Long-time behavior of solutions to a class of quasilinear parabolic equations with random coefficients, Annales de l'Institut Henri-Poincaré C, Analyse Non Linéaire, 15 (1998), 191-232. doi: 10.1016/S0294-1449(97)89299-2. [7] I. D. Chueshov and P. A. Vuillermot, Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Stratonovitch's case, Probability Theory and Related Fields, 112 (1998), 149-202. doi: 10.1007/s004400050186. [8] I. D. Chueshov and P. A. Vuillermot, Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Itô's case, Stochastic Analysis and Applications, 18 (2000), 581-615. doi: 10.1080/07362990008809687. [9] I. D. Chueshov and P. A. Vuillermot, Non-random invariant sets for some systems of parabolic stochastic partial differential equations, Stochastic Analysis and Applications, 22 (2004), 1421-1486. doi: 10.1081/SAP-200029487. [10] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1990. [11] A. Erdélyi, W. Magnus, F. Oberhettinger and G. Tricomi, Higher Transcendental Functions, II, McGraw-Hill, Inc., New York, 1953. [12] A. Galichon, Optimal Transport Methods in Economics, Princeton University Press, Princeton, 2016. [13] B. Jamison, Reciprocal processes, Zeitschrift f ür Wahrscheinlichkeitstheorie und Verwandte Gebiete, 30 (1974), 65-86. doi: 10.1007/BF00532864. [14] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, 113, Springer Verlag, New York, 1991. [15] A. Messiah, Quantum Mechanics, Dover Books on Physics, Dover, 2014. [16] M. Reed and B. Simon, Methods of Modern Mathematical Physics Ⅳ: Analysis of Operators, Academic Press, New York, 1978. [17] S. Roelly and M. Thieullen, A characterisation of reciprocal processes via an integration by parts formula on the path space, Probability Theory and Related Fields, 123 (2002), 97-120. doi: 10.1007/s004400100184. [18] E. Schrödinger, Sur la théorie relativiste de l'électron et l'interprétation de la mécanique quantique, Annales de l'Institut Henri Poincaré, 2 (1932), 269-310. [19] C. Villani, Optimal Transport: Old and New, Grundlehren der Mathematischen Wissenschaften, 338, Springer Verlag, New York, 2009. [20] P. A. Vuillermot, Global exponential attractors for a class of almost-periodic parabolic equations in $\mathbb{R}^{N}$, Proceedings of the American Mathematical Society, 116 (1992), 775-782. [21] P. A. Vuillermot and J. C. Zambrini, Bernstein diffusions for a class of linear parabolic partial differential equations, Journal of Theoretical Probability, 27 (2014), 449-492. doi: 10.1007/s10959-012-0426-3. [22] P. A. Vuillermot and J. C. Zambrini, On some Gaussian Bernstein processes in ${{\mathbb{R}}^{N}}$ and the periodic Ornstein-Uhlenbeck process, Stochastic Analysis and Applications, 34 (2016), 573-597. doi: 10.1080/07362994.2016.1156547.
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