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

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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.

Keywords: Bernstein processes, parabolic partial differential equations.

Mathematics Subject Classification: 35K15, 60G15, 65D15.

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.

show all references

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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