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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.   Google Scholar

[2]

B. BergéI. 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.  Google Scholar

[3]

S. BernfeldY. 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.  Google Scholar

[4]

S. Bernstein, Sur les liaisons entre les grandeurs aléatoires, Verhandlungen des Internationalen Mathematikerkongress, 1 (1932), 288-309.   Google Scholar

[5]

I. D. Chueshov, Monotone Random Systems -Theory and Applications, Lecture Notes in Mathematics, 1779, Springer Verlag, New York, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1990.  Google Scholar

[11]

A. Erdélyi, W. Magnus, F. Oberhettinger and G. Tricomi, Higher Transcendental Functions, II, McGraw-Hill, Inc., New York, 1953. Google Scholar

[12] A. Galichon, Optimal Transport Methods in Economics, Princeton University Press, Princeton, 2016.   Google Scholar
[13]

B. Jamison, Reciprocal processes, Zeitschrift f ür Wahrscheinlichkeitstheorie und Verwandte Gebiete, 30 (1974), 65-86.  doi: 10.1007/BF00532864.  Google Scholar

[14]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, 113, Springer Verlag, New York, 1991.  Google Scholar

[15]

A. Messiah, Quantum Mechanics, Dover Books on Physics, Dover, 2014. Google Scholar

[16] M. Reed and B. Simon, Methods of Modern Mathematical Physics Ⅳ: Analysis of Operators, Academic Press, New York, 1978.   Google Scholar
[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.  Google Scholar

[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.   Google Scholar

[19]

C. Villani, Optimal Transport: Old and New, Grundlehren der Mathematischen Wissenschaften, 338, Springer Verlag, New York, 2009.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[2]

B. BergéI. 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.  Google Scholar

[3]

S. BernfeldY. 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.  Google Scholar

[4]

S. Bernstein, Sur les liaisons entre les grandeurs aléatoires, Verhandlungen des Internationalen Mathematikerkongress, 1 (1932), 288-309.   Google Scholar

[5]

I. D. Chueshov, Monotone Random Systems -Theory and Applications, Lecture Notes in Mathematics, 1779, Springer Verlag, New York, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1990.  Google Scholar

[11]

A. Erdélyi, W. Magnus, F. Oberhettinger and G. Tricomi, Higher Transcendental Functions, II, McGraw-Hill, Inc., New York, 1953. Google Scholar

[12] A. Galichon, Optimal Transport Methods in Economics, Princeton University Press, Princeton, 2016.   Google Scholar
[13]

B. Jamison, Reciprocal processes, Zeitschrift f ür Wahrscheinlichkeitstheorie und Verwandte Gebiete, 30 (1974), 65-86.  doi: 10.1007/BF00532864.  Google Scholar

[14]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, 113, Springer Verlag, New York, 1991.  Google Scholar

[15]

A. Messiah, Quantum Mechanics, Dover Books on Physics, Dover, 2014. Google Scholar

[16] M. Reed and B. Simon, Methods of Modern Mathematical Physics Ⅳ: Analysis of Operators, Academic Press, New York, 1978.   Google Scholar
[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.  Google Scholar

[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.   Google Scholar

[19]

C. Villani, Optimal Transport: Old and New, Grundlehren der Mathematischen Wissenschaften, 338, Springer Verlag, New York, 2009.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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