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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 |
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.
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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