• Previous Article
    Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions
  • DCDS-B Home
  • This Issue
  • Next Article
    Viral infection model with diffusion and state-dependent delay: Stability of classical solutions
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. 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.

[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.

[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. 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.

[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.

[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.

[1]

Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231

[2]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221

[3]

Alan E. Lindsay. An asymptotic study of blow up multiplicity in fourth order parabolic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 189-215. doi: 10.3934/dcdsb.2014.19.189

[4]

Diogo A. Gomes, Gabriele Terrone. Bernstein estimates: weakly coupled systems and integral equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 861-883. doi: 10.3934/cpaa.2012.11.861

[5]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703

[6]

Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053

[7]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515

[8]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167

[9]

Herbert Koch. Partial differential equations with non-Euclidean geometries. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 481-504. doi: 10.3934/dcdss.2008.1.481

[10]

Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032

[11]

Dong Sun, V. S. Manoranjan, Hong-Ming Yin. Numerical solutions for a coupled parabolic equations arising induction heating processes. Conference Publications, 2007, 2007 (Special) : 956-964. doi: 10.3934/proc.2007.2007.956

[12]

Wolfgang Walter. Nonlinear parabolic differential equations and inequalities. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 451-468. doi: 10.3934/dcds.2002.8.451

[13]

Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1345-1360. doi: 10.3934/cpaa.2011.10.1345

[14]

Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1131-1143. doi: 10.3934/cpaa.2007.6.1131

[15]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295

[16]

Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031

[17]

Frédéric Gibou, Doron Levy, Carlos Cárdenas, Pingyu Liu, Arthur Boyer. Partial Differential Equations-Based Segmentation for Radiotherapy Treatment Planning. Mathematical Biosciences & Engineering, 2005, 2 (2) : 209-226. doi: 10.3934/mbe.2005.2.209

[18]

Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227

[19]

Seyedeh Marzieh Ghavidel, Wolfgang M. Ruess. Flow invariance for nonautonomous nonlinear partial differential delay equations. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2351-2369. doi: 10.3934/cpaa.2012.11.2351

[20]

Antonio Cañada, Salvador Villegas. Lyapunov inequalities for partial differential equations at radial higher eigenvalues. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 111-122. doi: 10.3934/dcds.2013.33.111

2016 Impact Factor: 0.994

Metrics

  • PDF downloads (51)
  • HTML views (171)
  • Cited by (0)

Other articles
by authors

[Back to Top]