# American Institute of Mathematical Sciences

• Previous Article
Viral infection model with diffusion and state-dependent delay: Stability of classical solutions
• DCDS-B Home
• This Issue
• Next Article
Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions
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. 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.  Google Scholar [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.  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. 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.  Google Scholar [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.  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
 [1] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [2] Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047 [3] Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073 [4] Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249 [5] Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050 [6] Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321 [7] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [8] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468 [9] Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 16: 331-348. doi: 10.3934/jmd.2020012 [10] Puneet Pasricha, Anubha Goel. Pricing power exchange options with hawkes jump diffusion processes. Journal of Industrial & Management Optimization, 2021, 17 (1) : 133-149. doi: 10.3934/jimo.2019103 [11] Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055 [12] Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252 [13] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [14] Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107 [15] Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 [16] Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297 [17] Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 [18] Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020104 [19] Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323 [20] Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076

2019 Impact Factor: 1.27