Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Doubly probabilistic representation for the stochastic porous media type equation

Viorel Barbu, Michael Röckner, and Francesco Russo

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The purpose of the present paper consists in proposing and discussing a doubly probabilistic representation for a stochastic porous media equation in the whole space $\mathbb{R}^{1}$ perturbed by a multiplicative colored noise. For almost all random realizations $\omega$, one associates a stochastic differential equation in law with random coefficients, driven by an independent Brownian motion.

Résumé

Cet article propose et discute une représentation doublement probabiliste pour une équation des milieux poreux stochatique dans l’espace tout entier $\mathbb{R}^{1}$, perturbée par un bruit multiplicatif coloré. Pour presque toute réalisation $\omega$ de l’aléa, on associe une équation différentielle stochastique en loi avec coefficients aléatoires, dirigée par un mouvement brownien indépendant.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 2043-2073.

Dates
Received: 15 May 2014
Revised: 4 August 2016
Accepted: 5 August 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1511773738

Digital Object Identifier
doi:10.1214/16-AIHP783

Mathematical Reviews number (MathSciNet)
MR3729647

Zentralblatt MATH identifier
06847074

Subjects
Primary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 60H15: Stochastic partial differential equations [See also 35R60] 60H30: Applications of stochastic analysis (to PDE, etc.) 60H10: Stochastic ordinary differential equations [See also 34F05] 60G46: Martingales and classical analysis 35C99: None of the above, but in this section 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 82C31: Stochastic methods (Fokker-Planck, Langevin, etc.) [See also 60H10]

Keywords
Stochastic partial differential equations Infinite volume Singular porous media type equation Doubly probabilistic representation Multiplicative noise Singular random Fokker–Planck type equation Filtering

Citation

Barbu, Viorel; Röckner, Michael; Russo, Francesco. Doubly probabilistic representation for the stochastic porous media type equation. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 2043--2073. doi:10.1214/16-AIHP783. https://projecteuclid.org/euclid.aihp/1511773738


Export citation

References

  • [1] V. Barbu. Analysis and Control of Nonlinear Infinite-Dimensional Systems. Mathematics in Science and Engineering. 190. Academic Press Inc., Boston, MA, 1993.
  • [2] V. Barbu. Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monographs in Mathematics. Springer, New York, 2010.
  • [3] V. Barbu, G. Da Prato and M. Röckner. Existence and uniqueness of nonnegative solutions to the stochastic porous media equation. Indiana Univ. Math. J. 57 (1) (2008) 187–211.
  • [4] V. Barbu, G. Da Prato and M. Röckner. Existence of strong solutions for stochastic porous media equation under general monotonicity conditions. Ann. Probab. 37 (2) (2009) 428–452.
  • [5] V. Barbu, G. Da Prato and M. Röckner. Stochastic porous media equations and self-organized criticality. Comm. Math. Phys. 285 (3) (2009) 901–923.
  • [6] V. Barbu, M. Röckner and F. Russo. Probabilistic representation for solutions of an irregular porous media type equation: The degenerate case. Probab. Theory Related Fields 151 (1–2) (2011) 1–43.
  • [7] V. Barbu, M. Röckner and F. Russo. A stochastic Fokker–Planck equation and double probabilistic representation for the stochastic porous media type equation. Preprint, 2014. Available at arXiv:1404.5120.
  • [8] V. Barbu, M. Röckner and F. Russo. Stochastic porous media equations in $\mathbb{R}^{d}$. J. Math. Pures Appl. (9) 103 (4) (2015) 1024–1052.
  • [9] N. Belaribi, F. Cuvelier and F. Russo. A probabilistic algorithm approximating solutions of a singular pde of porous media type. Monte Carlo Methods Appl. 17 (4) (2011) 317–369.
  • [10] N. Belaribi and F. Russo. Uniqueness for Fokker–Planck equations with measurable coefficients and applications to the fast diffusion equation. Electron. J. Probab. 17 (84) (2012) 28.
  • [11] S. Benachour, P. Chassaing, B. Roynette and P. Vallois. Processus associés à l’équation des milieux poreux. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (4) (1996) 793–832.
  • [12] P. Benilan, H. Brezis and M. G. Crandall. A semilinear equation in $L^{1}(\mathbb{R}^{N})$. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (4) (1975) 523–555.
  • [13] P. Benilan and M. G. Crandall. The continuous dependence on $\varphi$ of solutions of $u_{t}-\Delta\varphi(u)=0$. Indiana Univ. Math. J. 30 (2) (1981) 161–177.
  • [14] P. Blanchard, M. Röckner and F. Russo. Probabilistic representation for solutions of a porous media type equation. Ann. Probab. 38 (5) (2010) 1870–1900.
  • [15] H. Brezis and M. G. Crandall. Uniqueness of solutions of the initial-value problem for $u_{t}-\Delta\varphi(u)=0$. J. Math. Pures Appl. (9) 58 (2) (1979) 153–163.
  • [16] E. Häusler and H. Luschgy. Stable Convergence and Stable Limit Theorems. Probability Theory and Stochastic Modelling 74. Springer, Cham, 2015.
  • [17] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 2nd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer-Verlag, Berlin, 2003.
  • [18] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Graduate Texts in Mathematics 113. Springer-Verlag, New York, 1991.
  • [19] H. P. McKean Jr. Propagation of chaos for a class of non-linear parabolic equations. In Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967) 41–57. Air Force Office Sci. Res., Arlington, Va, 1967.
  • [20] É. Pardoux. Filtrage non linéaire et équations aux dérivées partielles stochastiques associées. In École d’Été de Probabilités de Saint-Flour XIX–1989 67–163. Lecture Notes in Math. 1464. Springer, Berlin, 1991.
  • [21] C. Prévôt and M. Röckner. A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics 1905. Springer, Berlin, 2007.
  • [22] J. Ren, M. Röckner and F.-Y. Wang. Stochastic generalized porous media and fast diffusion equations. J. Differential Equations 238 (1) (2007) 118–152.
  • [23] B. D. Ripley. The disintegration of invariant measures. Math. Proc. Cambridge Philos. Soc. 79 (2) (1976) 337–341.
  • [24] Röckner and F. Russo. Uniqueness for stochastic Fokker Planck and porous media equations in the sense of distributions. J. Evol. Equ. To appear. Available at http://arxiv.org/abs/1609.00165.
  • [25] F. Russo and P. Vallois. Elements of stochastic calculus via regularization. In Séminaire de Probabilités XL 147–185. Lecture Notes in Math. 1899. Springer, Berlin, 2007.
  • [26] R. E. Showalter. Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Mathematical Surveys and Monographs 49. American Mathematical Society, Providence, RI, 1997.
  • [27] D. W. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Processes. Classics in Mathematics. Springer-Verlag, Berlin, 2006. Reprint of the 1997 edition.
  • [28] J.-L. Vázquez. The Porous Medium Equation. Oxford Mathematical Monographs. Mathematical Theory. The Clarendon Press Oxford University Press, Oxford, 2007.