Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation
Download PDF
Download PDF
  • Published: June 1995

A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation

  • Peter Kotelenez1 

Probability Theory and Related Fields volume 102, pages 159–188 (1995)Cite this article

  • 380 Accesses

  • 50 Citations

  • Metrics details

Summary

A system ofN particles inR d with mean field interaction and diffusion is considered. Assuming adiabatic elimination of the momenta the positions satisfy a stochastic ordinary differential equation driven by Brownian sheets (microscopic equation), where all coefficients depend on the position of the particles and on the empirical mass distribution process. This empirical mass distribution process satisfies a quasilinear stochastic partial differential equation (SPDE). This SPDE (mezoscopic equation) is solved for general measure valued initial conditions by “extending” the empirical mass distribution process from point measure valued initial conditions with total mass conservation. Starting with measures with densities inL 2(R d,dr), wheredr is the Lebesgue measure, the solution will have densities inL 2(R d,dr) and strong uniqueness (in the Itô sense) is obtained. Finally, it is indicated how to obtain (macroscopic) partial differential equations as limits of the so constructed SPDE's.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Borkar, V.S.: Evolution of interacting particles in a Brownian medium. Stochastics14, 33–79 (1984)

    Google Scholar 

  2. Dalecky, Yu.L., Goncharuk, N.Yu.: On a quasilinear stochastic differential equation of parabolic type. Stoch. Anal. Appl.12 (1), 103–129 (1994)

    Google Scholar 

  3. Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. Cambridge: Cambridge University Press. 1992

    Google Scholar 

  4. Davies, E.B.: One-parameter semigroups. London, New York: Academic Press 1980

    Google Scholar 

  5. Dawson, D.A.: Stochastic evolution equations and related measure processes. J. Multivar. Anal.5, 1–52 (1975)

    Google Scholar 

  6. Dawson, D.A., Vaillancourt, J.: Stochastic McKean-Vlasov Equations. (Preprint-Technical Report No. 242, Carleton University Lab. Stat. Probab. 1994

  7. De Acosta, A.: Invariance principles in probability for triangular arrays ofB-valued random vectors and some applications. Ann. Probab.2, 346–373 (1982)

    Google Scholar 

  8. Dudley, R.M.: Real analysis and probability. Belmont, California: Wadsworth and Brooks 1989

    Google Scholar 

  9. Dynkin, E.B.: Markov processes. Vol. I. Berlin Heidelberg New York: Springer 1965

    Google Scholar 

  10. Fife, P.: Models for phase separation and their mathematics. In: Nonlinear Partial Differential Equations and Applications. Mimura, M., Nishida, T., (eds.) Tokyo: Kinokuniya Pubs., to appear

  11. Gärtner, J.: On the McKean-Vlasov limit for interacting diffusions. Math. Nachr.187, 197–248 (1988)

    Google Scholar 

  12. Il'in, A.M., Khasminskii, R.Z.: On equations of Brownian motion. Probab. Theory, Appl., Vol. IX, No. 3, (1964) (in Russian)

  13. Kotelenez, P.: On the semigroup approach to stochastic evolution equations. In: Arnold, L., Kotelenez, P., (eds.): Stochastic space-time models and limit theorems. Dordrecht Reidel, D., pp. 95–139 (1985)

    Google Scholar 

  14. Kotelenez, P.: Existence, uniqueness and smoothness for a class of function valued stochastic partial differential equations. Stochastics and Stochastic Rep.41, 177–199 (1992)

    Google Scholar 

  15. Kotelenez, P.: Comparison methods for a class of function valued stochastic partial differential equations. Probab. Theory Relat. Fields93, 1–19 (1992)

    Google Scholar 

  16. Kotelenez, P.: A stochastic Navier-Stokes equation for the vorticity of a two-dimensional fluid. (Preprint #92-115, Case Western Reserve University)

  17. Kotelenez, P., Wang, K.: Newtonian particle mechanics and stochastic partial differential equations. In: Dawson, D.A., (ed.) Measure Valued Processes, Stochastic Partial Differential Equations and Interacting Systems. Centre de Recherche Mathématiques, CRM Proceedings and Lecture Notes, Vol. 5, pp. 139–149 (1994)

  18. Krylov, N.V., Rozovskii, B.L.: On stochastic evolution eqaations. Itogi Nauki i tehniki, VINITI, 71–146 (1979)

  19. Lebowitz, J.L., Rubin, E.: Dynamical study of Brownian motion Phys. Rev.131, (6) 2381–2396 (1963)

    Google Scholar 

  20. Marchioro, C., Pulvirenti, M.: Hydrodynamics in two dimensions and vortex theory. Comm. Math. Phys.84, 483–503 (1982)

    Google Scholar 

  21. Metivier, M., Pellaumail, J.: Stochastic integration. New York: Academic Press 1980

    Google Scholar 

  22. Nelson, E.: Dynamical theories of Brownian motion. Princeton, N.J.: Princeton University Press 1972

    Google Scholar 

  23. Pardoux, E.: Equations aux derivees partielles stochastique non linearies monotones. Etude de solutions fortes de type Itô. These (1975)

  24. Triebel, H.: Interpolation theory, function spaces, differential operators. Berlin: VEB Deutscher Verlag der Wissenschaften 1978

    Google Scholar 

  25. Vaillancourt, J.: On the existence of random McKean-Vlasov limits for triangular arrays of exchangeable diffusions. Stoch. Anal. Appl. 1988

  26. Walsh, J.B.: An introduction to stochastic partial differential equations. In: Hennequin P.L. (ed.) Ecole d'Ete de Probabilities de Saint-Flour XIV-1984. Lecture notes in Mathematics 1180. Berlin Heidelberg New York: Springer 1986

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Case Western Reserve University, 44106, Cleveland, Ohio, USA

    Peter Kotelenez

Authors
  1. Peter Kotelenez
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was supported by NSF grant DMS92-11438 and ONR grant N00014-91J-1386

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kotelenez, P. A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation. Probab. Th. Rel. Fields 102, 159–188 (1995). https://doi.org/10.1007/BF01213387

Download citation

  • Received: 18 May 1994

  • Revised: 09 December 1994

  • Issue Date: June 1995

  • DOI: https://doi.org/10.1007/BF01213387

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification

  • 60H15
  • 81G99
  • 35K55
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature