Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Itô and Stratonovich stochastic partial differential equations: Transition from microscopic to macroscopic equations

Author: Peter M. Kotelenez
Journal: Quart. Appl. Math. 66 (2008), 539-564
MSC (2000): Primary 60H10, 60H05, 60H30, 60F17
Published electronically: July 2, 2008
MathSciNet review: 2445528
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information


We review the derivation of stochastic ordinary and quasi-linear stochastic partial differential equations (SODE's and SPDE's) from systems of microscopic deterministic equations in space dimension $ d\geq 2$ as well as the macroscopic limits of the SPDE's. The macroscopic limits are quasi-linear (deterministic) PDE's. Both noncoercive and coercive SPDE's, driven by Itô differentials with respect to correlated Brownian motions, are considered. For the solutions of semi-linear noncoercive SPDE's with smooth and homogeneous diffusion kernels we show that these solutions can be obtained as solutions of first-order SPDE's, driven by Stratonovich differentials and their macroscopic limit, and are solutions of a class of semi-linear second-order parabolic PDE's. Further, the space-time covariance structure of correlated Brownian motions is described and for space dimension $ d\geq 2$ the long-time behavior of the separation of two uncorrelated Brownian motions is shown to be similar to the independent case.

References [Enhancements On Off] (What's this?)

  • 1. Vivek S. Borkar, Evolution of interacting particles in a Brownian medium, Stochastics 14 (1984), no. 1, 33–79. MR 774584, https://doi.org/10.1080/17442508408833331
  • 2. D. A. Dawson, J. Vaillancourt, and H. Wang, Stochastic partial differential equations for a class of interacting measure-valued diffusions, Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 2, 167–180 (English, with English and French summaries). MR 1751657, https://doi.org/10.1016/S0246-0203(00)00121-7
  • 3. Andrey A. Dorogovtsev, One Brownian stochastic flow, Theory Stoch. Process. 10 (2004), no. 3-4, 21–25. MR 2329772
  • 4. D. Dürr, S. Goldstein, and J. L. Lebowitz, A mechanical model of Brownian motion, Comm. Math. Phys. 78 (1980/81), no. 4, 507–530. MR 606461
  • 5. Dynkin, E.B., Markov processes. Vol. I. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965.
  • 6. Albert Einstein, Investigations on the theory of the Brownian movement, Dover Publications, Inc., New York, 1956. Edited with notes by R. Fürth; Translated by A. D. Cowper. MR 0077443
  • 7. Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR 838085
  • 8. Avner Friedman, Stochastic differential equations and applications. Vol. 2, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Probability and Mathematical Statistics, Vol. 28. MR 0494491
  • 9. Jürgen Gärtner, On the McKean-Vlasov limit for interacting diffusions, Math. Nachr. 137 (1988), 197–248. MR 968996, https://doi.org/10.1002/mana.19881370116
  • 10. I. M. Gel′fand and N. Ya. Vilenkin, Generalized functions. Vol. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Applications of harmonic analysis; Translated from the Russian by Amiel Feinstein. MR 0435834
  • 11. Gikhman, I.I. and Skorokhod, A.V., Stochastic differential equations. Naukova Dumka, Kiev (in Russian - English Translation (1972): Stochastic Differential Equations. Springer-Verlag, Berlin).
  • 12. Nataliya Yu. Goncharuk and Peter Kotelenez, Fractional step method for stochastic evolution equations, Stochastic Process. Appl. 73 (1998), no. 1, 1–45. MR 1603842, https://doi.org/10.1016/S0304-4149(97)00079-3
  • 13. Hermann Haken, Advanced synergetics, Springer Series in Synergetics, vol. 20, Springer-Verlag, Berlin, 1983. Instability hierarchies of self-organizing systems and devices. MR 707096
  • 14. Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, 2nd ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. MR 1011252
  • 15. G. Jetschke, On the equivalence of different approaches to stochastic partial differential equations, Math. Nachr. 128 (1986), 315–329. MR 855965, https://doi.org/10.1002/mana.19861280127
  • 16. Kampen, N.G. van, Stochastic Processes in Physics and Chemistry. North-Holland Publ. Co., Amsterdam, New York, 1983.
  • 17. Peter Kotelenez, A stochastic Navier-Stokes equation for the vorticity of a two-dimensional fluid, Ann. Appl. Probab. 5 (1995), no. 4, 1126–1160. MR 1384369
  • 18. Peter Kotelenez, A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation, Probab. Theory Related Fields 102 (1995), no. 2, 159–188. MR 1337250, https://doi.org/10.1007/BF01213387
  • 19. Peter M. Kotelenez, From discrete deterministic dynamics to Brownian motions, Stoch. Dyn. 5 (2005), no. 3, 343–384. MR 2166985, https://doi.org/10.1142/S0219493705001511
  • 20. P. Kotelenez, Correlated Brownian motions as an approximation to deterministic mean-field dynamics, Ukraïn. Mat. Zh. 57 (2005), no. 6, 757–769 (English, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 57 (2005), no. 6, 900–912. MR 2208453, https://doi.org/10.1007/s11253-005-0238-z
  • 21. Kotelenez, P., Stochastic Ordinary and Stochastic Partial Differential Equations - Transition from Microscopic to Macroscopic Equations. Springer-Verlag, Berlin-Heidelberg-New York, 2007.
  • 22. Kotelenez, P., Leitman M. and Mann, J. Adin Jr., On the Depletion Effect in Colloids. Preprint.
  • 23. Kotelenez, P. and Kurtz, T.G., Macroscopic Limit for Stochastic Partial Differential Equations of McKean-Vlasov Type. (Preprint)
  • 24. Krylov, N.V., Private Communication.
  • 25. Krylov, N.V. and Rozovsky, B.L., On stochastic evolution equations. Itogi Nauki i tehniki, VINITI, 71-146, 1979 (in Russian).
  • 26. Hiroshi Kunita, Stochastic flows and stochastic differential equations, Cambridge Studies in Advanced Mathematics, vol. 24, Cambridge University Press, Cambridge, 1997. Reprint of the 1990 original. MR 1472487
  • 27. Thomas G. Kurtz and Jie Xiong, Particle representations for a class of nonlinear SPDEs, Stochastic Process. Appl. 83 (1999), no. 1, 103–126. MR 1705602, https://doi.org/10.1016/S0304-4149(99)00024-1
  • 28. Lifshits, E.M. and Pitayevskii, L.P., Physical Kinetics. Theoretical Physics X. Nauka, Moscow, 1979 (in Russian).
  • 29. Metivier, M. and Pellaumail, J., Stochastic Integration. Adademic Press, New York, 1980.
  • 30. Oelschläger, K., A Martingale Approach to the Law of Large Numbers for Weakly Interacting Stochastic Processes. Ann. Probab. 12 (1984), 458-479.
  • 31. E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics 3 (1979), no. 2, 127–167. MR 553909, https://doi.org/10.1080/17442507908833142
  • 32. B. L. Rozovskiĭ, Stochastic evolution systems, Mathematics and its Applications (Soviet Series), vol. 35, Kluwer Academic Publishers Group, Dordrecht, 1990. Linear theory and applications to nonlinear filtering; Translated from the Russian by A. Yarkho. MR 1135324
  • 33. Ya. G. Sinaĭ and M. R. Soloveĭchik, One-dimensional classical massive particle in the ideal gas, Comm. Math. Phys. 104 (1986), no. 3, 423–443. MR 840745
  • 34. Domokos Szász and Bálint Tóth, Towards a unified dynamical theory of the Brownian particle in an ideal gas, Comm. Math. Phys. 111 (1987), no. 1, 41–62. MR 896758
  • 35. C. Truesdell and W. Noll, The nonlinear field theories of mechanics, 2nd ed., Springer-Verlag, Berlin, 1992. MR 1215940
  • 36. Jean Vaillancourt, On the existence of random McKean-Vlasov limits for triangular arrays of exchangeable diffusions, Stochastic Anal. Appl. 6 (1988), no. 4, 431–446. MR 964251
  • 37. John B. Walsh, An introduction to stochastic partial differential equations, École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 265–439. MR 876085, https://doi.org/10.1007/BFb0074920
  • 38. Eugene Wong and Moshe Zakai, On the relation between ordinary and stochastic differential equations, Internat. J. Engrg. Sci. 3 (1965), 213–229 (English, with French, German, Italian and Russian summaries). MR 0183023

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 60H10, 60H05, 60H30, 60F17

Retrieve articles in all journals with MSC (2000): 60H10, 60H05, 60H30, 60F17

Additional Information

Peter M. Kotelenez
Affiliation: Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106
Email: pxk4@po.cwru.edu

DOI: https://doi.org/10.1090/S0033-569X-08-01102-6
Received by editor(s): May 15, 2007
Published electronically: July 2, 2008
Article copyright: © Copyright 2008 Brown University
The copyright for this article reverts to public domain 28 years after publication.

Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website