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
DOI:
https://doi.org/10.1090/S0033-569X-08-01102-6
Published electronically:
July 2, 2008
MathSciNet review:
2445528
Full-text PDF Free Access
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Abstract: 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
- Vivek S. Borkar, Evolution of interacting particles in a Brownian medium, Stochastics 14 (1984), no. 1, 33–79. MR 774584, DOI https://doi.org/10.1080/17442508408833331
- 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, DOI https://doi.org/10.1016/S0246-0203%2800%2900121-7
- Andrey A. Dorogovtsev, One Brownian stochastic flow, Theory Stoch. Process. 10 (2004), no. 3-4, 21–25. MR 2329772
- 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
- Dynkin, E.B., Markov processes. Vol. I. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965.
- 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
- 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
- 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
- Jürgen Gärtner, On the McKean-Vlasov limit for interacting diffusions, Math. Nachr. 137 (1988), 197–248. MR 968996, DOI https://doi.org/10.1002/mana.19881370116
- 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
- Gikhman, I.I. and Skorokhod, A.V., Stochastic differential equations. Naukova Dumka, Kiev (in Russian - English Translation (1972): Stochastic Differential Equations. Springer-Verlag, Berlin).
- Nataliya Yu. Goncharuk and Peter Kotelenez, Fractional step method for stochastic evolution equations, Stochastic Process. Appl. 73 (1998), no. 1, 1–45. MR 1603842, DOI https://doi.org/10.1016/S0304-4149%2897%2900079-3
- Hermann Haken, Advanced synergetics, Springer Series in Synergetics, vol. 20, Springer-Verlag, Berlin, 1983. Instability hierarchies of self-organizing systems and devices. MR 707096
- 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
- G. Jetschke, On the equivalence of different approaches to stochastic partial differential equations, Math. Nachr. 128 (1986), 315–329. MR 855965, DOI https://doi.org/10.1002/mana.19861280127
- Kampen, N.G. van, Stochastic Processes in Physics and Chemistry. North-Holland Publ. Co., Amsterdam, New York, 1983.
- 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
- 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, DOI https://doi.org/10.1007/BF01213387
- Peter M. Kotelenez, From discrete deterministic dynamics to Brownian motions, Stoch. Dyn. 5 (2005), no. 3, 343–384. MR 2166985, DOI https://doi.org/10.1142/S0219493705001511
- 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, DOI https://doi.org/10.1007/s11253-005-0238-z
- Kotelenez, P., Stochastic Ordinary and Stochastic Partial Differential Equations - Transition from Microscopic to Macroscopic Equations. Springer-Verlag, Berlin-Heidelberg-New York, 2007.
- Kotelenez, P., Leitman M. and Mann, J. Adin Jr., On the Depletion Effect in Colloids. Preprint.
- Kotelenez, P. and Kurtz, T.G., Macroscopic Limit for Stochastic Partial Differential Equations of McKean-Vlasov Type. (Preprint)
- Krylov, N.V., Private Communication.
- Krylov, N.V. and Rozovsky, B.L., On stochastic evolution equations. Itogi Nauki i tehniki, VINITI, 71-146, 1979 (in Russian).
- 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
- 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, DOI https://doi.org/10.1016/S0304-4149%2899%2900024-1
- Lifshits, E.M. and Pitayevskii, L.P., Physical Kinetics. Theoretical Physics X. Nauka, Moscow, 1979 (in Russian).
- Metivier, M. and Pellaumail, J., Stochastic Integration. Adademic Press, New York, 1980.
- Oelschläger, K., A Martingale Approach to the Law of Large Numbers for Weakly Interacting Stochastic Processes. Ann. Probab. 12 (1984), 458-479.
- E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics 3 (1979), no. 2, 127–167. MR 553909, DOI https://doi.org/10.1080/17442507908833142
- 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
- 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
- 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
- C. Truesdell and W. Noll, The nonlinear field theories of mechanics, 2nd ed., Springer-Verlag, Berlin, 1992. MR 1215940
- 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, DOI https://doi.org/10.1080/07362998808809160
- 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, DOI https://doi.org/10.1007/BFb0074920
- 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, DOI https://doi.org/10.1016/0020-7225%2865%2990045-5
References
- Borkar, V.S., Evolution of Interacting Particles in a Brownian Medium. Stochastics 14 (1984), 33-79. MR 774584 (86f:60073)
- Dawson, D.A., Vaillancourt, J., and Wang, H., Stochastic Partial Differential Equations for a Class of Interacting Measure-valued Diffusions. Ann. Inst Henri Poincaré, Probabilités et Statistiques 36 (2000), no. 2, 167-180. MR 1751657 (2002d:60053)
- Dorogovtsev, A., One Brownian Stochastic Flow. Theory of Stochastic Processes, 10 (2004), no. 3-4, 21-25. MR 2329772
- Dürr, D., Goldstein, S. and Lebowitz, J.L., A Mechanical Model of Brownian Motion. Commun. Math. Phys. 78 (1981), 507-530. MR 606461 (83d:60109)
- Dynkin, E.B., Markov processes. Vol. I. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965.
- Einstein, A., Über die von der molekularkinetischen Theorie der Wärme gefordete Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann.d.Phys. 17 (quoted from the English translation: (1956) Investigations on the Theory of Brownian Movement, Dover Publications, Inc., New York. MR 0077443 (17:1035g)
- Ethier, S.N. and Kurtz, T.G., Markov Processes. Characterization and Convergence. John Wiley & Sons, New York - Toronto, 1986. MR 838085 (88a:60130)
- Friedman, A., Stochastic differential equations and applications. Vol. 2, Academic Press. New York-San Francisco-London, 1976. MR 0494491 (58:13350b)
- Gärtner, J.: On the McKean-Vlasov limit for interacting diffusions. Math. Nachr. 137 (1988), 197-248. MR 968996 (90a:60184)
- Gel$’$fand, I.M. and Vilenkin, N. Ya., Generalized Functions. Vol. 4. Academic Press. New York - London, 1964. MR 0435834 (55:8786d)
- Gikhman, I.I. and Skorokhod, A.V., Stochastic differential equations. Naukova Dumka, Kiev (in Russian - English Translation (1972): Stochastic Differential Equations. Springer-Verlag, Berlin).
- Goncharuk, N. and Kotelenez, P., Fractional Step Method for Stochastic Evolution Equations. Stoch. Proc. Appl. 73 (1998), 1-45. MR 1603842 (99b:60090)
- Haken, H., Advanced Synergetics. Springer-Verlag, Berlin-Heidelberg, New York, Tokyo, 1983. MR 707096 (86h:00020)
- Ikeda, N. and Watanabe, S., Stochastic differential equations and diffusion processes. North-Holland Publishing Company. Amsterdam - New York, 1989. MR 1011252 (90m:60069)
- Jetschke, G., On the Equivalence of Different Approaches to Stochastic Partial Differential Equations. Math. Nachr. 128 (1986), 315-329. MR 855965 (87k:60156)
- Kampen, N.G. van, Stochastic Processes in Physics and Chemistry. North-Holland Publ. Co., Amsterdam, New York, 1983.
- Kotelenez, P., A Stochastic Navier-Stokes Equation for the Vorticity of a Two-dimensional Fluid. Ann. Applied Probab. 5 (1995), No. 4, 1126-1160. MR 1384369 (97f:60116)
- Kotelenez, P., A Class of Quasilinear Stochastic Partial Differential Equations of McKean-Vlasov Type with Mass Conservation. Probab. Theory Relat. Fields. 102 (1995), 159-188 MR 1337250 (96k:60157)
- Kotelenez, P., From Discrete Deterministic Dynamics to Stochastic Kinematics - A Derivation of Brownian Motions. Stochastics and Dynamics 5 (2005), Number 3, 343-384. MR 2166985
- Kotelenez, P., Correlated Brownian Motions as an Approximation to Deterministic Mean-Field Dynamics. Ukrainian Math. J. 57 (2005), no. 6, 757-769 MR 2208453 (2006k:82125)
- Kotelenez, P., Stochastic Ordinary and Stochastic Partial Differential Equations - Transition from Microscopic to Macroscopic Equations. Springer-Verlag, Berlin-Heidelberg-New York, 2007.
- Kotelenez, P., Leitman M. and Mann, J. Adin Jr., On the Depletion Effect in Colloids. Preprint.
- Kotelenez, P. and Kurtz, T.G., Macroscopic Limit for Stochastic Partial Differential Equations of McKean-Vlasov Type. (Preprint)
- Krylov, N.V., Private Communication.
- Krylov, N.V. and Rozovsky, B.L., On stochastic evolution equations. Itogi Nauki i tehniki, VINITI, 71-146, 1979 (in Russian).
- Kunita, H., Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, Cambridge, New York, Port Chester, Melbourne, Sydney, 1990. MR 1472487 (98e:60096)
- Kurtz, T.G. and Xiong, Jie, Particle Representations for a Class of Nonlinear SPDEs. Stochastic Process Appl. 83 (1999), 103-126. MR 1705602 (2000g:60108)
- Lifshits, E.M. and Pitayevskii, L.P., Physical Kinetics. Theoretical Physics X. Nauka, Moscow, 1979 (in Russian).
- Metivier, M. and Pellaumail, J., Stochastic Integration. Adademic Press, New York, 1980.
- Oelschläger, K., A Martingale Approach to the Law of Large Numbers for Weakly Interacting Stochastic Processes. Ann. Probab. 12 (1984), 458-479.
- Pardoux, E., Stochastic Partial Differential Equations and Filtering of Diffusion Processes. Stochastics Vol. 3, 1979, 127-167. MR 553909 (81b:60059)
- Rozovsky, B.L., Stochastic Evolution Systems. Nauka, Moscow (in Russian - English Translation (1990), Kluwer Academic, Dordrecht-Boston). MR 1135324 (92k:60136)
- Sinai, Ya. G. and Soloveichik, M.R., One-Dimensional Classical Massive Particle in the Ideal Gas. Commun. Math. Phys. 104 (1986), 423-443. MR 840745 (88h:82014)
- Szász, D. and Tóth, B., Towards a Unified Dynamical Theory of the Brownian Particle in an Ideal Gas. Commun. Math. Phys. 111 (1986), 41-62. MR 896758 (89d:82021)
- Truesdell, C. and Noll, W., Encyclopedia of Physics - Volume III/3 - The Nonlinear Field Theories of Mechanics. Springer-Verlag, Berlin-Heidelberg-New York, 1965. MR 1215940 (94c:73002)
- Vaillancourt, J., On the Existence of Random McKean-Vlasov limits for Triangular Arrays of Exchangeable Diffusions. Stoch. Anal. Appl. 6(4) (1988), 431-446. MR 964251 (90a:60142)
- Walsh, J.B., An Introduction to Stochastic Partial Differential Equations. Ecole d’Eté de Probabilité de Saint Fleur XIV. Lecture Notes in Math. 1180. Springer, Berlin, 1986, pp. 265-439. MR 876085 (88a:60114)
- Wong, E. and Zakai, M., On the Relation between Ordinary and Stochastic Differential Equations, Int. J. Eng. Sci. 3 (1965), 213-229. MR 0183023 (32:505)
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Additional Information
Peter M. Kotelenez
Affiliation:
Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106
Email:
pxk4@po.cwru.edu
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.