Summary
In this paper we consider the parabolic equation with random coefficients:
We show that uε(t,x) converges to the solution uo(t,x) of the averaging equation:
Also, the fluctuation process \(y^\varepsilon (t,x)\left( { \equiv \left( {u^\varepsilon (t,x)--u^0 (t\user2{,}x)} \right)/\sqrt \varepsilon } \right)\) converges weakly to a generalized Ornstein-Uhlenbeck process on L′.
References
Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic analysis for periodic structure, pp. 516–533. Amsterdam: North-Holland 1978
Brodskiy, Ya.S., Lakacher, B.Ya.: Fluctuation in averaging scheme for differential equations with random right hand side. (Russian) Theory of random process 12, 8–17 (1984)
Davydov, Yu.A.: Convergence of distributions generated by stationary stochastic processes. Theory Probab. Appl. 13, 691–696 (1968)
Eidel'man, S.D.: Parabolic system. Amsterdam: North-Holland 1969
Friedman, A.: Partial differential equations of parabolic type. Prentice Hall, Englewood Cliffs, N.J., 1964
Geman, S.: A method of averaging for random differential equations with applications to stability and stochastic approximations. In: Bharuch Reid, A.T. (ed.) Approximate solution of random equation, pp. 37–85, New York: North-Holland 1979
Ibragimov, I.A., Linnik, Yu.V.: Independent and stationary sequences of random variables. Groningen: Wolter-Noordhoff 1971
Itô, K.: Infinite dimensional Ornstein-Uhlenbeck processes in stochastic analysis. pp. 197–224. Tokyo: Kinokuniya 1984
Kesten, H., Papanicolaou, G.C.: A limit theorem for turbulent diffusion. Commun. Math. Phys. 65, 97–128 (1979)
Khas'minskii, R.Z.: Principle of averaging for parabolic and elliptic differential equations and for Markov processes with small diffusion. Theory Probab. Appl. 8, 1–21 (1963)
Khas'minskii, R.Z.: On the stochastic processes defined by differential equations with a small parameter. Theory Probab. Appl. 11, 211–222 (1966)
Krylov, N.M., Bogoliubov, N.N.: Introduction to nonlinear mechanics. Princeton: Princeton University Press 1947
Kushner, H.J.: A martingale method for the convergence of a sequence of processes to a jumpdiffusion process. Z. Wahrscheinlichkeitstheor. Verw. geb. 53, 209–219 (1980)
Mitoma, I.: On the sample continuity of L′-processes. J. Math. Soc. Japan 35, 629–636 (1983)
Mitoma, I.: Tightness of probabilities on C([0, 1]; L′) and D([0, 1]; L′). Ann. Probab. 11, 989–999 (1983)
Nash, Jr., J.F.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)
Porper, F.O., Eidel'man, S.D.: Two-sided estimates of fundamental solutions of second-order parabolic equation and some applications. Russ. Math. Surv. 39, 119–178 (1984)
Stroock, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. Berlin Heidelberg New York: Springer 1979
Watanabe, H.: Fluctuations in certain dynamical systems with averaging. Stochastic Processes Appl. 21, 147–157 (1985)
Watanabe, H.: Averaging and fluctuations of certain stochastic equations. In: Albeverio, S., Blanchard, Ph., Streit, L. (eds.) Stochastic Processes Mathematics and Physics. Bielefeld 1985. (Lect. Notes Math., vol. 1250, pp. 348–355) Berlin Heidelberg New York: Springer 1987
Zhikov, V.V., Kozlov, S.M., Oleinik, O.A.: Averaging of parabolic operators. Trans. Mosc. Math. Soc. 45, 189–241 (1982)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Watanabe, H. Averaging and fluctuations for parabolic equations with rapidly oscillating random coefficients. Probab. Th. Rel. Fields 77, 359–378 (1988). https://doi.org/10.1007/BF00319294
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00319294
Keywords
- Stochastic Process
- Probability Theory
- Statistical Theory
- Parabolic Equation
- Average Equation