Abstract
The formulas for the limiting covariance matrix obtained for the class of diffusions considered in Chap. 9 are not explicit in terms of their coefficients. In this chapter we give upper and lower bounds for the quadratic form corresponding to the limiting covariance matrix. They are formulated in terms of variational principles and can be used to obtain further information about properties of the limiting normal law.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Avellaneda M, Majda AJ (1991) An integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows. Commun Math Phys 138(2):339–391
Cherkaev AV, Gibiansky LV (1994) Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli. J Math Phys 35(1):127–145
Fannjiang A (1998) Boundary value problem with unbounded, fast oscillatory random flows. J Differ Equ 143(2):327–359
Fannjiang A, Papanicolaou G (1994) Convection enhanced diffusion for periodic flows. SIAM J Appl Math 54(2):333–408
Fannjiang A, Papanicolaou G (1996) Diffusion in turbulence. Probab Theory Relat Fields 105(3):279–334
Olla S (1994b) Lectures on homogenization of diffusion processes in random fields. Publications de l’école doctorale de l’école polytechnique
Rhodes R (2006) Homogénéisation en milieu aléatoire. These de l’universite de provence, ecole doctorale de mathématiques et informatique de Marseille
Yaglom AM (1962) An introduction to the theory of stationary random functions. Prentice-Hall, Englewood Cliffs. Revised English edition. Translated and edited by Richard A Silverman
Zeidler E (1995) Applied functional analysis: main principles and their applications. Applied mathematical sciences, vol 109. Springer, New York
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Komorowski, T., Landim, C., Olla, S. (2012). Variational Principles for the Limiting Variance. In: Fluctuations in Markov Processes. Grundlehren der mathematischen Wissenschaften, vol 345. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29880-6_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-29880-6_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-29879-0
Online ISBN: 978-3-642-29880-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)