Central limit theorem for diffusion processes in an anisotropic random environment

• Autorzy: Nieznaj E.

Identyfikatory

DOI

10.4064/ba53-2-8

• Języki publikacji: EN

Abstrakty

EN

We prove the central limit theorem for symmetric diffusion processes with non-zero drift in a random environment. The case of zero drift has been investigated in e.g. [18], [7]. In addition we show that the covariance matrix of the limiting Gaussian random vector corresponding to the diffusion with drift converges, as the drift vanishes, to the covariance of the homogenized diffusion with zero drift.

Słowa kluczowe

EN

diffusions; random fields; effective diffusivity

PL

dyfuzja; pole losowe; współczynnik dyfuzji efektywny

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2005
• Tom: Vol. 53, no 2
• Strony: 187--205
• Opis fizyczny: Bibliogr. 19 poz.

Twórcy

autor

Nieznaj E.

• Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, Nadbystrzycka 38 A, 20-618 Lublin, Poland,

Bibliografia

• [1] P. Billingsley, Convergence of Probability Measures, 2nd ed., Wiley, 1999.
• [2] S. Ethier and T. Kurtz, Markov Processes, Wiley, New York, 1986.
• [3] A. Fannjiang and T. Komorowski, An invariance principle for diffusion in turbulence, Ann. Probab. 27 (1999), 751-781.
• [4] I. S. Heiland, Central limit theorems for martingales with discrete or continuous time, Scand. J. Statist. 9 (1982), 79-94.
• [5] T. Komorowski and S. Olla, A note on the central limit theorem for two-fold stochastic random walks in a random environment, Bull. Polish Acad. Sci. Math. 51 (2003), 217-232.
• [6] —, —, On homogenization of time-dependent random flows, Probab. Theory Related Fields 121 (2001), 98-116.
• [7] S. M. Kozlov, Averaging of random operators, Math. USSR-Sb. 37 (1980), 167-180.
• [8] —, Conductivity of two-dimensional random media, Russian Math. Surveys 34 (1979), no. 4, 168-169.
• [9] —, The method of averaging and walks in inhomogeneous environments, ibid. 40 (1985), no. 2, 73-145.
• [10] S. M. Kozlov, S. M. Zhikov and O. A. Oleĭnik, Averaging of parabolic operators, Trudy Moskov. Mat. Obshch. 45 (1982), 182-236 (in Russian).
• [11] U. Krengel, Ergodic Theorems, de Gruyter, Berlin, 1985.
• [12] C. Landim, S. Olla and H. T. Yau, Convection-diffusion equation with space-time ergodic random flow, Probab. Theory Related Fields 112 (1998), 203-220.
• [13] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101-134.
• [14] S. Olla, Homogenization of diffusion processes in random fields, École doctorale de École Polytechnique, 1994.
• [15] —, Central limit theorems for tagged particles and for diffusion in random environment, in: Random Media, Panorama et Synthèses 12, Soc. Math. France, 2001, 75-100.
• [16] K. Oelschläger, Homogenization of a diffusion process in a divergence-free random field, Ann. Probab. 16 (1988), 1084-1126.
• [17] H. Osada, Homogenization of diffusion processes with random stationary coefficients, in: Probability Theory and Mathematical Statistics, Lecture Notes in Math. 1021, Springer, Berlin, 1983, 507-517.
• [18] G. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, in: Random Fields (Esztergom, 1979), Colloq. Math. Soc. János Bolyai 27, North-Holland, Amsterdam, 1982, 835-873.
• [19] L. C. G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales, Vol. 1, Wiley, 1994.