Abstract
We consider heat conduction in a periodic body which is composed of finitely many different components. The effective conductivity is represented in terms of skew Brownian motion. The representation formula is a fluctuation-dissipation relation. The dissipation term in this formula is related to the transmission of heat through the surface separating the different components of the body; it is described by the skew reflections of Brownian motion at these surfaces. The problems caused by the discontinuity of the microscopic conductivity are handled in the framework of Dirichlet forms.
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References
S. V. Anulova, Diffusion processes with singular characteristics, inStochastic Differential Systems. Filtering and Control, B. Grigelionis, ed. (Springer, Berlin, 1980), pp. 264–269.
R. F. Bass and P. Hsu, The semimartingale structure of reflecting Brownian motion,Proc. Am. Math. Soc. 108:1007–1010 (1990).
R. F. Bass and P. Hsu, Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains,Ann. Prob. 19:486–508 (1991).
J. R. Baxter and G. A. Brosamler, Energy and the law of the iterated logarithm,Math. Scand. 38:115–136 (1976).
A. De Masi, P. A. Ferrari, S. Goldstein, and D. W. Wick, An invariance principle for reversible Markov processes: Applications to random motions in random environments,J. Stat. Phys. 55:787–855 (1989)
M. Fukushima,Dirichlet Forms and Markov Processes (North-Holland, Amsterdam, 1980).
K. Golden and G. Papanicolaou, Bounds for effective parameters of heterogeneous media by analytic continuation,Comm. Math. Phys. 90:473–491 (1983).
P. Hsu, On excursions of reflecting Brownian motion,Trans. Am. Math. Soc. 296:239–264 (1986).
K. Itô and H. P. McKean, Brownian motions on a half line,Illinois J. Math. 7:181–231 (1963).
R. Lang, Une représentation probabiliste de la conductivité effective,C. R. Acad. Sci. Paris 317 (Ser. I):967–969 (1993).
K. Lichtenecker, Die Dielektrizitätskonstante natürlicher und künstlicher Mischkörper,Phys. Z. 27:115–158 (1926).
H. Osada, Homogenization of diffusion processes with random stationary coefficients, inProceedings of the 4th Japan-USSR Symposium on Probability Theory (Springer, Berlin, 1983), pp. 507–517.
H. Osada and T. Saitoh, An invariance principle for non-symmetric Markov processes and reflecting diffusions in random domains,Prob. Theory Related Fields (1995), to appear.
G. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating coefficients, inRandom Fields, J. Fritz, J. L. Lebowitz, and D. Szász, eds. (North-Holland, Amsterdam, 1981), pp. 835–853.
D. Revuz and M. Yor,Continuous Martingales and Brownian Motion, (Springer, Berlin, 1991).
H. Spohn,Large Scale Dynamics of Interacting Particles (Springer, Berlin, 1991).
V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik,Homogenization of Differential Operators and Integral Functionals (Springer, Berlin, 1994).
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Lang, R. Effective conductivity and skew Brownian motion. J Stat Phys 80, 125–146 (1995). https://doi.org/10.1007/BF02178356
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DOI: https://doi.org/10.1007/BF02178356