Abstract
We prove a limit theorem for a process in a random one-dimensional medium, which has been considered before as a model for hopping conduction in a disordered medium. To the edge between the two integersj and (j+ 1) a rate λj > 0 is attached. Theseλ j :j integral are taken as independent, identically distributed random variables, and represent the medium. For given values λj, X(t) is a Markov chain in continuous time which jumps fromj to (j + 1) and from (j + 1) toj at the same rate λj. We show that in many cases there exists normalizing constants y(t) (which tend to oo witht) such that the distribution of X(t)/γ(t), or more generally of the whole processX(st)/γ(t) S⩾0, converges to a limit as t→ ∞. The limit process is continuous and self-similar.
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References
S. Alexander, J. Bernasconi, W. R. Schneider, and R. Orbach, Excitation dynamics in random one-dimensional system,Rev. Mod. Phys. 53:175–198 (1981).
V. V. Anshelevic, K. M. Khanin, and Ya. G. Sinai, Symmetric random walks in random environments,Commun. Math. Phys. 85:449–470 (1982).
V. V. Anshelevich and A. V. Vologodskii, Laplace operator and random walk on onedimensional nonhomogeneous lattice,J. Stat. Phys. 25:419–430 (1981).
J. Bernasconi and H. U. Beyeler, Some comments on hopping in random one-dimensional systems,Phys. Rev. B 21:3745–3747 (1980).
J. Bernasconi, W. R. Schneider, and W. Wyss, Diffusion and hopping conductivity in disordered one-dimensional lattice systems,Z. Phys. B 37:175–184 (1980).
J. Bernasconi and W. R. Schneider, Classical hopping conduction in random one-dimensional systems: non-universal limit theorems and quasi-localization effects,Phys. Rev. Lett. 47:1643–1647 (1981).
J. Bernasconi and W. R. Schneider, Diffusion in one-dimensional lattice system with random transfer rates, inLecture Notes in Physics, No. 153 (Springer-Verlag, Berlin, 1982), pp. 389–393.
J. Bernasconi and W. R. Schneider, Diffusion in random one-dimensional systems,J. Stat. Phys. 30:355–362 (1983).
P. Billingsley,Convergence of Probability Measures (John Wiley & Sons, New York, 1968).
R. M. Blumenthal and R. K. Getoor,Markov Processes and Potential Theory (Academic Press, New York, 1968).
B. Derrida and Y. Pomeau, Classical diffusion in a random chain,Phys. Rev. Lett. 48:627–630 (1982).
R. M. Dudley, Distances of probability measures and random variables,Ann. Math. Statist. 39:1563–1572 (1968).
W. Feller,An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed. (John Wiley & Sons, New York, 1971).
K. Ito, Stochastic processes, Lecture notes, No. 16, Aarhus University (1969).
T. Kaijser, A note on random continued fractions, inProbability and Mathematical Statistics: Essays in Honor of Carl-Gustav Esseen, A. Gut and L. Holst, eds. (Uppsala University, Uppsala, 1983), pp. 74–83.
S. Karlin and H. M. Taylor, A first course in stochastic processes, 2nd ed. (Academic Press, New York, 1975).
S. M. Kozlov, Averaging of random operators,Mat. Sborn. 113:302–308 (1980) (translated inMath. USSR, Sbornik 37:167–180).
R. Künnemann, The diffusion limit for reversible jump processes on ℤd with ergodic random bond coefficients,Commun. Math. Phys. 90:27–68 (1983).
T. Lindvall, Weak convergence of probability measures and random functions in the function space D[0, ∞),J. Appl. Prob. 10:109–121 (1973).
G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients,Coll. Math. Soc. János Bolyai, 27, Random Fields, Vol. 2 (North-Holland, Amsterdam, 1981), pp. 835–873.
W. R. Schneider, Hopping transport in disordered one-dimensional lattice systems: random walk in a random medium, inLecture Notes in Physics, No. 173 (Springer-Verlag, Berlin, 1982), pp. 289–303.
S. Schumacher, Diffusions with random coefficients, Ph.D. thesis, University of California, Los Angeles, 1984.
A. V. Skorohod, Limit theorems for stochastic processes,Theory Prob. Appl. 1:262–290 (1956) (English transi.).
M. J. Stephen and R. Kariotes, Diffusion in a one-dimensional disordered system,Phys. Rev. B 26:1917–1925 (1982).
C. J. Stone, Limit theorems for random walks, birth and death processes and diffusion processes,Il. J. Math. 7:638–660 (1963).
C. Stone, Weak convergence of stochastic processes defined on semi-infinite time intervals,Proc. Am. Math. Soc. 14:694–696 (1963).
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Kawazu, K., Kesten, H. On birth and death processes in symmetric random environment. J Stat Phys 37, 561–576 (1984). https://doi.org/10.1007/BF01010495
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DOI: https://doi.org/10.1007/BF01010495