Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models

Alessandra Faggionato (University La Sapienza, Rome)


We consider a   sequence  $X^{(n)}$, $n \geq 1 $,   of continuous-time nearest-neighbor random walks on the one dimensional lattice $\mathbb{Z}$.  We reduce  the spectral analysis of the Markov generator of $X^{(n)}$ with Dirichlet conditions outside $(0,n)$ to the analogous problem  for  a suitable generalized second order differential operator $-D_{m_n} D_x$, with Dirichlet conditions outside a giveninterval. If  the measures $dm_n$ weakly converge to some measure $dm_\infty$,  we prove a limit theorem for the eigenvalues and eigenfunctions of $-D_{m_n}D_x$ to the corresponding spectral quantities of $-D_{m_\infty}  D_x$.  As second result,  we prove the Dirichlet-Neumann bracketing for the operators  $-D_m D_x$ and, as a consequence, we establish lower and upper bounds for the asymptotic annealed eigenvalue counting functions in the case that $m$ is a self-similar stochastic process.  Finally, we apply the above results to investigate the spectral structure of some classes of  subdiffusive random trap and barrier models coming from one-dimensional physics.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-36

Publication Date: February 24, 2012

DOI: 10.1214/EJP.v17-1831


  • Alexander, S.; Bernasconi, J.; Schneider, W. R.; Orbach, R. Excitation dynamics in random one-dimensional systems. Rev. Modern Phys. 53 (1981), no. 2, 175--198. MR0611317
  • Bertoin, Jean. Lévy processes. Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. x+265 pp. ISBN: 0-521-56243-0 MR1406564
  • Ben Arous, Gérard; Černý, Jiří. Bouchaud's model exhibits two different aging regimes in dimension one. Ann. Appl. Probab. 15 (2005), no. 2, 1161--1192. MR2134101
  • Ben Arous, Gérard; Černý, Jiří. Dynamics of trap models. Mathematical statistical physics, 331--394, Elsevier B. V., Amsterdam, 2006. MR2581889
  • Boivin, D.; Depauw, J. Spectral homogenization of reversible random walks on $\Bbb Z^ d$ in a random environment. Stochastic Process. Appl. 104 (2003), no. 1, 29--56. MR1956471
  • J.P. Bouchaud, L. Cugliandolo, J. Kurchan, M. Mézard, Out--of--equilibrium dynamics in spin--glasses and other glassy systems. In Spin--Glasses and Random Fields (A.P. Young, ed.), Singapore, Word Scientific (1998).
  • J.-P. Bouchaud, D.S. Dean, Aging on Parisi's tree. J. Phys. I France 5, 265-286 (1995).
  • Bovier, Anton; Faggionato, Alessandra. Spectral characterization of aging: the REM-like trap model. Ann. Appl. Probab. 15 (2005), no. 3, 1997--2037. MR2152251
  • Bovier, Anton; Faggionato, Alessandra. Spectral analysis of Sinai's walk for small eigenvalues. Ann. Probab. 36 (2008), no. 1, 198--254. MR2370603
  • Courant, R.; Hilbert, D. Methods of mathematical physics. Vol. I. Interscience Publishers, Inc., New York, N.Y., 1953. xv+561 pp. MR0065391
  • Dym, H.; McKean, H. P. Gaussian processes, function theory, and the inverse spectral problem. Probability and Mathematical Statistics, Vol. 31. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. xi+335 pp. MR0448523
  • Dieudonné, J. Foundations of modern analysis. Enlarged and corrected printing. Pure and Applied Mathematics, Vol. 10-I. Academic Press, New York-London, 1969. xviii+387 pp. MR0349288
  • E.B. Dynkin, Markov processes, Volume II, Grundlehren der mathematischen Wissenschaften 122, Berlin, Springer Verlag (1965). MR0193671
  • Doyle, Peter G.; Snell, J. Laurie. Random walks and electric networks. Carus Mathematical Monographs, 22. Mathematical Association of America, Washington, DC, 1984. xiv+159 pp. ISBN: 0-88385-024-9 MR0920811
  • Faggionato, A.; Jara, M.; Landim, C. Hydrodynamic behavior of 1D subdiffusive exclusion processes with random conductances. Probab. Theory Related Fields 144 (2009), no. 3-4, 633--667. MR2496445
  • Fontes, L. R. G.; Isopi, M.; Newman, C. M. Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension. Ann. Probab. 30 (2002), no. 2, 579--604. MR1905852
  • Freiberg, Uta. Spectral asymptotics of generalized measure geometric Laplacians on Cantor like sets. Forum Math. 17 (2005), no. 1, 87--104. MR2110540
  • Fukushima, Masatoshi; Ōshima, Yōichi; Takeda, Masayoshi. Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 1994. x+392 pp. ISBN: 3-11-011626-X MR1303354
  • Hambly, B. M. On the asymptotics of the eigenvalue counting function for random recursive Sierpinski gaskets. Probab. Theory Related Fields 117 (2000), no. 2, 221--247. MR1771662
  • Itô, Kiyosi; McKean, Henry P., Jr. Diffusion processes and their sample paths. Second printing, corrected. Die Grundlehren der mathematischen Wissenschaften, Band 125. Springer-Verlag, Berlin-New York, 1974. xv+321 pp. MR0345224
  • I.S. Kac, M.G. Krein, On the spectral functions of the string. Amer. Math. Soc. Transl. (2), Vol. 103, 19--102 (1974).
  • Kasahara, Yuji. Spectral theory of generalized second order differential operators and its applications to Markov processes. Japan. J. Math. (N.S.) 1 (1975/76), no. 1, 67--84. MR0405615
  • Kawazu, Kiyoshi; Kesten, Harry. On birth and death processes in symmetric random environment. J. Statist. Phys. 37 (1984), no. 5-6, 561--576. MR0775792
  • Kigami, Jun; Lapidus, Michel L. Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals. Comm. Math. Phys. 158 (1993), no. 1, 93--125. MR1243717
  • Kodaira, Kunihiko. The eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of $S$-matrices. Amer. J. Math. 71, (1949). 921--945. MR0033421
  • Kong, Q.; Zettl, A. Eigenvalues of regular Sturm-Liouville problems. J. Differential Equations 131 (1996), no. 1, 1--19. MR1415044
  • Küchler, Uwe. Some asymptotic properties of the transition densities of one-dimensional quasidiffusions. Publ. Res. Inst. Math. Sci. 16 (1980), no. 1, 245--268. MR0574035
  • Lapidus, Michel L. Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture. Trans. Amer. Math. Soc. 325 (1991), no. 2, 465--529. MR0994168
  • Löbus, J.-U. Generalized second order differential operators. Math. Nachr. 152 (1991), 229--245. MR1121236
  • Löbus, Jörg-Uwe. Constructions and generators of one-dimensional quasidiffusions with applications to self-affine diffusions and Brownian motion on the Cantor set. Stochastics Stochastics Rep. 42 (1993), no. 2, 93--114. MR1275814
  • Mandl, Petr. Analytical treatment of one-dimensional Markov processes. Die Grundlehren der mathematischen Wissenschaften, Band 151 Academia Publishing House of the Czechoslovak Academy of Sciences, Prague; Springer-Verlag New York Inc., New York 1968 xx+192 pp. MR0247667
  • G. Métivier, Valeurs propres de problèmes aux limites elliptiques irrégulier. Bull. Soc. Math. France, Mém. 51-- 52, 125--219 (1977). MR0473578
  • Mourrat, Jean-Christophe. Principal eigenvalue for the random walk among random traps on $\Bbb Z^ d$. Potential Anal. 33 (2010), no. 3, 227--247. MR2718254
  • Ogura, Yukio. One-dimensional bi-generalized diffusion processes. J. Math. Soc. Japan 41 (1989), no. 2, 213--242. MR0984748
  • Reed, Michael; Simon, Barry. Methods of modern mathematical physics. I. Functional analysis. Academic Press, New York-London, 1972. xvii+325 pp. MR0493419
  • Reed, Michael; Simon, Barry. Methods of modern mathematical physics. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. xv+396 pp. ISBN: 0-12-585004-2 MR0493421
  • Resnick, Sidney I. Extreme values, regular variation, and point processes. Applied Probability. A Series of the Applied Probability Trust, 4. Springer-Verlag, New York, 1987. xii+320 pp. ISBN: 0-387-96481-9 MR0900810
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 MR1725357
  • Stone, Charles. Limit theorems for random walks, birth and death processes, and diffusion processes. Illinois J. Math. 7 1963 638--660. MR0158440
  • Uno, Toshio; Hong, Imsik. Some consideration of asymptotic distribution of eigenvalues for the equation $d^{2}u/dx^{2}+\lambda \rho (x)u=0$. Japan. J. Math. 29 1959 152--164. MR0118891
  • Yosida, Kôsaku. Lectures on differential and integral equations. Pure and Applied Mathematics, Vol. X Interscience Publishers, New York-London 1960 ix+220 pp. MR0118869
  • H. Weyl. Über die asymptotische Verteilung der Eigenwerte. Gött. Nach., 110--117 (1911).
  • Weyl, Hermann. Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). (German) Math. Ann. 71 (1912), no. 4, 441--479. MR1511670
  • Zettl, Anton. Sturm-Liouville theory. Mathematical Surveys and Monographs, 121. American Mathematical Society, Providence, RI, 2005. xii+328 pp. ISBN: 0-8218-3905-5 MR2170950

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.