zbMATH — the first resource for mathematics

Homogenization of a bond diffusion in a locally ergodic random environment. (English) Zbl 1075.60129
A nearest neighbour random walk on \(\mathbb Z\) with random jump rates is studied. It is supposed that the jump rate from state \(x\) to state \(x+1\) is a strictly positive bounded random variable \(\eta (x)\), \(0<c_ {-}\leq \eta (x)\leq c_ {+}<\infty \), and \(\eta (x)\) equals also to the jump rate from \(x+1\) to \(x\). Let \(\Omega = [c_ {-},c_ {+}] ^ {\mathbb Z}\) be the space of configurations, and \((\tau _ {z},\; z\in \mathbb Z)\) the group of translations on \(\Omega \), \(\tau _ {z}\eta (x) = \eta (x+z)\). For a fixed environment \(\eta \in \Omega \), let \((x^ \eta _ {t})_ {t\geq 0}\) be the corresponding random walk with \(x^ \eta _ 0 = 0\); denote by \(X^ {\eta ,\varepsilon }\) a rescaled process defined by \(X^ {\eta , \varepsilon }_ {t} = \varepsilon x^ \eta _ {\varepsilon ^ {-2}t}\). Further it is assumed that \((\mu _ \varepsilon )_ {\varepsilon >0}\) is a locally ergodic system of probability measures on \(\Omega \), that is, there exists a family \((\bar \mu _ {y},\; y\in \mathbb R)\) of probability measures which are ergodic with respect to the translation group \((\tau _ {z})\) and such that \[ \varepsilon \sum _ {z} g(\varepsilon z)f(\varepsilon z,\tau _ {z} \eta ) \longrightarrow \iint g(y)f(y,\omega )\,d\bar \mu _ {y} (\omega )\,dy \quad \text{as \(\varepsilon \downarrow 0\)} \] in \(\mu _ \varepsilon \)-probability for any bounded measurable function \(f:\mathbb R\times \Omega \to \mathbb R\) continuous in the first variable and local in the second one, and for any compactly supported continuous function \(g:\mathbb R\to \mathbb R\). Finally, it is supposed that the function \(y\mapsto \int f\,d\bar \mu _ {y}\) is locally integrable on \(\mathbb R\) for any bounded local function \(f\) on \(\Omega \), and the function \(a: y\mapsto (\int \eta (0)^ {-1} \,d\bar \mu _ {y})^ {-1}\) satisfies \(a\in C^ 2(\mathbb R)\) and has bounded derivatives. Under these hypotheses it is proven that the distribution of \(X^ {\eta ,\varepsilon }\) converges in \(\mu _ \varepsilon \)-probability to the law of a diffusion process on \(\mathbb R\) with the generator \(Lf = {d\over dy}(a(y){df\over dy})\), \(f\in C^ 2(\mathbb R)\).

60K37 Processes in random environments
60F17 Functional limit theorems; invariance principles
82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
Full Text: DOI
[1] Anshelevich, V.V.; Vologodskii, A.V., Laplace operator and random walk on one-dimensional nonhomogeneous lattice, J. statist. phys., 25, 3, 419-430, (1981) · Zbl 0512.60059
[2] Anshelevich, V.V.; Khanin, K.M.; Sinai, Y.G., Symmetric random walks in random environments, Comm. math. phys., 85, 449-470, (1982) · Zbl 0512.60058
[3] Aronson, D.G., Bound on the fundamental solution of a parabolic equation, Bull. amer. math. soc., 73, 890-896, (1967) · Zbl 0153.42002
[4] Bensoussan, A.; Lions, J.L.; Papanicolaou, G., Asymptotic analysis for periodic structures, (1978), North-Holland Amsterdam · Zbl 0411.60078
[5] Bourgeat, A.; Mikelik, A.; Wright, S., Stochastic two-scale convergence in the Mean and applications, J. reine angew. math., 456, 19-51, (1994) · Zbl 0808.60056
[6] Carlen, E.; Kusuoka, S.; Stroock, D.W., Upper bounds for symmetric Markov transition functions, Ann. inst. H. Poincaré probab. statist., 23, 2, 245-287, (1987) · Zbl 0634.60066
[7] De Masi, A.; Ferrari, P.; Goldstein, S.; Wick, W.D., An invariance principle for reversible Markov processes. applications to random motions in random environments, J. statist. phys., 55, 787-855, (1989) · Zbl 0713.60041
[8] Fabes, E.; Stroock, D.W., The de giorgi – moser Harnack principle via the old ideas of Nash, Arch. rational mech. anal., 96, 327-338, (1987) · Zbl 0652.35052
[9] Giacomin, G.; Olla, S.; Spohn, H., Equilibrium fluctuations for ∇ φ interface model, Ann. probab., 29, 3, 1138-1172, (2001) · Zbl 1017.60100
[10] Grigorescu, I., Self-diffusion for Brownian motions with local interaction, Ann. probab., 27, 3, 1208-1267, (1999) · Zbl 0961.60099
[11] Kipnis, C.; Landim, C., Scaling limit of interacting particle systems, (1999), Springer Berlin · Zbl 0927.60002
[12] Kipnis, C.; Varadhan, S.R.S., Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Comm. math. phys., 104, 1-19, (1986) · Zbl 0588.60058
[13] Kozlov, S.M., The method of averaging and walks in inhomogeneous environments, Russian math. surveys, 40, 2, 73-145, (1985) · Zbl 0615.60063
[14] Kunnemann, R., The diffusion limit for reversible jump processes on \(Z\^{}\{d\}\) with ergodic random bond conductivities, Comm. math. phys., 90, 27-68, (1983)
[15] Papanicolaou, G.; Varadhan, S.R.S., Boundary value problems with rapidly oscillating random coefficients, Collq. math. soc. János bolyai, random fields, 27, 835-873, (1979)
[16] Siri, P., Asymptotic behaviour of a tagged particle in an inhomogeneous zero-range process, Stochastic process. appl., 77, 139-154, (1998) · Zbl 0935.60081
[17] Stroock, D.W.; Zheng, W., Markov chain approximations to symmetric diffusions, Ann. inst. H. Poincaré probab. statist., 33, 5, 619-649, (1997) · Zbl 0885.60065
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.