Gaussian estimates for spatially inhomogeneous random walks on Zd

Abstract

It is shown in this paper that the transition kernel corresponding to a spatially inhomogeneous random walk on Z d admits upper and lower Gaussian estimates.

Rigorous Scaling Law for the Heat Current in Disordered Harmonic Chain

“…In particular, little appears to be known on lower bounds of the type (5.1b) for non-reversible Markov chains. Results for Markov chains on a lattice [16], or for differential equations in non-divergence form [10], do not adapt straightforwardly (and maybe not at all) to our case. Instead, since we consider only the case w → 0, it has been possible to treat the left hand side of (5.1a) and (5.1b) as a perturbation of quantities that can be computed explicitly.…”

“…There exist K, K ′ , ǫ ′ > 0 such that, for every (y, n) ∈ H(ǫ, 1), and for every ξ ∈ R satisfying |ξw| ≤ ǫ ′ , one hase −Knw 2 ξ 2 ≤ |Λ n (ξ)| ≤ e −K ′ nw 2 ξ 2 , (A.13) |Λ ′ n (ξ)| ≤ Knw 2 (1 + |ξ|)e −K ′ nw 2 ξ 2 , (A.14) |Λ ′′ n (ξ)| ≤ Knw 2 (1 + nw 2 + nw 2 ξ 2 )e −K ′ nw 2 ξ 2 , (A.15)| arg(Λ n (ξ))| ≤ Knw 2 (|ξ| + w|ξ| 3 ). (A 16).…”

Moments of Exit Times from Wedges for Non-homogeneous Random Walks with Asymptotically Zero Drifts

“…random integer-component vectors, is classical and extensive; see for example [4,13,23]. For random walks that are not spatially homogeneous the theory is less complete, and many techniques available for the study of homogeneous random walks can no longer be applied, or are considerably complicated; see, for instance, [12,20]. In the present paper we study angular properties of non-homogeneous random walks, specifically exit times from cones and existence of limiting directions.…”

Combinatorics Meets Potential Theory

“…This principle generalizes to Π-caloric functions. The proof in [ [35], §5.1, §5.2 and §5.3], given for nonnegative L-caloric functions in cylindrical domains, is based only on the maximum and Harnack principles and can be readily extended to Π-caloric functions.…”

“…However, our method has a more general scope and applies in higher dimension and for models with longer steps. It relies on a systematic use of tools and techniques from discrete potential theory developed in [24,35,36]. It assumes a centering condition and requires the construction of an appropriate harmonic function whose existence is difficult to establish in full generality.…”