We establish the quenched local limit theorem for reversible random walk on Zd (with d >= 2) among stationary ergodic random conductances that permit jumps of arbitrary length. The proof is based on the weak parabolic Harnack inequalities and on-diagonal heat-kernel estimates for long-range random walks on general ergodic environments. In particular, this partly solves (Probab. Theory Related Fields 180 (2021) 847-889), Open Problem 2.7, where the quenched invariance principle was obtained. As a byproduct, we prove the maximal inequality with an extra tail term for long-range reversible random walks, which in turn yields the everywhere sublinear property for the associated corrector.

In this paper, we establish higher-order convergence rates of the periodic homogenizatio for symmetric Lévy-type operators, encompassing the subcritical α-stable regime, critical regime, and supercritical diffusive regime. To this end, we develop a systematic framework to decompose the contributions of the underlying jumping kernel across small, intermediate, and large spatial scales – a strategy tailored to all the aforementioned regimes. To the best of our knowledge, this work represents the first comprehensive study of higher-order convergence rates in the homogenization of non-local operators.

In this paper we consider a time-continuous random walk in $\mathbb{Z}^d$ in a dynamical random environment with symmetric jump rates to nearest neighbours. We assume that these random conductances are stationary and ergodic and, moreover, that they are bounded from below but unbounded from above with finite first moment. We derive sharp on-diagonal estimates for the annealed first and second discrete space derivative of the heat kernel which then yield local limit theorems for the corresponding kernels. Assuming weak algebraic off-diagonal estimates, we then extend these results to the annealed Green function and its first and second derivative. Our proof which extends the result of Delmotte and Deuschel (2005) to unbounded conductances with first moment only, is an adaptation of the recent entropy method of Benjamini et. al. (2015).

In two dimensions, the l-level Sierpinski gasket SG(l) is obtained by splitting an equilateral triangle into a collection of l2 equilateral triangles of equal size and with the same total area, retaining only the l(l + 1)/2 triangles with the same orientation as the original triangle, and then iterating this procedure indefinitely. We show that the canonical diffusions on the spaces SG(l), l >= 2, can be rescaled to yield Brownian motion on the initial triangle. Our argument also applies to the analogous higher-dimensional Sierpinski gaskets. Moreover, we prove a local central limit theorem for the associated transition densities. Key to this is the derivation of a Poincar & eacute; inequality, in the proof of which we exploit the Euclidean-type mixing that occurs between the bottlenecks present at each scale of the fractal.

We establish quantitative homogenization results for time-dependent random conductance models with stable-like long range jumps on ^d, where the transition probability from x to y is given by w_t, x,y|x-y|^-d-α with α∈ (0,2). In particular, time-dependent random coefficients {w_t,x,y: t∈_+, (x,y)∈ E} are uniformly bounded from above (but may be degenerate), and satisfy the Kolmogorov continuous condition, where E={(x, y): x ≠ y ∈^d} is the set of all unordered pairs on ^d. The proofs are based on L^2-estimates and energy estimates for solutions to regionalparabolic equations and multi-scale Poincaré inequalities associated with time-dependent symmetric stable-like random walks with random coefficients.

In the context of a metric measure space \((X,d,\mu)\), we explore the potential-theoretic implications of having a finite-dimensional Besov space. We prove that if the dimension of the Besov space \(B^\theta_{p,p}(X)\) is \(k>1\), then \(X\) can be decomposed into \(k\) number of irreducible components (Theorem 1.1). Note that \(\theta\) may be bigger than \(1\), as our framework includes fractals. We also provide sufficient conditions under which the dimension of the Besov space is 1. We introduce critical exponents \(\theta_p(X)\) and \(\theta_p^{\ast}(X)\) for the Besov spaces. As examples illustrating Theorem 1.1, we compute these critical exponents for spaces \(X\) formed by glueing copies of \(n\)-dimensional cubes, the Sierpiński gaskets, and of the Sierpiński carpet.

Let ( K , d ) (K,d) be a connected compact metric space and p ∈ ( 1 , ∞ ) p\in (1, \infty ) . Under the assumption of Kigami [Conductive homogeneity of compact metric spaces and construction of p-energy, Memoirs of the European Mathematical Society, vol. 5, Europea Mathematical Society (EMS), Berline, 2023, Assumption 2.15] and the conductive p p -homogeneity, we show that W p ( K ) ⊂ C ( K ) \mathcal {W}^p(K)\subset C(K) holds if and only if p > dim A R ⁡ ( K , d ) p>\operatorname {dim}_{AR}(K,d) , where W p ( K ) \mathcal {W}^p(K) is Kigami’s ( 1 , p ) (1,p) -Sobolev space and dim A R ⁡ ( K , d ) \operatorname {dim}_{AR}(K,d) is the Ahlfors regular dimension.

Homogenization for non-local operators in periodic environments has been studied intensively. So far, these works are mainly devoted to the qualitative results, that is, to determine explicitly the operators in the limit. To the best of authors' knowledge, there is no result concerning the convergence rates of the homogenization for stable-like operators in periodic environments. In this paper, we establish a quantitative homogenization result for symmetric α-stable-like operators on ^d with periodic coefficients. In particular, we show that the convergence rate for the solutions of associated Dirichlet problems on a bounded domain D is of order ε^(2-α)/2_{α∈ (1,2)}+ε^α/2_{α∈ (0,1)}+ε^1/2|log|^2_{α=1}, while, when the solution to the equation in the limit is in C^2_c(D), the convergence rate becomes ε^2-α_{α∈ (1,2)}+ε^α_{α∈ (0,1)}+ε |log|^2_{α=1}. This indicates that the boundary decay behaviors of the solution to the equation in the limit affects the convergence rate in the homogenization.

We consider random conductance models with long range jumps on $\Z^d$, where the one-step transition probability from $x$ to $y$ is proportional to $w_{x,y}|x-y|^{-d-\alpha}$ with $\alpha\in (0,2)$. Assume that $\{w_{x,y}\}_{(x,y)\in E}$ are independent, identically distributed and uniformly bounded non-negative random variables with $\Ee w_{x,y}=1$, where $E$ is the set of all unordered pairs on $\Z^d$. We obtain a quantitative version of stochastic homogenization for these random walks, with explicit polynomial rates up to logarithmic corrections.

We present on-diagonal heat kernel estimates and quantitative homogenization statements for the one-dimensional Bouchaud trap model. The heat kernel estimates are obtained using standard techniques, with key inputs coming from a careful analysis of the volume growth of the invariant measure of the process under study. As for the quantitative homogenization results, these include both quenched and annealed Berry-Esseen-type theorems, as well as a quantitative quenched local limit theorem. Whilst the model we study here is a particularly simple example of a random walk in a random environment, we believe the roadmap we provide for establishing the latter result in particular will be useful for deriving quantitative local limit theorems in other, more challenging, settings.

In this paper, we study the local boundedness of local weak solutions to the following parabolic equation associated with fractional p-Laplacian type operators ∂_t u(t,x)-p.v.∫_^d|u(t,y)-u(t,x)|^p-2(u(t,y)-u(t,x))J(t;x,y) dy=0, (t,x)∈×^d, where p.v. means the integral in the principal value sense, p∈(1,∞) and J(t;x,y) is comparable to the kernel of the fractional p-Laplacian operator |x-y|^-d-sp with s∈(0,1) and uniformly in (t;x,y)∈×^d×^d. Unlike existing results in the literature, the local boundedness of the solutions obtained in this paper extends the known results for the linear case (i.e., the case that p=2), in particular with a nonlocal parabolic tail that uses the L^1-norm in time for all p∈ (1,∞). The proof is based on a new level set truncation in the De Giorgi-Nash-Moser iteration and a careful choice of iteration orders, as well as a general Caccioppoli-type inequality that is efficiently applied to fractional p-Laplacian type operators with all p>1.

Let (K, d) be a connected compact metric space and p is an element of(1,infinity).Under the assumption of Kigami [Conductive homogeneity of compact metricspaces and construction of p-energy, Memoirs of the European Mathematical Society, vol. 5, Europea Mathematical Society (EMS), Berline, 2023, Assumption 2.15] and the conductivep-homogeneity, we show that W-p(K)subset of C(K)holds if and only if p>d im (AR)(K, d), where W-p(K) is Kigami's (1,p)-Sobolevspace and dim (AR)(K, d) is the Ahlfors regular dimension

This chapter introduces the notions of polynomial coordinate systems and approximate group dilations relative to such coordinate systems. Rescaling via suitable dilation structures is key to the formulation of limit theorems for random walks on groups. One of the main tools used in this book is the notion of approximate group dilations. The limit group structures that appear when one uses rescaling associated with approximate group dilations are discussed.

This chapter anticipates on later results that show how the limit theorems of Chaps. 5 and 6 apply to certain long-range random walks. It focuses on the problem of identifying the limit Lévy process that is obtained through these limit theorems when applied to a given explicit long-range random walk. This is done by drawing interesting links between variations on established results regarding approximations of Lévy processes on Lie groups on the one hand and the limit theorems obtained in Chaps. 5 and 6 on the other hand. Several examples are given to illustrate the limit theorems explicitly using this approach. Examples discussed in Chap. 1 are also revisited in this new light.

This chapter is devoted to our main functional limit theorem for symmetric long-range random walks on finitely generated torsion-free nilpotent groups. A set of technical conditions are identified as sufficient conditions to establish a functional limit theorem by adapting established techniques in the present setting. These conditions are not only phrased mostly in terms of the given random walk but also involve the existence of an appropriate approximate dilation. All together, they are rather strong conditions and finding ways to work under less stringent hypotheses is an interesting open problem.

Let $(K,d)$ be a connected compact metric space and $p\in (1, \infty)$. Under the assumption of \cite[Assumption 2.15]{Ki2} and the conductive $p$-homogeneity, we show that $\mathcal{W}^p(K)\subset C(K)$ holds if and only if $p>\operatorname{dim}_{AR}(K,d)$, where $\mathcal{W}^p(K)$ is Kigami's $(1,p)$-Sobolev space and $\operatorname{dim}_{AR}(K,d)$ is the Ahlfors regular dimension.

Continuing in the same vein as in the previous chapter where the functional limit theorems are treated, sufficient conditions on the original long-range random walk are provided in order to apply and adapt existing local limit theorems to the problems considered here. The local theorem presented in this chapter, Theorem 6.1, is one of the central results of this monograph.

For any torsion-free finitely generated nilpotent group $$\Gamma $$ , this short but essential chapter introduces the set $$\mathcal {S}\mathcal {M}(\Gamma )$$ , a set of stable-like probability measures on $$\Gamma $$ . For each measure $$\mu $$ in $$\mathcal {S}\mathcal {M}(\Gamma )$$ , a particular “geometry” associated with $$\mu $$ is defined. This geometry will later be the key needed to understand how to define norms and appropriate approximate dilations adapted to the measure $$\mu $$ in order to apply the limit theorems of Chaps. 5 and 6 .

This short chapter is devoted to a key technical result which consists in passing from the vague convergence of the family of rescaled measures associated with the driving probability measure of a long-range random walk to the vague convergence of the associated jump kernels. This involves taking into account the change of group law induced by the rescaling of space through an approximate group dilation.

We consider a natural class of long range random walks on torsion free nilpotent groups and develop limit theorems for these walks. Given the original discrete group $\Gamma$ and a random walk $(S_n)_ {n\ge1}$ driven by a certain type of symmetric probability measure $\mu$, we construct a homogeneous nilpotent Lie group $G_\bullet(\Gamma,\mu)$ which carries an adapted dilation structure and a stable-like process $(X_t)_{ t\ge0}$ which appears in a Donsker-type functional limit theorem as the limit of a rescaled version of the random walk. Both the limit group and the limit process on that group depend on the measure $\mu$. In addition, the functional limit theorem is complemented by a local limit theorem.