Anomalous threshold behavior of long range random walks

We consider weighted graphs satisfying sub-Gaussian estimate for the natural random walk. On such graphs, we study symmetric Markov chains with heavy tailed jumps. We establish a threshold behavior of such Markov chains when the index governing the tail heaviness (or jump index) equals the escape time exponent (or walk dimension) of the sub-Gaussian estimate. In a certain sense, this generalizes the classical threshold corresponding to the second moment condition.


Introduction
This work concerns a new threshold behavior of random walks on graphs driven by low moment measures. As the title suggests, this work combines two lines of research that have been actively pursued: anomalous random walks and long range random walks. The graphs were are interested in have a nearest neighbor random walk that satisfies sub-Gaussian estimates. Sub-Gaussian estimates for nearest neighbor random walks are typical of many regular fractals like Sierpinski gaskets, carpets and the Viscek graphs. See [22] for a recent survey on such anomalous random walks. Another line of work that has received much attention recently is the long term behavior of random walks with heavy tailed jumps. For example [5], [10], [11], [2], [4] are just a few works in this direction. In much of the existing literature the 'jump index' β is assumed to be in (0, 2). Our work is a modest attempt to understand the behavior of such random walks when β ∈ (0, ∞).
The motivation for our work comes from a recent work by the second author and Zheng [24]. In [24], the behavior of long range random walks on groups was investigated for the full range of the jump index β ∈ (0, ∞). For random walks on groups there is a threshold behavior at β = 2. For graphs satisfying a sub-Gaussian heat kernel estimate, we show that the threshold behavior happens when the jump index β equals the escape time exponent.
Let Γ be an infinite, connected, locally finite graph endowed with a weight µ xy . The elements of the set Γ are called vertices. Some of the vertices are connected by an edge, in which case we say that they are neighbors. The weight is a symmetric non-negative function on Γ × Γ such that µ xy > 0 if and only if x and y are neighbors (in which case we write x ∼ y). We call the pair (Γ, µ) a weighted graph.
The weight µ xy on the edges induces a weight µ(x) on the vertices and a measure µ on subsets A ⊂ Γ defined by for all x ∈ Γ and r ≥ 0. We assume that the measure µ is comparable to the counting measure in the sense that there exists C µ ∈ [1, ∞) such that µ x = µ ({x}) satisfies We consider weighted graphs (Γ, µ) satisfying the following uniform volume doubling assumption: there exists V h : [0, ∞) → (0, ∞), a strictly increasing continuous function and constants C D , C h > 1 such that for all r > 0 and for all x ∈ M and for all r > 0. It can be easily seen from (2) that for all 0 < r ≤ R and for all α ≥ log 2 C D . For the rest of the work, we will assume that our weighted graph (Γ, µ) satisfies (1), (2) and (3).
Remark. If (Γ, µ) satisfies (2) and (3), we may assume that V h (n) = V (x 0 , n) for some fixed x 0 and for all natural numbers n. For non-integer values we can extend it by linear interpolation. Since the graph is connected, infinite and locally finite, the function V h defined above is continous, strictly increasing on [0, ∞).
There is a natural random walk X n on (Γ, µ) associated with the edge weights µ xy . The Markov chain is defined by the following one-step transition probability We will assume that there exists p 0 > 0 such that P (x, y) ≥ p 0 for all x, y such that x ∼ y. We also consider P as a Markov operator which acts on functions of Γ by For any non-negative integer n, the n-step transition probability P n is defined by P n (x, y) = P(X n = y | X 0 = x) = P x (X n = y). Define the heat kernel of weighted graph (Γ, µ) by This Markov chain is symmetric with respect to the measure µ, that is p n (x, y) = p n (y, x) for all x, y ∈ Γ and for all n ∈ N. We assume that there exists γ > 1 such that the following sub-Gaussian estimates are true for the heat kernel p n . There exist constants c, C > 0 such that, for all x, y ∈ Γ and Let ·, · denote the inner product in 2 (Γ, µ). For the Markov operator P , define the corresponding Dirichlet form E P by for all f ∈ 2 (Γ, µ). For any two sets A, B ⊂ Γ, the resistance R P (A, B) is defined by [21,Theorem 3.1], we have the following estimate for the resistance. There exist constants C R , M > 1 such that for all x ∈ Γ and for all r ≥ 1. Other related work that characterizes the sub-Gaussian estimates (5) and (6) are [20] and [3]. The parameter γ in (5) and (6) is sometimes called the 'escape time exponent' or 'anomalous diffusion exponent' or 'walk dimension'. It is known that γ ≥ 2 necessarily (see for instance [13,Theorem 4.6]). For any α ∈ [1, ∞) and for any γ ∈ [2, α + 1], Barlow constructs graphs of polynomial volume growth satisfying V (x, r) (1 + r) α and sub-Gaussian estimates (5) and (6)  . Moreover, these are the complete range of α and γ for which sub-Gaussian estimates with escape rate exponent γ could possibly hold for graphs of polynomial growth with growth exponent α.
Let φ : [0, ∞) → [1, ∞) be a continuous, regularly varying function of positive index. We say a Markov operator K satisfies (J φ ) if it has symmetric kernel k with respect to the measure µ and if there exists a constant C φ > 0 such that for all x, y ∈ M . Let k n (x, y) denote the kernel of the iterated power K n with respect to the measure µ. If K satisfies (J φ ) and if φ is regularly varying with index β > 0, then we say that β is the jump index of the random walk driven by K. We demonstrate threshold behavior as the jump index β varies by analyzing the function as n → ∞ (see [17] for a proof of (8)). The following theorem gives bounds on ψ K (n) that are sharp up to constants.
Example. We write φ in Theorem 1.1 as φ(t) = ((1 + t)l(t)) β where l is a slowly varying function (we refer the reader to [8, Chap. I] for a textbook introduction on slowly and regularly varying functions). The function ζ of Theorem 1.1 can be described more explicitly as follows: φ(t) and ζ is essentially the asymptotic inverse of φ, namely where l # is the de Bruijn conjugate of l. For instance, if l has the property that l(t a ) l(t) for all a > 0, then l # 1/l.
Remark 1. (a) Let φ in Theorem 1.1 be regularly varying with index β > 0. If β ∈ (0, 2) we know matching two sided estimates on k n (x, y) for all n ∈ N and for all x, y ∈ M . Assume that φ(t) = ((1 + t)l(t)) β where l is a slowly varying function. The main result of [15] states that where l # is the de Bruijn conjugate of l. (b) We conjecture that the two-sided estimate (10) is true for any β ∈ (0, γ), where γ is the escape time exponent for the sub-Gaussian estimate in (5) and (6). The proof of (10) in [15] doesn't seem to work if β ∈ [2, γ). In particular, the use of Davies' method to prove off-diagonal upper bounds does not seem to work directly. (c) The conclusion of Theorem 1.1 can be strengthened for random walks on groups for all values of β (γ is necessarily 2 for random walks on groups). See [24, Theorem 1.5] for more. (d) Another intriguing question is to find matching two-sided estimates k n (x, y) for the case β ≥ γ for appropriate range of d(x, y). In light of [24, Theorem 1.5] for random walks on groups, we conjecture that for all n ∈ N * and for all x, y ∈ M such that d(x, y) ≤ ζ(n). Definition 1.2. For a Markov operator K on Γ and any number r > 0, we define the r-moment of random walk driven by K as where (X n ) n∈N is a random walk driven by the Markov operator K and d r Here is a corollary of Theorem 1.1 that illustrates moment threshold behavior of random walks. It states that the asymptotic behavior of ψ K is same as ψ P corresponding to the natural random walk if and only if K has finite γ-moment. (a) K has finite γ-moment, that is M γ,K < ∞.
(b) There exists a constant C > 0 such that for all n ∈ N.
Remark. For random walks on groups one must have γ = 2 and such a second moment threshold behavior is known in greater generality [16, Theorem 1.4 and Corollary 1.5]. See [6], [7] and [24] for extensions and generalizations of such moment threshold behavior for random walks on groups. It is an interesting open problem to formulate and prove a γ-moment threshold in greater generality without the assumption (J φ ).

Proof of Corollary 1.3. By Theorem 1.1, (b) holds if and only if
∞ 0 for some fixed x ∈ Γ. It is well-known that the volume doubling property (2) and (3) implies a reverse volume doubling property which has the following consequence: There exists an integer A ∈ N * and c 1 > 0 such that for all r ≥ 1/2 (Proof of [19,Proposition 3.3] goes through with minor modifications). There exists c 2 , c 3 > 0 such that for all x ∈ M . Now we show a reverse inequality of (15).There exists C 1 , C 2 > 0 such that for all x ∈ M . The second line above follows from (2) and Potter's bound [8, To show (a) implies (b), we use (13), (15) and a generalization of Cauchy condensation test due to Schlömilch to obtain (12). To show (b) implies (a), we use (12), Cauchy condensation test and (16) to obtain (13) which implies (b). Theorem 1.1 and Corollary 1.3 suggests that for spaces with sub-Gaussian estimates and a scaling structure (for example regular fractals), one might be able to formulate and prove a central limit theorem with a γ + moment condition.
1.1. Analytic preliminaries on Markov operator and Dirichlet form. Let (Γ, µ) be a countable, weighted graph. Let K be a Markov operator, symmetric with respect to the measure µ. Denote the kernel of the iterated operator K n with respect to µ by k n (x, y), that is K n f (x) = y∈Γ k n (x, y)f (y)µ(y). We will collect some useful facts about the operator K. For any p ∈ [1, ∞], we denote by f p the norm of f in p (Γ, µ) and by ·, · the inner product in 2 (Γ, µ). A fundamental property of K is that it is a contraction in p (Γ) for any p ∈ [1, ∞], that is and for all f ∈ p (Γ, µ). By the symmetry k 1 (x, y) = k 1 (y, x), we have that K is self-adjoint in 2 (Γ, µ), that is for all f, g ∈ 2 (Γ, µ). For any n ∈ N, we denote by The following useful lemma compares Dirichlet form of a Markov operator K with its iterated power K n . Lemma 1.4 (Folklore). Let K be a Markov operator on Γ symmetric with respect to the measure µ. Then for any f ∈ 2 (Γ, µ) and for any n ∈ N * Proof. We verify this using spectral theory. Let E λ be the spectral resolution of K. Therefore The result follows from the observation that 1 − λ n − n + nλ ≤ 0 for all λ ∈ [−1, 1] and for all n ∈ N * . Lemma 1.5 (Folklore). Let K be a Markov operator on Γ symmetric with respect to the measure µ and let f ∈ 2 (Γ, µ) be a non-zero function. Then the function Proof. We use self-adjointness (17) and Cauchy-Schwarz inequality to get which gives the desired result.

Pseudo-Poincaré inequality using Discrete subordination
Pseudo-Poincaré inequality provides an efficient way to prove Nash inequality which in turn gives upper bounds on ψ K (n). For a function f : Γ → R and R > 0, we define a function f R : Γ → R by In other words, f R (x) is the µ-average of f in B(x, R). The main result of the section is the following pseudo-Poincaré inequality.
We introduce a discrete subordination of the natural random walk on (Γ, µ) whose kernel is comparable to the kernel of K in Proposition 2.5. We introduce a new Markov operator which has a symmetric kernel q(x, y) = 1 2 (p 1 (x, y) + p 2 (x, y)) with respect to µ. Let q k denote the kernel of the Markov operator Q k . For a Markov operator Q k , let E Q k (f, f ) := (I − Q k )f, f denote the corresponding Dirichlet form. Let R Q denote the resistance defined using the Dirichlet form E Q . We will now compare kernels of P k and Q k .
Remark. The advantage of working with the kernel q n is that it satisfies as stronger sub-Gaussian lower estimate (22) in comparison to (6) satisfied by p n . This makes subordination of kernel Q preferable(as opposed to P ) and technically easier.
Lemma 2.2. The kernel q k satisfies the following improved sub-Gaussian estimates: there exist constants c, C > 0 such that, for all x, y ∈ Γ and Proof. Observe that q n (x, y) = n k=0 2 −n n k p n+k (x, y). This along with (5), (2) gives the desired upper bound (21).
Note that, there exists C 1 > 1 such that for all n ∈ N * and for all k ∈ N such that n 4 ≤ k ≤ 3n 4 . There exists c 1 , c 2 > 0 for all x, y ∈ M and for all n ∈ N such that n ≥ 1 ∨ d(x, y). The second line above follows from (23), the third line follows from (6) and (2) and the last line follows from weak law of large numbers.
The operators P and Q have comparable Dirichlet forms and resistances. Lemma 2.3. The resistances R Q and R P are comparable by the following inequality for all subsets A, B ⊂ Γ.
Proof. It suffices to compare the corresponding Dirichlet forms E Q and E P . Note We have the following pseudo-Poincaré inequality for iterated powers of Q.
for all f ∈ 2 (Γ, µ) and for all k ∈ N and R ∈ R satisfying 1 ≤ k ≤ R.
Proof. There exists C 2 > 0 such that The first line follows from Jensen's inequality, the second line follows from the lower bound (22) of Lemma 2.2, (4) and (3), the last line follows from the µ-symmetry of Q. Since Q is a contraction on 2 (Γ, µ), we have Since Q is a contraction on 2 (Γ, µ), we have By (29) and (30), we get Combining (28) and (31) gives the desired inequality.
The following subordinated kernel satisfying (J φ ) is a useful tool to study the behavior of long range random walks.
Then Q φ has a symmetric kernel q φ with respect to µ and there exists C > 0 such that .
Proof of Proposition 2.1. By Proposition 2.5, the Markov operators K and Q φ have comparable Dirichlet forms. Hence it suffices to consider the case K = Q φ . If R < 1, then f ≡ f R which in turn implies the pseudo-Poincaré inequality (19).
Hence we assume that R ≥ 1. There exists c 1 > 0 such that for all f ∈ 2 (Γ, µ) and for all R > 0 which is the desired inequality. In the second line above, we used Lemma 2.4 and in the last line we used that φ is a positive continuous regularly varying function which satisfies the Potter's bound (35).

Nash inequality and Ultracontractivity.
In this section, we use pseudo-Poincaré inequality (19) to obtain a Nash inequality and on-diagonal upper bounds. A polished treatment of the relationship between Nash inequalities and ultracontractivity is presented in [9]. It is wellknown that pseudo-Poincaré inequality along with assumptions on volume growth gives a Sobolev-type inequality (see [23,Theorem 2.1] for an early reference to this approach).
The following function η which appears in (19) plays a crucial role in this work. Define the function η : for R > 0 and η(0) = γφ(0) so that η is a continuous function. We also need the following modification of η defined asη : so thatη is a non-decreasing function. It is known that [8, Theorem 1.5.3]η is asymptotically equivalent to η, that is lim t→∞η (t)/η(t) = 1. If φ is regularly varying with positive index, so is η. We now compute the index of η and list some of its basic properties.
(c) The existence of an asymptotic inverse which is regularly varying of index 1/(β∧γ) follows from (a) and [8,Proposition 1.5.12]. The fact that ζ can be chosen to be continuous, bounded below by 1 and non-decreasing follows from Theorem 1.8.2, Proposition 1.5.1 and Theorem 1.5.3 of [8] respectively. The existence of C > 0 satisfying (43) follows from the definition of asymptotic inverse and continuity of ζ,η andη and lim t→∞η (t)/η(t) = 1. with symmetric kernel k with respect to the measure µ. Then there exist constants C 1 , C 2 > 0 such that for all f ∈ 1 (Γ, µ), whereη is given by (41) and (42).
We now estimate the functions h(t) and its inverse m(t).
where C 3 = C 1 (C D − 1). In the third line above, we used thatη • V −1 h is a non-decreasing function.

Lower bound on ψ K
The lower bound on ψ K follows from a test function argument due to Coulhon and Grigor'yan [13,Theorem 4.6]. However we need a good test function for that argument to work. Such a test function can be obtained from the resistance estimate in (7).
Proof of Theorem 1.1. The upper bound and lower bound follows from Theorems 3.3 and 4.1 respectively.

Stable subordination and the case β < γ
In this section, we provide evidence to the conjecture in Remark 1(b) and (e). Let (Γ, µ) be a weighted graph satisfying the volume doubling condition: there exists C D > 0 such that V (x, 2r) ≤ C D V (x, r) (68) for all x ∈ M and for all r > 0. Similar to (4), there is a volume comparision estimate for any x ∈ M , for all 0 < s ≤ r and for all α ≥ log 2 C D . As before let P and p n denote the Markov operator corresponding to the natural random walk and the heat kernel respectively. We assume that the heat kernel satisfies the following sub-Gaussian estimates. There exist constants c, C > 0 such that, for all x, y ∈ Γ and Consider a random walk X n driven by the operator Q defined in (20). We consider the continous time Markov chain Y β 0 (t) = X N (S β 0 (t)) where N (t) and S β 0 are independent Poisson process and β 0 -stable subordinator for some β 0 ∈ (0, 1). Let k t,β 0 denote the kernel of Y β 0 (t) with respect to the measure µ. By definition of k t,β 0 , we have for all t ≥ 0 and for all x, y ∈ M , where A β 0 (t, i) := P(N (S β 0 (t)) = i). Let q i denote the kernel of the iterated operator Q i with respect to the measure µ for i ∈ N. By the same proof as Lemma 2.2, we get similar sub-Gaussian estimates for the more general volume doubling setup. We assume that the kernel q n satisfies the following sub-Gaussian estimates: There exist constants c, C > 0 such that, for all x, y ∈ Γ q n (x, y) ≤ C V (x, n 1/γ ) exp − d(x, y) γ Cn and q n (x, y) ≥ c V (x, n 1/γ ) exp − d(x, y) γ cn 1 γ−1 , ∀n ≥ 1 ∨ d(x, y). (74) Using estimates on the stable subordinator S β 0 and the estimates on the kernel q n similar to Lemma 2.2, we show the following: for all β 0 ∈ (0, 1), for all i ∈ N * and for all t > 0 such that i ≥ max 6, 4t 1/β 0 . We used (79) in the first line i ≥ max 6, 4t 1/β 0 in the second line and (82) and Stirling asymptotics for Gamma function in the last line. We need the following estimate to prove the desired diagonal upper bound.
Lemma 5.2. Under the doubing assumption (68), there exists C 1 > 0 such that for all x ∈ M and for all u ≥ 0.