LOCAL SUB-GAUSSIAN ESTIMATES ON GRAPHS: THE STRONGLY RECURRENT CASE

This paper proves upper and lower o(cid:11)-diagonal, sub-Gaussian transition probabilities estimates for strongly recurrent random walks under su(cid:14)cient and necessary conditions. Several equivalent conditions are given showing their particular role inﬂuence on the connection between the sub-Gaussian estimates, parabolic and elliptic Harnack inequality.


The origin of the problem
In a recent paper ( [16]) a complete characterization was given of polynomially growing (strongly) transient graphs (with volume growth V (x, R) R α ) possessing sub-Gaussian behavior with mean exit time E(x, R) R β ( α > β ≥ 2). In this setting the classical Gaussian estimates are replaced with the so called sub-Gaussian estimates which have the form for n ≥ d(x, y) if and only if the volume growth is polynomial and the Green function decays polynomially as well. The β > 2 case has the sub-Gaussian name to reflect the sub-diffusive character of the diffusion process.
The aim of this paper is to prove the strongly recurrent counterpart ( α < β ) of the result ( [16] where α > β) . In fact this paper proves more. It shows a local (or as it is sometimes, a called relative) version assuming volume doubling instead of polynomial growth. This setting brings two new difficulties. One is the local formalism, the other is that due to the recurrence there is no global Green function (contrary to the transient case of [16]) and all the analysis is based on the local Green function, the Green function of the process killed on exiting from a finite set. This technique was developed in [25], [26] and in [27].

Basic objects
Let Γ be an infinite connected graph and µ x,y the weight function on the connected vertices x ∼ y, x, y ∈ Γ, inducing a measure µ on Γ. The measure µ(x) is defined for an x ∈ Γ by µ(x) = It is clear that volume doubling implies V (x, R) ≤ CR α with The random walk is defined by the weights via the one-step transition probabilities P (x, y) = µ x,y µ(x) , P(X n+1 = y|X n = x) = P (x, y) and P n (x, y) = P(X n+1 = y|X 0 = x) while the transition probability kernel is p n (x, y) = 1 µ(y) P n (x, y).

Definition 1.2 The transition probability kernel satisfies the local sub-Gaussian estimates if
there are c, C > 0 such that for all x, y ∈ Γ and n ∈ N where p n = p n + p n+1 .

Definition 1.3 A weighted graph satisfies (β−parabolic or simply) parabolic Harnack inequality
if for any given profile C there is a constant C H (C) > 0 for which the following is true. Assume that u is the solution of the equation for k, R ∈ N, then on the smaller cylinders defined by and taking (n − , x − ) ∈ U − , (n + , x + ) ∈ U + , d(x − , x + ) ≤ n + − n − the inequality holds, where u n = u n + u n+1 .
It is standard that if the (classical) parabolic Harnack inequality holds for a given profile, then it holds for any other profile as well, provided the volume doubling condition holds. It is clear that the same holds for the β−parabolic Harnack inequality.
The elliptic Harnack inequality is direct consequence of the β-parabolic one as it is true in the classical case.

Definition 1.4
The graph satisfies the elliptic Harnack inequality if there is a C > 0 such that for all x ∈ Γ, R > 1 and v > 0 harmonic function on B(x, 2R) which means that the following inequality holds max The notation a ξ b ξ will be used in the whole sequel if there is a C > 1 such that 1/Ca ξ ≤ b ξ Ca ξ for all possible ξ.

The result in brief
The main result presents a strongly recurrent counterpart (α < β) of the result of [16] (where α > β) and goes beyond it on one hand giving local version of the sub-Gaussian estimate and on the other hand providing a set of equivalent conditions to it (given later in Section 2 as well as the definition of strong recurrence.).

Theorem 1.1 For strongly recurrent graphs with the property that for all
the following statements are equivalent 3. Γ satisfies (P H β ) Remark 1. 1 We shall see that the implications 2. =⇒ 3. =⇒ 1. hold for all random walks on weighted graphs. The details will be given in Section 2.
Additionally it is proved that for the same graphs (P H β ) implies the β-Poincaré inequality which is defined below.

Definition 1.7
The generalized Poincaré inequality in our setting is the following. For for all To our best knowledge the results of Theorem 1.1 is new for β = 2 as well. It is a generalization of several works having the Gaussian estimates (β = 2) ( [29], [9], [17] and their bibliography).
Results on sub-diffusive behavior are well-known in the fractal settings but only in the presence of strong local symmetry and global self-similarity (c.f. [1] and its bibliography) We recall a new result from [17,Theorem 5.2] which is in some respect generalization of [12] [13], [24], [23] and [11]. In fact [17] provides new and simple proof of this which involves scale-invariant local Sobolev inequality eliminating the difficult part of the Moser's parabolic iterative method. A similar result for graphs with the classical method was given by [9].
These findings are partly extended in [17,Section 5.] to the sub-Gaussian case, (non-classical case as it is called there), showing that on Dirichlet spaces with proper metric (UE β ) and (LE β ) =⇒ (P H β ) and (D) which is exactly 2. =⇒ 3. in Theorem 1.1 in the context of the paper [17]. Let us point out that Theorem 1.1 uses the usual shortest path metric without further assumption.
Our paper is confined to graphs, but from the definitions, results and proof it will be clear that they generalize in measure metric spaces and in several cases the handling of continuous space and time would be even easier.

Acknowledgments
The author is indebted to Professor Alexander Grigor'yan for the useful discussions and friendly support. Thanks are also due to Gábor Elek for his permanent encouragement and for useful discussions.
The author is grateful to the London Mathematical Society for a visiting grant, and to the Mathematics Department of the Imperial Collage for perfect working conditions.

Basic Definitions
In this section we give the necessary definitions and formulate the main result in detail.

Condition 1 During the whole paper for all
is a standing assumption.
The analysis of the random walk needs some basic elements of potential theory( [10]). For any finite subgraph, say for a ball A = B(w, R), w ∈ Γ, R > 0 the definition of the resistance (on the subgraph induced on A ) ρ(B, C) = ρ A (B, C) between two sets B, C ⊂ A is a well defined quantity if µ −1 x.y is the resistance associated to the edge x s y. Thanks to the monotonicity principle (c.f. [10]) this can be extended to the infinite graph, but we do not need it here. For the sake of short notation we shall introduce for x ∈ Γ, R > r ≥ 1 and ρ(x, r, R) = ρ(B(x, r), S(x, R)) for the resistance of the annulus.

Definition 2.1 We say that the random walk (or the graph) is strongly recurrent if there is a
Remark 2.1 It is evident that from (SR) it follows that there is a δ > 0 and c > 0 for which ρ(x, R) > cR δ (δ = log 2 (1+c ρ )). It is well known that a random walk is recurrent if ρ(x, R) → ∞ (c.f. [21], [10]), which means that strongly recurrent walks are recurrent.
The weakly recurrent case (i.e. the random walk is recurrent but (SR) is not true) is not dealt with in the present paper. In this case, a similar result is expected along very similar arguments, but the appearance of slowly varying functions brings in extra technical difficulties.

Definition 2.2 For
A ⊂ Γ, P A = P A (y, z) = P (y, z)| A×A is a sub-stochastic matrix, the restriction of P to the set A. It's iterates are denoted by P A k and it defines also a random walk, killed at the exiting from the ball.
is the local Green function (and Green kernel respectively). The notation P R = P x,R = P B(x,R) (y, z) will be used for A = B(x, R) and for the corresponding Green function by G R .

Remark 2.2
It is well-known that (c.f. [25]) where we have used the notation ∂A for the boundary of A : ∂A = {z ∈ Γ\A : ∃y ∈ A and y ∼ z}

Definition 2.4
We say that the graph has regular (relative to the volume) resistance growth if there is a µ > 0 such that for all x ∈ Γ, R > 0 .

Definition 2.5 The annulus resistance growth rate is defined similarly. It holds if there is a
The Laplace operator of finite sets is ∆ A = I − P A = (I − P )| A×A or particularly for balls is The smallest eigenvalue is denoted in general by λ(A) and for A = B(x, R) by λ = λ(x, R) = λ(B(x, R)). For variational definition and properties see [8].
Definition 2. 6 We shall say that the graph has regular eigenvalue property if there is a ν > 0 such that for all x ∈ Γ,

Statement of the results
The main result is the following In fact we show more in the course of the proof, namely. The proof of Theorem 2.1 follows the pattern shown below.
The idea, that in statement 1. of Theorem 2.1, the conditions regarding time, resistance and eigenvalue might be equivalent is due to A. Grigor'yan, as well as the suggestion that the R β −parabolic Harnack inequality could be inserted as a third equivalent statement.
The proof of the lower estimate is basically the same as it was given in [16]. The proof of the upper estimate and the equivalence of the conditions need several steps and new arguments. Corollary 4.6 and Theorem 4.1, collect some scaling relations. Theorem 5.1 uses the λ−resolvent technique (c.f. [5], [27]) while Theorem 6.1 is a generalization of [13].
During the whole paper several constants should be handled. To make their role transparent we introduce some convention. For important constants like C V we introduce a separate notation, for unimportant small (< 1) constants we will use c and big (> 1) constants will be denoted by C. The by-product constants of the calculation will be absorbed into one.

The exit time
Let us introduce the notation

Definition 3.1 The graph satisfies the center-point condition if there is a C > 0 such that
for all x ∈ Γ and R > 0.

Proposition 3.1 For all graphs (E β ) implies (E) and
The next Lemma has an important role in the estimate of the exit time and in the estimate of the λ−resolvent introduced later.

Lemma 3.1 For all
.
Proof. Denote n = t and observe that we obtain, by the strong Markov property, The following Theorem is taken from [16], see also [27], [28]. (Ψ)

Some potential theory
Before we start the potential analysis we ought to recall some properties of the measure and volume.
Proposition 4.1 If (p 0 ) holds then, for all x ∈ Γ and R > 0 and for some C = C(p 0 ), Proof. Let x ∼ y. Since P (x, y) = µxy µ(x) and µ xy ≤ µ(y), the hypothesis (p 0 ) implies p 0 µ(x) ≤ µ(y). Similarly, p 0 µ(y) ≤ µ(x). Iterating these inequalities, we obtain, for arbitrary x and y, Another consequence of (p 0 ) is that any point x has at most p −1 0 neighbors. Therefore, any ball The volume doubling has a well-known consequence, the so-called covering principle, which is the following Proof. The proof is elementary and well-known, hence it is omitted. The only point which needs some attention is that for R < 2 condition (p 0 ) has to be used.
We need some consequences of (D). The volume function V acts on Γ × N and has some further remarkable properties ( [8, Lemma 2.2]).
The graph has property (HG) if the local Green functions displays regular behavior in the following sense. There is a constant The analysis of the local Green function starts with the following Lemma which has been proved in [16,Lemma 9.2]. where Proof. The following potential-theoretic argument is borrowed from [6]. Denote for an X ⊂ Γ X = X ∪ ∂X. Given a non-negative harmonic function u in B 2 , denote by S u the following class of superharmonic functions: Define the function w on U by In particular, it suffices to prove (4.4) for w instead of u.
. Let us recall that the function E x (U ) solves the following boundary value problem in U : Using this and the strong minimum principle, v is superharmonic and strictly positive in U . Hence, for a large enough constant C, we have Cv ≥ u in B 1 whence Cv ∈ S u and w ≤ Cv.
Since v = 0 in U \ U , this implies w = 0 in U \ U and w ∈ c 0 (U ).
Denote f := ∆w. Since w ∈ c 0 (U ), we have, for any x ∈ U , Next we will prove that f = 0 outside A so that the summation in (4.8) can be restricted to z ∈ A. Given that much, we obtain, for all x, y ∈ B, We are left to verify that w is harmonic in B 1 and outside Consider the function w which is equal to w everywhere in U except for the point x, and w at x is defined to satisfy Hence, by the definition (4.6) of w, w ≤ w in U which contradicts w(x) > w (x). Proof. The proof of (HG) =⇒ (H) is just an application of the above lemma setting . The opposite direction follows by finitely many repetition of (H) using the balls covering B(x, A 2 R)\B(x, A 1 R) provided by the covering principle.

Proposition 4.3 If (SR) and (H) holds then there is a
Let us choose w ∈ S(x, R) which maximize ρ(Γ y , S x,M R ). From the maximum principle and the choice of w it follows that ρ(x, Γ y ) is minimized and on the other hand (c.f. [25]) ρ(Γ w , S(x, M R)) = 1 µ(x) G A (w, x), and using (HG) it follows that where the last inequality is a consequence of (SR). Finally it follows that We included this corollary for sake of completeness in order to connect our definition of strong recurrence with the usual one. The proof is easy, we give it in brief.

Corollary 4.4 If (SR) and (H) holds then there is a
Proof. Let us use Proposition 4.3.
where the last inequality follows from the maximum principle. The potential level of the vertex w maximizing G x,M R (., x) runs inside of B(x, R) and w ∈ S(x, R). Here we assume that R ≥ 3 and apply (HG) with A 0 = 1/3, For R ≤ 2 we use (p 0 ) adjusting the constant C.
The next proposition 1 is an easy adaptation of [25]. More precisely there is a constant c > 0 such that for all x ∈ Γ, R > 0 In addition (ρ β ) holds if and only if (E β ) holds.

Proof. The upper estimate is trivial
The lower estimate is almost as simple as the upper one.
at this point one can use (CG) to get from which the statement follows for all R = M i . For intermediate values of R the statement follows using R > M i trivial lover estimate and decrease of the leading constant as well as for R < M using (p 0 ).
The first eigenvalue of the Laplace operator I − P A for a set A ⊂ Γ is one of the key objects in the study of random walks (c.f. [8] ). Since it turned out that the other important tools are the resistance properties, it is worth finding a connection between them. Such connection was already established in [26] and [27]. Now we present some elementary observations which will be used in the rest of the proofs, and are interesting on their own.

Lemma 4.3 For all random walks on (Γ, µ) and for all
Proof. Assume that f ≥ 0 is the eigenfunction corresponding to λ = λ(A), the smallest eigenvalue of the Laplace operator ∆ A = I − P A on A and let f be normalized so that max y∈A f (y) = f (x) = 1. It is clear that which gives the statement.

Lemma 4.4 For all random walks on (Γ, µ) it is obvious that
and Proof.
The second statement follows from the first one taking maximum for x ∈ A on both sides.   [27], [28])For all random walks on weighted graphs and R ≥ 2

Proof. Apply Proposition 4.5 with
which provides the statement. Proof. The idea of the proof is based on [15] and [26]. Consider u(y) harmonic function on B defined by the boundary values u(x) = 1 on x ∈ A, u(y) = 0 for y ∈ Γ\B. This is the capacity potential for the pair A, B. It is clear that 1 ≥ u ≥ 0 by the maximum principle. From the variational definition of λ it follows that

Proposition 4.6 For all recurrent random walks and for all
where we have used the Ohm law, which says that the unit potential drops from 1 to 0 between ∂A to B results I ef f = 1/R ef f = 1/ρ(A, B), incoming current through ∂A and the outgoing "negative" current through ∂B. It is clear that (4.13) is just a particular case of (4.12), (4.14) follows from (4.13) using (D) finally, (4.15) can be seen applying Corollary (4.2).
The above results have an important consequence. It is useful to state it separately.

Corollary 4.6 If (p 0 ), (SR) and (H) holds then for all
where the arguments (x, R) are suppressed and ρ A = ρ(x, R, 2R). and . (4.20) Proof. This Theorem shows that the alternatives under the first condition in Theorem 2.1 are equivalent.

The diagonal estimates
The on-diagonal estimates basically were given in [26]. There the main goal was to get a Weyl type result by controlling of the spectral density via the diagonal upper (and lower) bounds of the process, killed at leaving B(x, R). The result immediately extends to the transition probabilities of the original chain.
and furthermore if n ≤ c 3 R β then .
The (DLE) follows from the next simple observation Proof. ¿From the condition using Lemma 3.1 it follows, that if n ≤ 1 2 E w (A) then which was to be shown.
Proof. The statement follows from Proposition 5.1.

(5.22)
Proof. We can apply Proposition 5.1 for A = B(x, R), to get (5.22) with w = x and having (E) thanks to Proposition 3.1.

Definition 5.1 Let us define the λ−resolvent and recall the local Green function as follows
and The starting point of the proof of the (DUE) is the following lemma (from [27]) for the λ−resolvent without any change.
Proof. The proof is elementary. It follows from the eigenfunction decomposition that P x) is non-increasing in n (c.f. [16] or [27]). For R > 2n P , x), hence the monotonicity holds for P 2n (x, x) in the 2n < R time range. But R is chosen arbitrarily, hence P 2n is non-increasing and we derive Proof. The argument is taken from [26,Lemma 6.4]. Let ξ λ be a geometrically distributed random variable with parameter a e −λ . One can see easily that Here P (T R ≥ ξ λ ) can be estimated thanks to Lemma 3.1 R) and (E) holds.

Proof of Theorem 5.1. Combining the previous lemmas with
Now let us recall from Remark 2.2 , that G R (x, x) = µ(x)ρ(x, R) and let us use the conditions and by Lemma 5.1 and 5.2 . ¿From this it follows that P 2n+1 (x, x) ≤ cµ(x) V (x, n 1 β ) −1 and with Cauchy-Schwartz and the standard argument (c.f. [8]) one has that This proves (DU E) and (P UE) and (DLE) follows from Proposition 5.2.

Off-diagonal estimates
In this section we deduce the off-diagonal estimates based on the diagonal ones.

Upper estimate
The upper estimate uses an idea of [13].  (w, z)) .
Proof. Let us observe first that the triangular inequality implies using the Jensen inequality that d(z, y))) .

Corollary 6.1 For all random walks and
The next step towards to the proof of (UE β ) is to get an estimate of E D (w, n). Lemma 6.2 For all w ∈ Γ, n ∈ N (P UE) and (Ψ) implies Proof. Let us assume first that d(w, z) < n 1 β in the summa of E D . In this case . (6.25) Let us consider the sum of "far away" vertices and denote δ = 1 The max can be handled as usual using (P UE) Applying this in the sum S(w, r)) exp εα 2 r β n Here P n (w, S(w, r)) can be estimated using (Ψ) to get further upper bound by where the last sum is evidently bounded by a constant depending only on c Ψ and β. The estimates in (6.25) and (6.27) provide the statement.
Proof of Theorem 6.1. Now we collect our findings. By Proposition 3.1 (E β ) implies (Ψ) while (P UE) is given by Theorem 5.1, consequently we can apply Lemma 6.2 in Lemma 6.1. The final step is standard to replace V (y, n Let us remark that this proof is considerably simpler than those given in [13], [8] with the aid of integral estimates and mean value inequalities while here we have the full power of (Ψ).

Lower estimate
For the lower estimate it is common to use the upper estimate of the time derivative of the heat kernel. The following statements are taken from [16]. Proposition 6.1 ([16,Proposition 11.2]) Assume that the elliptic Harnack inequality (H) holds on (Γ, µ). Let u ∈ c 0 (B(x, R)) satisfy in B(x, R) the equation ∆u = f . Then for any ε > 0, there exists σ = σ(ε, H) < 1 such that for any positive r < R,  . (6.33) By (H) and Proposition 6.1, we have, for any 0 < r < R and for some σ = σ(ε 2 ) ∈ (0, 1), As it is derived in Proposition 3. .
. This can be applied to get The elliptic Harnack inequality evidently follows from the β−parabolic one. The proof of the rest is via proving (DUE) and (PLE), namely for all x ∈ Γ, R > 0, A = B(x, 2R) if d(x, y) < R and 4 9 R β ≤ n ≤ 5 9 R β . .
Let us remark that the particular choice of the profile has no real importance, the only point is to ensure that n − = n, n + = 2n can be chosen. This will be applied repeatedly without any further comment.
The next step is to show (P H β ) =⇒ (7.35) This can be seen again from (P H β ) applied to P A n getting P A n (z, y) ≤ C H P A 2n (x, y) for z ∈ B(x, R) and for an other solution u of a parabolic equation with boundary conditions as follows. Let u k (w) defined on [0, R β ] × B(x, 2R) and (P H β ) provides for y ∈ B(x, R/2) This proves (7.35) for R ≥ 9, for small R − s the statement follows from (p 0 ). We have got (P LE) for 2n and it follows for 2n + 1 using (p 0 ) in the one step decomposition. It is clear that (NLE) follows from (7.35) imposing the condition d(x, y) ≤ n 1 β (which is stronger than the assumed d(x, y) ≤ R, n ≤ 5 9 R β ).
Proof. The statement direct consequence of (4.13) and Proposition 7.3.
The lower estimate of E is quite simple in the possession of the (7.35) which is a consequence again of (P H β ) by Proposition 7.1.

Proposition 7.4
If (p 0 ) and (7.35) (variant of (P LE) ) holds, then there is a c > 0 such that for all x ∈ Γ, R ≥ 0 E(x, R) ≥ cR β Proof. We assumed that for all x ∈ Γ, R ≥ 1

Poincaré Inequality
In this section we show that a Poincaré type inequality follows from the parabolic Harnack inequality. The opposite direction is not clear. There are some indications that it might be generally not true.