Anchored heat kernel upper bounds on graphs with unbounded geometry and anti-trees

We derive Gaussian heat kernel bounds on graphs with respect to a fixed origin for large times under the assumption of a Sobolev inequality and volume doubling on large balls. The upper bound from our previous work [KR22] is affected by a new correction term measuring the distance to the origin. The main result is then applied to anti-trees with unbounded vertex degree, yielding Gaussian upper bounds for this class of graphs for the first time. In order to prove this, we show that isoperimetric estimates with respect to intrinsic metrics yield Sobolev inequalities. Finally, we prove that anti-trees are Ahlfors regular and that they satisfy an isoperimetric inequality of a larger dimension.

The focus of the present note lies on Gaussian upper bounds for heat kernels on graphs with possibly unbounded geometry.Seminal works go back to Davies and Delmotte for graphs with bounded geometry, [Dav93a,Del99].In contrast, we will be dealing with graphs which do not satisfy any a priori boundedness assumption on the local geometry of the graph.First results in this direction have been obtained in [Fol11, ADS16, BC16, BHY17, CKW20].Recently, the authors obtained Gaussian bounds for heat kernels for large times on graphs with unbounded geometry.The assumptions include Sobolev inequalities and volume doubling in large balls in terms of intrinsic metrics [KR22].Here, we derive an anchored version of these recent Gaussian upper bounds of the heat kernel p of locally finite weighted discrete graphs over a discrete measure space (X, m), cf.Section 2. Applied to anti-trees, our results yield for the first time Gaussian upper bounds for their heat kernels and, hence, provide explicit examples of graphs with unbounded vertex degree.
In order to highlight our contribution it is instructive to recall a simplified version of the main result Theorem 6.1 from [KR22].Assume that the graph is equipped with an intrinsic metric ρ of jump size one.Moreover, all large balls around x, y ∈ X are supposed to satisfy a Sobolev inequality with exponent n > 2 and volume doubling.Then, the authors obtained Gaussian upper bounds of the form p t (x, y) ≤ f (x, y, t) 1 ∨ ( t 2 + ρ(x, y) 2 − t) for large t > 0. The function ζ 1 is optimal as observed by Davies and satisfies meaning that the fraction of the left-and right-hand side converges to one.Moreover, the polynomial correction term has different behavior as t → 0 than the corresponding term on manifolds, cf.[Gri09].The function f measures the unboundedness of the geometry about x and y.It is bounded if, e.g., m is the normalizing measure, or the L p -means of vertex degree and inverted measure in balls grow at most exponentially.Note that no a priori boundedness of the vertex degree or the measure is assumed.
The aim of this note is twofold.First, we will relax the conditions on the vertices.Instead of assuming that large balls around the vertices x and y fulfill Sobolev and volume doubling, we rather impose these properties about a root vertex o ∈ X.This yields Gaussian upper bounds for all vertices with an additional polynomial correction term measuring the distance to o.In fact, Theorem 2.1 states that for all x, y ∈ X and t > 0 large The function f o measures the unboundedness of the geometry with respect to o, incorporating the distance from x and y.The function f o is bounded if m is the normalizing measure or weighted L p -means of the vertex degree and inverted measure about o grow at most exponentially.The above statements require Sobolev and volume doubling for all large balls.Our main result Theorem 2.1 even yields a localized version involving the spectral bottom of the Laplacian.To the best of our knowledge, this is the first variant of this result in this form on graphs.
For degenerating parabolic equations in R n , Zhikov imposed conditions at the origin and derived off-diagonal Gaussian upper bounds, [Zhi13].A similar approach was used for elliptic operators on Z n with unbounded but integrable weights in [MO16,ADS16].However, the latter articles do not provide the optimal Gaussian estimate.
Figure 1: First spheres of an anti-tree.
Our second aim concerns the application of our results to anti-trees.These graphs play a prominent role in the theory of graphs with unbounded geometry as they provide examples for similarities and disparities between graphs and manifolds.Specifically, considering the combinatorial graph distance, they includeexamples of polynomial growth which are stochastically incomplete [Woj11] and have a spectral gap [KLW13].This disparity was later resolved by the use of intrinsic metrics [Fol14, HKW13, Hua11, HKS20, KLW21].
To illustrate the anti-trees we are considering here, let γ ≥ 0, X an anti-tree with root o an which has s k−1 = [k γ ] vertices in the sphere with combinatorial distance k ≥ 1 to o.In contrast to trees where each vertex has only one backwards neighbor, anti-trees have complete bipartite graphs between spheres.Furthermore, let ρ the intrinsic path degree metric, cf.Section 4. Then if γ = 0, the anti-tree is isomorphic to N, for γ ∈ (0, 2) it has polynomial volume growth, for γ = 2 the volume growth is exponential, and for γ > 2 the intrinsic diameter is bounded (with respect to the intrinsic metric ρ).
So far, there were no results on heat kernel bounds of the above form for antitrees as their vertex degree is not bounded.For γ ∈ (0, 2), we denote d = 2(γ+1) 2−γ which appears as the volume growth dimension.Then there is C 0 > 0 such that for all x, y ∈ X with either ρ(o, x) = ρ(o, y) or x = y and t ≥ 2 • 72 2 we have Replacing the intrinsic metric ρ with the combinatorial graph metric, we denote the combinatorial distance of x to the root o by |x|.Then, we can further estimate this heat kernel bound to obtain the following for large t > 0 In the proof we show that such anti-trees satisfiy certain isoperimetric estimates in all distance balls which then in turn implies the required Sobolev inequalities.Albeit the latter implication is classical for manifolds, we could not find a reference on general graphs with respect to intrinsic metrics.Therefore, we generalize the proof of the Cheeger inequality to our setting from [KLW21].Similar Gaussian bounds for the heat kernel on anti-trees could be derived from the main results in [KR22] in conjunction with our developments on isoperimetric constants.However, this bound would not include the first polynomial correction term displayed above and would only hold for larger times, cf.Theorem 4.10.
The outline of this paper is as follows.In Section 1, we introduce weighted graphs, their Laplacian, intrinsic metric, and the heat kernel.Moreover, we recall the Sobolev and volume doubling properties we are working with.In Section 2 we formulate and prove our main result.The implication that isoperimetric estimates yield Sobolev inequalities is proven in Section 3. Section 4 is devoted to the study of anti-trees.We show that their heat kernels are spherically symmetric and that their study can therefore be reduced to the study of the heat kernel on a particular model graph on N 0 .It will be shown that these models satisfy isoperimetry and Ahlfors regularity, i.e., polynomial volume growth, if the sphere function is chosen as above.

The set-up
Let X be a countable set, m : X → (0, ∞) a measure on X, and b a locally finite connected graph over (X, m).We denote by C(X) the set of continuous functions on X, and where we interpret the right-hand side as divergence form operator.Note that ∆ is non-negative and symmetric in ℓ 2 (X, m).By abusing notation, we also denote by ∆ ≥ 0 iits Friedrichs extension.We let be the infimum of the spectrum of ∆.The heat kernel p is the minimal integral kernel of the heat semigroup (P t ) t≥0 , i.e., P t = e −t∆ , t ≥ 0, and for all f ∈ ℓ 2 (X, m) and t ≥ 0 we have Moreover, for any f ∈ ℓ 2 (X, m), the map t → P t f solves the heat equation In the following, ρ will always be a non-trivial intrinsic (pseudo-) metric with finite jump size For r ≥ 0 and o ∈ X, we denote We abbreviate B(r) = B o (r) if there is no confusion about the role of o.A standing assumption on the intrinsic metric is the following.
Assumption 1.1.The distance balls with respect to the intrinsic metric are compact and the jump size is finite. .
The combinatorial interior of a set A ⊂ X will be denoted by The results presented in here are obtained by assuming the classical Sobolev and volume doubling assumptions.We encode them for weighted graph Laplacians in terms of intrinsic metrics.The big difference to the assumptions on manifolds is that we require these properties only on annuli with positive inradius instead of all small balls.
(ii) The volume doubling property ), however, with different constants.Due to this asymmetry in the radii the volume doubling assumption

Anchored Gaussian upper bounds
In order to formulate our main result, we recall the following weighted means of the degree and the inverse measure introduced in [KR22].The weighted vertex degree is given, for x ∈ X, by For R ≥ 0, and p ∈ (1, ∞), we define Define the error-function Γ(r) := Γ o (r, p, n, β) ≥ 0 by Our main theorem reads as follows.
where r(t, x) As initially observed by Davies, [Dav93b], instead of the Gaussian e −r 2 /4t known from manifolds, for graphs the function e −ζ S (r,t) with for r ≥ 0, t > 0, appears, where S is the jump size of the intrinsic metric.
In order to prove Theorem 2.1, we use two results from [KR22] based on Davies' method.Here we combine these two propostions in a different way than in [KR22] to obtain anchored estimates.To obtain estimates of solutions of the heat equation, we investigate properties of solutions of the ω-heat equation where ∆ ω := e ω ∆e −ω is a sandwiched Laplacian for ω ∈ ℓ ∞ (X).The following proposition is the first ingredient of the proof of the main theorem.It provides an ℓ 2 -mean value inequality for non-negative solutions of the ω-heat equation.The displacement of the solutions with respect to the heat equation is measured in terms of the function , and all non-negative solutions v ≥ 0 of the ω-heat equation on Proposition 2.2 suffices to derive off-diagonal heat kernel bounds as shown by the following proposition, our second ingredient, in points where ℓ 2 -mean value inequalities are available.To this end, observe that, denoting P ω t := e ω P t e −ω , the semigroup (P ω t ) t≥0 acts on ℓ 2 (X, m).Moreover, the map t → P ω t f solves the ω-heat equation for f ∈ ℓ 2 (X, m) and ω ∈ ℓ ∞ (X).
Proof of Theorem 2.1.
Now we choose τ ∈ (0, 1] and estimate the remaining terms.We let satisfying τ ∈ (0, 1] and immediately get a similar estimate for r(t, x) replaced by r(t, y), and Moreover, we obtain Now, we estimate τ − n 2 .By the definition of r(t, z), z ∈ {x, y}, we have we obtain Plugging in the above estimates into the upper bound for p t (x, y) yields the claim.

Isoperimetric and Sobolev inequalities
In this section we show that isoperimetric inequalities formulated in terms of intrinsic metrics imply Sobolev inequalities.These results will be used in Section 4 in order to obtain heat kernel upper bounds for anti-trees.For a graph b over (X, m), an intrinsic metric ρ, U ⊂ X and n ∈ (2, ∞], let , where Note that h U,∞ is also known as the Cheeger constant of U .It was shown in [BKW15] that the bottom of the spectrum can be bounded in terms of h U,∞ .These considerations directly extend to bound the first Dirichlet eigenvalue of a finite set.In the following we adapt this proof in order to obtain a lower bound on the Sobolev constant instead.In order to do so, we collect some lemmas.
The following lemma is well-known, we included a proof in the Appendix A for the reader's convenience.

Lemma 3.1. For a decreasing function
As usual, we denote for a function f : The following lemma is found in [KLW21, Lemma 10.8].
Lemma 3.2 (Co-area formula).For w : X × X → [0, ∞) and f : X → R, we have The next lemma is an adapted version of the area formula found in [KLW21, Lemma 10.9].This corresponds to the special case α = 1.
This settles the claim.
Theorem 3.4 (Sobolev inequality and isoperimety).For n ∈ (2, ∞], φ ∈ C c (U ) and C > 0 we have Proof.We employ the area formula with α = (n − 2)/n, the basic inequality from Lemma 3.1, the co-area formula, the definition of h U,n , and the fact that ρ is intrinsic to obtain for φ ∈ C c (U ) This proves the claim.

Anti-trees
In this section we show that anti-trees of a certain growth fulfill the assumptions of Theorem 2.1.This is can be achieved by reducing the analysis on the anti-tree to a one-dimensional problem.
We consider here graphs b with standard weights that is b(x, y) ∈ {0, 1}, x, y ∈ X.For a connected graph b over X, we denote by the distance spheres with respect to the combinatorial graph distance d to the root o ∈ X.Clearly, S −k = ∅ for k ∈ N. We furthermore denote For a so-called sphere function s : N −→ N the corresponding anti-tree is a graph b with standard weights over a set X such that for the distance spheres we have for k ≥ 0 and b(x, y) = 0 otherwise.For convenience later on, we set s 0 = 1 and s −1 = 0.Moreover, we equip X with the counting measure m = 1 and denote for k ∈ Z the combinatorial balls by Lemma 4.1 (Characterization anti-trees).Let b be connected a graph with standard weights such that there are no edges within spheres.The following statements are equivalent: (i) b is an anti-tree.
(ii) For any vertices x, y, x ′ , y ′ such that |x| = |x ′ | and |y| = |y ′ |, there is a graph isomorphism which interchanges x and x ′ as well as y and y ′ .
Proof.(i) =⇒ (ii): For an anti-tree the map which interchanges x with x ′ and y with y ′ and leaves every other vertex invariant is a graph isomorphism since x and x ′ as well as y and y ′ have the same neighbors each.(ii) =⇒ (i): The forward graph F x of a vertex x with respect to a root o is a subgraph of b induced by the vertices z ∈ k≥|x| S k such that d(x, z) = k for z ∈ S k .
Clearly, a graph isomorphism which interchanges x and x ′ with |x| = |x ′ | also has to map the forward graph F x to F x ′ and vice versa.If b is not an anti-tree, then there is a vertex x which is not connected to some y ∈ S |x|+1 , i.e., y ∈ F x .Furthermore, there is x ′ ∈ S |x| which is connected to y, i.e., y ∈ S |x ′ |+1 .Thus, there is no graph isomorphism interchanging x and x ′ and which keeps y = y ′ invariant: if there was such a ι, then 0 For a spherically symmetric function f , the corresponding Laplacian acts as To a given anti-tree with sphere function s, we can associate a one-dimensional weighted graph b over ( X, m) with for k ≥ 0 and b(k, l) = 0 otherwise.For the corresponding Laplacian ∆ and a spherically symmetric function f and x ∈ X we have Lemma 4.2.For an anti-tree b over (X, m) with heat kernel p and corresponding graph b over (N 0 , m) with heat kernel p, we have Proof.By the characterization of anti-trees for any x, y, x ′ , y ′ such that |x| = |x ′ | and |y| = |y ′ | (and y = y ′ if and only if x = x ′ ), there is a graph isomorphism ι interchanging x and x ′ as well as y and y ′ .This means that ι extends to a unitary operator on ℓ 2 (X, m) via f → f • ι which commutes with the Laplacian.Thus, it commutes with the semigroup.Hence, p t (x, y) = p t (x ′ , y ′ ) for all such x, y, x ′ , y ′ .In particular, the heat kernel is spherically symmetric about o in both variables, so there is a function q such that p t (x, y) = q t (|x|, |y|), x, y ∈ X with x = y if |x| = |y|, t ≥ 0. Specifically, q solves the heat equation for b, i.e., and q 0 (|x|, |y|) = 1 |y| (|x|).Since (t, x) → p t (x, y) is the smallest positive solution to the heat equation for b, so is q for b and, hence, q is the heat kernel for b.
From now on, we will choose the intrinsic metric ρ to be the intrinsic degree metric.Recall that for a graph b over (X, m) the intrinsic degree metric is given by for x, y ∈ X, where for antitrees the weighted degree is given by For an antitree b and the associated one-dimensional graph b, we denote the corresponding intrinsic degree metric ρ for b over (N 0 , m).Proof.Observe that for the weighted degree Deg of b we have Moreover, for any path between x and y with either |x| = |y| or x = y there is a unique path on N 0 .On the other hand, all the different paths in X corresponding to the same path on N 0 have the same length.This settles the claim.
We denote the closed distance balls about x ∈ X and |x| ∈ X with respect to ρ and ρ by B x (r) and Bk (x) for r ∈ R.Moreover, we denote the open balls in X and X by B • x (r) = {y ∈ X | ρ(x, y) < r} and B• |x| (r) = {|y| ∈ X | ρ(|x|, |y|) < r} respectively.From now on for given γ ∈ [0, 2), we consider the anti-tree with the sphere function The intrinsic and combinatorial distance can be estimated against each other which is elaborated in [Hua11].For convenience, we recall the argument here.For two real valued functions f and g, we denote In particular, there are c, C ≥ 0 such for every r there is c ≤ θ ≤ C such that This proves the first statement.The "in particular" statement is a direct consequence.
The following constant can be understood as the dimension of the antitree with parameter γ ∈ (0, 2).In particular, for we have V (d, R, ∞) in every x ∈ X with R ≥ ρ(x, o).
Proof.The equalities follow directly from the definitions.Since for N ∈ N 0 , Thus, by the lemma above The statement for r ≥ r x is clear and the statement for r ≤ r where the last estimate follows as Br (|x|) = Br 0 +r (0) for r ≥ r 0 , i.e. we have the bounds m( Br (|x|)) ≍ r 2(γ+1)/(2−γ) .This finishes the proof.
We apply the estimates to obtain Sobolev inequalities by Theorem 3.4.
Next we estimate the averaged degree functions Proof.The equality follows by definition.Recall that This proves the asymptotics for p < ∞.For p = ∞, one clearly obtains the bounds D ∞ (x, r) = s θ(r+r 0 ) 2/(2−γ) ≍ r 2γ/(2−γ) .Now, we collected all the ingredients to prove heat kernel estimates on anti-trees.
Proof.Again by Lemma 4.3 and Lemma 4.2, it suffices to prove the statement for the one-dimensional kernel p and ρ.
We apply [KR22, Theorem 6.1].We choose β = 1 + 1/n, p = ∞ and R 0 = 72.Then, for x, y we have SV (r, ∞) for r ≥ r x ∨ r y ∨ R 0 by Lemma 4.5 and Proposition 4. Acknowledgements.The authors acknowledge the financial support of the DFG.

A Appendix
Proof of Lemma 3.

.
x follows by the mean value theorem after factoring out r on the right hand side.The claim about volume doubling follows since m(B r (x)) = m(B r+r 0 (o)) for all r ≥ r 0 = ρ(x, o).Next, we estimate the isoperimetric constant of distance balls which we need to obtain a Sobolev inequality via Theorem 3.4.For the one dimensional graph b associated to an antitree and a finite set B ⊆ X recall that hB,n = inf Lemma 4.6 (Isoperimetic inequality).For γ ∈ [0, 2), n ≥ 2d = 4(γ+1) 2−γ ≥ 2, x ∈ X and r ≥ 1 ∨ ρ(o, x), we have h Br (|x|),n ≍ m( Br (|x|)) Since r ≥ 1 ∨ ρ(o, x) =: r 0 , we have that o ∈ B r (x).So, to determine the infimum in h Br(|x|),n , it suffices to consider balls B R = B R (o) with R ≤ r 0 + r and estimate bρ( BR ) m( BR )