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arXiv:1711.11119v1 [math.PR] 29 Nov 2017
HEAT KERNEL ESTIMATES AND INTRINSIC METRIC FOR RANDOM WALKS
WITH GENERAL SPEED MEASURE UNDER DEGENERATE CONDUCTANCES
SEBASTIAN ANDRES, JEAN-DOMINIQUE DEUSCHEL, AND MARTIN SLOWIK
ABS TR AC T. We establish heat kernel upper bounds for a continuous-time random
walk under unbounded conductances satisfying an integrability assumption, where
we correct and extend recent results in [3] to a general class of speed measures.
The resulting heat kernel estimates are governed by the intrinsic metric induced
by the speed measure. We also provide a comparison result of this metric with the
usual graph distance, which is optimal in the context of the random conductance
model with ergodic conductances.
1. INTRODU CT IO N
Let G= (V, E )be an infinite, connected, locally finite graph with vertex set V
and (non-oriented) edge set E. We will write x∼yif {x, y} ∈ E. Consider a family
of positive weights ω={ω(e)∈(0,∞) : e∈E} ∈ Ω, where Ω = RE
+is the set of all
possible configurations. We also refer to ω(e)as the conductance of the edge e. With
an abuse of notation, for x, y ∈Vwe set ω(x, y) = ω(y, x) = ω({x, y})if {x, y} ∈ E
and ω(x, y) = 0 otherwise. Let us further define measures µωand νωon Vby
µω(x):=X
y∼x
ω(x, y)and νω(x):=X
y∼x
1
ω(x, y).
Given a speed measure θ:V→(0,∞)we consider a continuous time continuous
time Markov chain, X={Xt:t≥0}, on Vwith generator Lω
θacting on bounded
functions f:V→Ras
Lω
θf)(x) = 1
θ(x)X
y∼x
ω(x, y)f(y)−f(x).(1.1)
Then the Markov chain, X, is reversible with respect to the speed measure θ, and
regardless of the particular choice of θthe jump probabilities of Xare given by
pω(x, y):=ω(x, y)/µω(x),x, y ∈V, and the various random walks corresponding
to different speed measures will be time-changes of each other. The maybe most
natural choice for the speed measure is θ≡θω=µω, for which we obtain the
constant speed random walk (CSRW) that spends i.i.d. Exp(1)-distributed waiting
times at all visited vertices. Another frequently arising choice for θis the counting
measure, i.e. θ(x) = 1 for all x∈V, under which the random walk waits at x
Date: December 1, 2017.
2010 Mathematics Subject Classification. 39A12; 60J35; 60K37;82C41.
Key words and phrases. random walk; heat kernel; chemical distance.
1
2 SEBASTIAN ANDRES, JEAN-DOMINIQUE DEUSCHEL, AND MARTIN SLOWIK
an exponential time with mean 1/µω(x). Since the law of the waiting times does
depend on the location, Xis also called the variable speed random walk (VSRW).
For any choice of θwe denote by Pω
xthe law of the process Xstarting at the
vertex x∈V. For x, y ∈Vand t≥0let pω
θ(t, x, y)be the transition densities of X
with respect to the reversible measure (or the heat kernel associated with Lω
θ), i.e.
pω
θ(t, x, y):=Pω
xXt=y
θ(y).
As our first main result we establish upper bounds on the heat kernel under a certain
integrability condition on the conductances, see Theorem 2.5 below. The resulting
bounds are of Gaussian type apart from an additional factor which may vanish for
specific choices of the speed measure or the conductances (see Remark 2.6 below).
It is well known that Gaussian bounds hold, for instance, for the CSRW on locally
finite graphs in the uniformly elliptic case, that is c−1≤ω(e)≤cfor all e∈Efor
some c≥1, see [12]. More recently, Folz showed in [15] upper Gaussian estimates
for elliptic random walk for general speed measures that need to be bounded away
from zero, provided on-diagonal upper bounds at two vertices are given. In [3]
we weakened the strict ellipticity condition and showed heat kernel upper bounds
for the CSRW and VSRW under a similar integrability condition as in Theorem 2.5,
while in the present paper we extend this result to general speed measures. Notice
that some integrability assumption on the conductances is necessary for Gaussian
bounds to hold. In fact, it is well known that due to a trapping phenomenon under
random i.i.d. conductances with sufficiently heavy tails at the zero the subdiffusive
heat kernel decay may occur, see [6,7] and cf. [8]. For the proof of Theorem 2.5
we use the same strategy as in [3] which is based on a combination of Davies’
perturbation method (cf. e.g. [10,11,9]) with a Moser iteration following an idea
in [20]. We refer to [3, Section 1.2] for a more detailed outline of the method.
Naturally, the heat kernel upper bounds in Theorem 2.5 are governed by the
distance function dω
θon V×Vdefined by
dω
θ(x, y):= inf
γ∈Γxy (lγ−1
X
i=0 1∧θ(zi)∧θ(zi+1)
ω(zi, zi+1)1/2),(1.2)
where Γxy is the set of all nearest-neighbor paths γ= (z0,...,zlγ)connecting x
and y(cf. [11,5,15,17,3]). Note that dω
θis a metric which is adapted to the
transition rates and the speed measure of the random walk. Further, for the CSRW,
i.e. θ≡θω=µω, the metric dω
θcoincides with the usual graph distance d, and for a
VSRW dω
θbecomes the so-called chemical distance. In general, dω
θcan be identified
with the intrinsic metric generated by the Dirichlet form associated with Lω
θand X,
see Proposition 2.3 below. Further, notice that dω
θ(x, y)≤d(x, y)for all x, y ∈V.
In fact, the distance dω
θcan become much smaller than the graph distance, see [3,
Lemma 1.12] for an example in the context of a VSRW under random conductances.
As our second main result stated in Theorem 3.3 below, for any x, y ∈Vsufficiently
HEAT KERNEL AND INTRINSIC METRIC FOR RANDOM WALKS WITH GENERAL SPEED MEASURE 3
far apart, we provide under a suitable integrability condition on ωa lower bound
on dω
θ(x, y)in terms of a certain power of d(x, y). This lower bound turns out to be
optimal within our general framework up to an arbitrarily small correction in the
exponent.
The rest of the paper is organised as follows. In Section 2we show the heat
kernel upper bounds. The lower bound on the chemical distance in terms of the
graph distance is proven in Section 3and in Section 4we discuss its optimality
by providing an example in the context of the random conductance model on Zd.
Throughout the paper we write cto denote a positive constant which may change on
each appearance. Constants denoted Ciwill be the same through each argument.
2. HEAT KERNEL UPPER BOUND S
2.1. Preliminaries. The graph Gis endowed with the counting measure, i.e. the
measure of A⊂Vis simply the number |A|of elements in A. Further, we denote by
B(x, r)the closed ball with center xand radius rwith respect to the natural graph
distance d, that is B(x, r):={y∈V|d(x, y)≤r}. Throughout the paper we will
make the following assumption on G.
Assumption 2.1. The graph Gsatisfies the following conditions.
(i) Uniformly bounded vertex degree, that is there exists Cdeg ∈[1,∞)such that
|{y:y∼x}| ≤ Cdeg,∀x∈V. (2.1)
(ii) Volume regularity of order dfor large balls, that is there exist d≥2and
Creg ∈(0,∞)such that for all x∈Vthere exists N1(x)<∞with
C−1
reg nd≤ |B(x, n)| ≤ Creg nd,∀n≥N1(x).(2.2)
(iii) Local Sobolev inequality (S1
d′)for large balls, that is there exists d′≥dand
CS1∈(0,∞)such that for all x∈Vthe following holds. There exists N2(x)<
∞such that for all n≥N2(x),
X
y∈B(x,n)|u(y)|d′
d′−1!d′−1
d′
≤CS1n1−d
d′X
y∨z∈B(x,n)
{y,z}∈Eu(y)−u(z)(2.3)
for all u:V→Rwith supp u⊂B(x, n).
Remark 2.2.The Euclidean lattice, (Zd, Ed), satisfies the Assumption 2.1 with d′=d
and N1(x) = N2(x) = 1.
For f:V→Rwe define the operator ∇by
∇f:E→R, E ∋e7−→ ∇f(e):=f(e+)−f(e−),
where for each non-oriented edge e∈Ewe specify one of its two endpoints as
its initial vertex e+and the other one as its terminal vertex e−. Further, the corre-
sponding adjoint operator ∇∗F:V→Racting on functions F:E→Ris defined
4 SEBASTIAN ANDRES, JEAN-DOMINIQUE DEUSCHEL, AND MARTIN SLOWIK
in such a way that h∇f , F iℓ2(E)=hf, ∇∗Fiℓ2(V)for all f∈ℓ2(V)and F∈ℓ2(E).
Notice that in the discrete setting the product rule reads
∇(fg) = av(f)∇g+ av(g)∇f, (2.4)
where av(f)(e):=1
2(f(e+) + f(e−)). On the weighted Hilbert space ℓ2(V, θ )the
Dirichlet form associated with Lω
θis given by
Eω(f, g):=f, −Lωgℓ2(V ,θ)=∇f, ω∇gℓ2(E)=1,dΓω(f, g)ℓ2(E),(2.5)
where dΓω(f, g):=ω∇f∇gand Eω(f) = Eω(f, f ).
As a first step, we identify the metric dθas the intrinsic metric of the Dirichlet
form Eωon ℓ2(V, θ).
Proposition 2.3. For every x, y ∈V,
dω
θ(x, y) = sup nψ(y)−ψ(x) : k∇ψk∞≤1,dΓω(ψ, ψ)(e)≤θ(e+)∧θ(e−), e ∈Eo.
Proof. We follow the argument in [17, Proposition 10.4]. For any x, y ∈Vset
∆ω
θ:= sup nψ(y)−ψ(x) : k∇ψk∞≤1,dΓω(ψ, ψ)(e)≤θ(e+)∧θ(e−), e ∈Eo.
Then, for any function ψ:V→Rwith the properties that k∇ψk∞≤1and
dΓω(ψ, ψ)(e)≤θ(e+)∧θ(e−)for all e∈Ewe obtain
∇ψ(e)≤1∧θ(e+)∧θ(e−)
ω(e)1/2
.
Let γ∈Γx,y be a nearest neighbour path connecting xand y. By summing over all
consecutive vertices in γ, we get that ψ(y)−ψ(x) = Plγ−1
i=0 ψ(zi+1)−ψ(zi). Thus,
∆ω
θ(x, y)≤dω
θ(x, y).
In order to obtain ∆ω
θ(x, y)≥dω
θ(x, y), set ψ(z):=dω
θ(x, z)for all z∈V. Then,
for any edge e∈Ean application of the triangle inequality and the definition of dω
θ
yields
∇ψ(e)≤dω
θ(x, e+)−dω
θ(x, e−)≤dω
θ(e+, e−)≤1.
Likewise, it follows that, for any e∈E,
dΓω(ψ, ψ)(e)≤ω(e)dω
θ(e+, e−)2≤θ(e+)∧θ(e−).
Thus, ψsatisfies the requirements in the definition of ∆ω
θ(x, y). Since ψ(x) = 0 we
finally have that dω
θ(x, y)≤∆ω
θ(x, y).
For some φ:V→[0,∞),p∈[1,∞)and any non-empty, finite B⊂V, we define
space-averaged weighted ℓp-norms on functions f:B→Rby
f
p,B,φ :=1
|B|X
x∈B|f(x)|pφ(x)1/p
and
f
∞,B := max
x∈B|f(x)|.
If φ≡1, we simply write kfkp,B :=kfkp,B,φ.
HEAT KERNEL AND INTRINSIC METRIC FOR RANDOM WALKS WITH GENERAL SPEED MEASURE 5
2.2. Result. Our main objective in this section is to prove Gaussian-like upper
bound on the heat kernel pθin term of the intrinsic distance dθ. For that purpose,
we impose the following assumption on the integrability of the conductances.
Assumption 2.4. Let d′≥2. For p, q, r ∈(1,∞]with
1
r+1
p·r−1
r+1
q<2
d′(2.6)
there exists Cint ∈[1,∞)such that for all x∈Vthere exists N3(x, ω)<∞such that
for all n≥N3(x, ω),
1∨µω/θ
p,B(x,n),θ ·
1∨νω
q,B(x,n)·
1∨θ
r,B(x,n)·
1∨1/θ
1,B(x,n)≤Cint.
(2.7)
Theorem 2.5. Suppose that ω∈Ωsatisfies Assumption 2.4. Then, there exist con-
stants ci=ci(d, p, q, Cint)and γ=γ(d, p, q, Cint)such that for any given tand xwith
√t≥N1(x)∨N2(x)∨N3(x, ω)and all y∈Vthe following hold.
(i) If dω
θ(x, y)≤c1tthen
pω
θ(t, x, y)≤c2t−d/21 + d(x, y)
√tγ
exp−c3
dω
θ(x, y)2
t.
(ii) If dω
θ(x, y)≥c5tthen
pω
θ(t, x, y)≤c2t−d/21 + d(x, y)
√tγ
exp−c4dω
θ(x, y)1∨log dω
θ(x, y)
t.
Remark 2.6.(i) If the distance dω
θand the graph distance dare comparable, as
for instance in the case of CSRW, the estimates in Theorem 2.5 turn into Gaussian
upper bounds since then the additional term (1 + d(x, y)/√t)γcan be absorbed by
the exponential term into a constant.
(ii) In the case of CSRW or VSRW Theorem 2.5 has been established in [3].
However, the term (1 + d(x, y)/√t)γis erroneously missing in the result for the
VSRW in [3, Theorem 1.10].
(iii) The on-diagonal decay t−d/2corresponds to 1/B(x, √t). In general we
expect a stronger decay to hold resulting from the volume of a ball with radius √t
w.r.t. the distance dω
θunder the speed measure θ.
In the remainder of this section we explain how the proof of [3, Theorem 1.6]
needs to be adjusted in order to prove Theorem 2.5, that is to obtain Gaussian-like
upper bounds on the the heat kernel for a larger class of speed measures θ. We
also take the opportunity to streamline the arguments in [3] and to correct some
technical mistakes leading to the error mentioned in Remark 2.6.
2.3. Maximal inequality for the perturbed Cauchy problem. We consider the
following Cauchy problem
(∂tu− Lω
θu= 0,
u(t= 0,·) = f, (2.8)
6 SEBASTIAN ANDRES, JEAN-DOMINIQUE DEUSCHEL, AND MARTIN SLOWIK
for some function f:V→R. Recall that for any given y∈Zd, the function
(t, x)7→ pω
θ(t, x, y)solves the heat equation (2.8) with f=
1
{y}/θ(y). For any
positive function φon Vsuch that φ, φ−1∈ℓ∞(V)we define the operator Lω
θ,φ
acting on bounded functions g:V→Ras
(Lω
θ,φ g)(x):=φ(x)(Lω
θφ−1g)(x).
As a first step we establish the following a-priori estimate.
Lemma 2.7. Suppose that f∈ℓ2(V, θω)and usolves the corresponding Cauchy prob-
lem (2.8). Further, set v(t, x):=φ(x)u(t, x)for a positive function φon Vsuch that
φ, φ−1∈ℓ∞(V). Then
v(t, ·)
ℓ2(V,θ)≤ehω
θ(φ)t
φf
ℓ2(V,θ),(2.9)
where
hω
θ(φ):= max
x∈V
1
2θ(x)X
y∼xdΓω(φ, φ−1)({x, y}).
Proof. This can be shown by the similar arguments as in [3, Lemma 2.1].
Our next aim is to derive a maximal inequality for the function v. For that purpose
we will adapt the arguments given in [2, Section 4] and set up a Moser iteration
scheme. For any finite interval I⊂R, finite, connected B⊂Vand p, p′∈(0,∞),
let us introduce a space-time-averaged norm on functions u:R×V→Rby
u
p,p′,I×B,θ :=1
|I|ZI
ut
p′
p,B,θ dt1/p′
and
u
p,∞,I×B,θ := sup
t∈I
ut
p,B,θ ,
where ut=u(t, .),t∈R.
Lemma 2.8. Suppose that Q=I×B, where I= [s1, s2]⊂Ris an interval and
B⊂Vis finite and connected. For a given φ > 0with φ, φ−1∈ℓ∞(V), let vt≥0be a
solution of ∂tv− Lω
θ,φv≤0on Q. Further, let η:V→[0,1] and ζ:R→[0,1] be two
cutoff functions with
supp η⊂Band η≡0on ∂B,
supp ζ⊂Iand ζ(s1) = 0.
Then, there exists C1<∞such that for α≥1and p, p∗∈(1,∞)with 1/p + 1/p∗= 1,
1
|I|
ζ(ηvα)2
1,∞,Q,θ +1
|I|ZI
ζ(t)Eω(ηvα
t)
|B|dt
≤C1α2
µω/θ
p,B,θ
∇η
2
ℓ∞
(E)
v2α
p∗,1,Q,θ +
ζ′
L
∞(I)+hω
θ(φ)
v2α
1,1,Q,θ.
(2.10)
Proof. Fix some α≥1. Since v≥0satisfies ∂tv+Lω
θ,φv≤0on Q, a summation by
parts yields
1
2α∂t
ηvα
t
2
ℓ2(V,θ)≤ −∇(η2φv2α−1
t), ω∇(φ−1vt)ℓ2(E)(2.11)
HEAT KERNEL AND INTRINSIC METRIC FOR RANDOM WALKS WITH GENERAL SPEED MEASURE 7
for any t∈I. By applying the product rule (2.4), we obtain
∇(η2φv2α−1
t), ω∇(φ−1vt)ℓ2(E)
=av(η2),dΓω(φv2α−1
t, φ−1vt)ℓ2(E)+av(φv2α−1
t),dΓω(η2
, φ−1vt)ℓ2(E)
=:T1+T2.(2.12)
Let us first focus on the term T1. Again, an application of the product rule (2.4)
together with the fact that (∇φt)(∇φ−1
t)≤0and −av(φ−1)(∇φ) = av(φ)(∇φ−1),
yields the following lower bound
dΓω(φv2α−1
t, φ−1vt)≥av(φ) av(φ−1) dΓω(v2α−1
t, vt) + av(v2α
t) dΓω(φ, φ−1)
+ av(φ)av(vt) dΓω(v2α−1
t, φ−1)−av(v2α−1
t) dΓω(vt, φ−1),
where we used that by H¨older’s inequality, av(vα1
t) av(vα2
t)≤av(vα1+α2
t)for any
α1, α2≥0. Further, by [3, Lemma B.1], we have that
dΓω(v2α−1
t, vt)≥2α−1
α2dΓω(vα
t, vα
t),
and
av(vt)(e)∇v2α−1
t(e)−av(v2α−1
t)(e)∇vt(e)
=v2α−1
t(e+)vt(e−)−v2α−1
t(e−)vt(e+)≤2(α−1)
αav(vα
t)(e)∇vα
t(e)
(2.13)
for all e∈E. Thus, by combining the estimates above and using that
av(φ)∇φ−1=pav(φ) av(φ−1)·p−(∇φ)(∇φ−1),(2.14)
an application of Young’s inequality, that reads |ab| ≤ 1
2(εa2+b2/ε), with ε= 1/(2α)
results in
T1≥3α−1
2α2av(η2) av(φ) av(φ−1),dΓω(vα
t, vα
t)ℓ2(E)−2α|B|hω
θ(φ)
v2α
t
1,B,θ .
Let us now address the term T2. Observe that
av(φv2α−1
t) dΓω(φ−1vt, η2)
= 2 av(φv2α−1
t) av(η)av(φ−1) dΓω(vt, η) + av(vt) dΓω(φ−1, η)
≥ −4 av(η) av(φ) av(v2α−1
t)av(φ−1)dΓω(vt, η)+ av(vt)dΓω(φ−1, η).
Since av(v2α−1
t) av(vt)≤av(v2α), an application of the Young inequality yields
4 av(η) av(φ) av(v2α
t)dΓω(φ−1, η)
(2.14)
≤8 av(φ) av(φ−1) av(v2α
t) dΓω(η, η)−1
2av(η2) av(v2α
t) dΓω(φ, φ−1).
8 SEBASTIAN ANDRES, JEAN-DOMINIQUE DEUSCHEL, AND MARTIN SLOWIK
On the other hand,
av(v2α−1
t)(e)(∇vt)(e)
≤av(vα
t)(e)(∇vα
t)(e) + 1
2v2α−1
t(e+)vt(e−)−v2α−1
t(e−)vt(e+)
(2.13)
≤2α−1
αav(vα
t)(e)∇vα
t(e).
Thus, by applying again Young’s inequality with ε= 1/(4α), we get
4 av(η) av(φ) av(φ−1) av(v2α−1
t)dΓω(vt, η)
≤42α−1
αav(φ) av(φ−1) av(η) av(vα
t)dΓω(vα
t, η)
≤av(φ) av(φ−1)2α−1
2α2av(η2) dΓω(vα
t, vα
t) + 8(2α−1) av(v2α
t) dΓω(η, η)
Hence, the estimates above together with the fact that
av(φ−1) av(φ) = 1 −1
4(∇φ)(∇φ−1)
give rise to the following lower bound
T2≥ −2α−1
2α2av(η2) av(φ) av(φ−1),dΓω(vα
t, vα
t)ℓ2(E)
−16α|B|
µω/θ
p,B,θ
∇η
2
ℓ∞(E)
v2α
t
p∗,B,θ −5α hω
θ(φ)|B|
v2α
t
1,B,θ .
Since av(φ) av(φ−1)≥1and
av(η2) dΓω(vα
t, vα
t)≥dΓω(ηvα
t, ηvα
t)−av(v2α
t) dΓω(η, η)
we obtain that there exists C1<∞such that
T1+T2≥1
2αEω(ηvα
t)
−C1
2α|B|
µω/θ
p,B,θ
∇η
2
ℓ∞(E)
v2α
t
p∗,B,θ +hω
θ(φ)
v2α
t
1,B,θ .
Hence,
∂t
(ηvα
t)2
1,B +Eω
t(ηvα
t)
|B|
≤C1α2
µω/θ
p,B,θ
∇η
2
ℓ∞(E)
v2α
t
p∗,B,θ +hω(φ)
v2α
t
1,B,θ .(2.15)
Finally, since ζ(s1) = 0,
Zs
s1
ζ(t)∂t
(ηvα
t)2
1,B dt=Zs
s1∂tζ(t)
(ηvα
t)2
1,B−ζ′(t)
(ηvα
t)2
1,Bdt
≥ζ(s)
(ηvα
s)2
1,B − kζ′kL
∞
(I)|I|
v2α
p∗,1,Q
for any s∈(s1, s2]. Thus, by multiplying both sides of (2.15) with ζ(t)and integrat-
ing the resulting inequality over [s1, s]for any s∈I, the assertion (2.10) follows by
an application of the H¨older inequality.
HEAT KERNEL AND INTRINSIC METRIC FOR RANDOM WALKS WITH GENERAL SPEED MEASURE 9
For any x0∈V,δ∈(0,1) and n≥1, we write Qδ(n)≡[0, δn2]×B(x0, n)to
denote the corresponding space-time cylinder, and we set
Qδ,σ(n):=(1 −σ)s′,(1 −σ)s′′ +σδn2×B(x0, σn), σ ∈(0,1],
where s′=εδn2and s′′ = (1 −ε)δn2for some fixed ε∈(0,1/4).
Proposition 2.9. For x0∈V,δ∈(0,1] and n≥N1(x0)∨N2(x0), let v > 0be such
that ∂tv− Lω
θ,φv= 0 on Q(x0, n). Then, for any p, q, r ∈(1,∞]satisfying (2.6)there
exists C2≡C2(d, p, q, r)<∞and κ=κ(d′, p, q, r)<∞such that
max
(t,x)∈Qδ,1/2(n)v(t, x)≤C2
nd/2mω(n)
δκ
e2(1−ε)h(φ)δn2
φf
ℓ2(V,θ),(2.16)
where
mω(n):=
1∨µω
θ
p,B(x0,n),θ ·
1∨νω
q,B(x0,n)·
1∨θ
r,B(x0,n)·
1∨1
θ
1,B(x0,n).
Proof. We will follow similar arguments as in the proof of [2, Proposition 4.2]. Fix
some 1/2≤σ′< σ ≤1. For p, r ∈(1,∞), let p∗:=p/(p−1) and r∗:=r/(r−1) be
the H¨older conjugate of pand r, respectively. For any k∈N0set αk:=αk, where
α:= 1 + 1
p∗−r∗
ρand ρ:=d′
d′−2 + d′/q .
Notice that for any p, q, r ∈(1,∞)satisfying (2.6) we have α > 1. In particular,
r∗/ρ + 1/p < 1. Further, for
σk=σ′+ 2−k(σ−σ′)and τk= 2−k−1(σ−σ′), k ∈N0,
we write Ik:= [(1 −σk)s′,(1 −σk)s′′ +σkδn2],Bk:=B(x0, σkn)and Qk:=Qδ,σk(n)
to lighten notation. Note that |Ik|/|Ik+1 | ≤ 2and |Bk|/|Bk+1| ≤ 2dC2
reg. Moreover,
for any k∈N0let ηkbe a cut-off functions in space and ζk∈C∞(R)be a cut-
off function in time such that supp ηk⊂Bk,ηk≡1on Bk+1,ηk≡0on ∂Bk,
∇ηk
ℓ∞(E)≤1/(τkn)and supp ζk⊂Ik,ζk≡1on Ik+1,ζk((1 −σk)s′) = 0 and
ζ′
k
L
∞([0,δn2]) ≤1/(τkδn2).
The constant c∈(0,∞)appearing in the computations below is independent of
nbut may change from line to line. First, by using H¨older’s and Young’s inequality,
v2αk
αp∗,α,Qk+1,θ ≤
v2αk
1,∞,Qk+1,θ +
v2αk
ρ/r∗,1,Qk+1,θ
≤c
ζk(ηkvαk)2
1,∞,Qk,θ +
ζk(ηkvαk)2
ρ/r∗,1,Qk,θ,
(cf. [16, Lemma 1.1]). Further, by Assumption 2.1(iii) we may apply the Sobolev
inequality for functions with compact support in [1, Equation (28)] to obtain
ζk(ηkvαk)2
ρ/r∗,1,Qk,θ ≤c n2
νω
q,Bk
θ
r∗/ρ
r,Bk
1
|Ik|ZIk
ζk(t)Eω(ηkvαk
t)
|Bk|dt.
10 SEBASTIAN ANDRES, JEAN-DOMINIQUE DEUSCHEL, AND MARTIN SLOWIK
Hence,
ζk(ηkvαk)2
1,∞,Qk,θ +
ζk(ηkvαk)2
ρ/r∗,1,Qk,θ
≤c n21
|Ik|
ζk(ηkvαk)2
1,∞,Qk,θ +
νω
q,Bk
θ
r∗/ρ
r,Bk
|Ik|ZIk
ζk(t)Eω(ηkvαk)
|Bk|dt
(2.10)
≤c α2
k
mω(n)
1∨1/θ
1,Bk1
δτ 2
k
+n2hω
θ(φ)
v2αk
p∗,1,Qk,θ.(2.17)
Thus, by combining the estimates above, we get
v
2αk+1p∗,2αk+1 ,Qk+1,θ =
v2αk
1/(2αk)
αp∗,α,Qk+1,θ
≤c22kα2
k1 + δn2hω
θ(φ)
δ(σ−σ′)2
mω(n)
1∨1/θ
1,Bk1/(2αk)
v
2αkp∗,2αk,Qk,θ.(2.18)
Further, observe that |BK+1|1/αK≤cuniformly in nfor any Ksuch that αK≥ln n.
Hence,
max
(t,x)∈Qδ,σ′(n)
v(t, x)≤max
(t,x)∈QK+1
v(t, x)
≤ |BK+1|1/αK
1/θ
1/(2αK)
1,BK
ζK(ηKvαK)2
1/(2αK)
1,∞,QK,θ.
By neglecting the second term on the left-hand side of (2.17), we obtain
max
(t,x)∈Qδ,σ′(n)v(t, x)
≤c22Kα2
K1 + δn2hω
θ(φ)mω(n)
δ(σ−σ′)21/(2αK)
v
2αKp∗,2αK,QK,θ.
Thus, by iterating the inequality (2.18) and using that P∞
k=0 k/αk<∞, we get
max
(t,x)∈Qδ,σ′(n)v(t, x)≤c1 + δn2hω
θ(φ)mω(n)
δ(σ−σ′)2κ/p∗
v
2p∗,2,Qδ,σ(n),θ ,
where κ/p∗:=1
2P∞
k=0 1/αk. By using similar arguments as in [19, Theorem 2.2.3]
or [1, Corollary 3.9], there exists c≡c(p, q, r, d′)<∞such that
max
(t,x)∈Qδ,1/2(n)v(t, x)≤c1 + δn2hω
θ(φ)mω(n)
δκ
v
2,∞,Qδ(n),θ
(2.9)
≤c C1/2
reg
nd/21 + δn2hω
θ(φ)mω(n)
δκ
ehω
θ(φ)δn2
φf
ℓ2(V,θ).
Since for any ε∈(0,1/2) there exists c(ε)<∞such that for all n≥1and δ∈(0,1],
1 + δn2hω
θ(φ)κe−(1−2ε)hω
θ(φ)δn2≤c(ε)<∞,
the claim follows.
HEAT KERNEL AND INTRINSIC METRIC FOR RANDOM WALKS WITH GENERAL SPEED MEASURE 11
2.4. Heat kernel bounds.
Proposition 2.10. Suppose that Assumption 2.4 hold and let x0∈Vbe fixed. Then,
for any given x∈Vand twith √t≥N1(x0)∨N2(x0)∨N3(x0, ω)the solution uof
the Cauchy problem in (2.8)satisfies
|u(t, x)| ≤ C3t−d/2X
y∈V1 + d(x0, x)
√tγ1 + d(x0, y)
√tγφ(y)
φ(x)e2hω
θ(φ)tf(y)
with γ:= 2κ−d/2and C3=C3(d, p, q, Cint).
Proof. Given (2.16) this follows as in the proof of [3, Proposition 2.7].
Proof of Theorem 2.5.First, notice that the heat kernel (t, x)7→ pω
θ(t, x, y)solves the
Cauchy problem (2.8) with f=
1
{y}/θ(y). Further, let x0∈Vbe arbitrary but
fixed and consider the function φ= eψwith ψ(z):=−λmin dω
θ(x, z), dω
θ(x, y)for
λ > 0. Then, for sufficiently large t, an application of Proposition 2.10 yields
pω
θ(t, x, y)≤C3t−d/21 + d(x0, x)
√tγ1 + d(x0, y)
√tγ
eψ(y)−ψ(x)+2hω
θ(φ)t.
Next we optimise over λ > 0. Since
∇ψ(e)≤λdω
θ(x, e+)−dω
θ(x, e−)
(1.2)
≤λ1∧θ(e+)∧θ(e−)
ω(e)1/2
and acosh(x)−1≤cosh(√ax)−1for all x∈Rand any a≥1, we get
hω
θ(φ) = max
x∈VX
y∼x
ω(x, y)
θ(x)cosh ∇ψ({x, y})−1≤Cdegcosh(λ)−1.
Hence,
expψ(y)−ψ(x) + 2hω
θ(φ)t≤expdω
θ(x, y)−λ+2Cdegt
dω
θ(x, y)cosh(λ)−1.
By setting
F(s) = inf
λ>0−λ+s−1cosh(λ)−1,
we finally get
pω
θ(t, x, y)≤C3
td/21 + d(x0, x)
√tγ1 + d(x0, y)
√tγ
expdω
θ(x, y)Fdω
θ(x, y)
2Cdegt.
(2.19)
Further, notice that
F(s) = s−1(1 + ss)1/2−1−log s+ (1 + s2)1/2
and F(s)≤ −s/2(1 −s2/10) for s > 0(see [5] and [11, page 70]). Hence, if s≤3,
then F(s)≤ −s/20 whereas if s≥e, then
F(s)≤1−log(2s) = −log(2s/e).
12 SEBASTIAN ANDRES, JEAN-DOMINIQUE DEUSCHEL, AND MARTIN SLOWIK
Now, choose x=x0. In view of (2.19) we find suitable constants c1,...,c3such
that if dω
θ(x0, y)≤c1tthen
pω
θ(t, x0, y)≤c2t−d/21 + d(x0, y )
√tγ
exp−2c3dω
θ(x0, y)2/t .
This finishes the proof of (i). In the case dω
θ(x0, y)≥c1tstatement (ii) can be
obtained from (2.19) by similar arguments.
3. COMPARISON R ESULT FOR TH E IN TR IN SIC METR IC
In this section we show that on a large scale the metric dω
θcan be bounded from
below by a certain power of the graph distance d. The required integrability condi-
tion will be formulated in terms of an Orlicz-norm which we introduce first.
Definition 3.1 (Young function and Orlicz-norm [18, Section 1.3]).A function Φ :
[0,∞)→[0,∞]is a Young function, if Φis convex, non-decreasing with Φ(0) = 0,
limr→∞ Φ(r) = ∞and there exists s∈(0,∞)such that Φ(s)<∞. Its Legendre-
Fenchel dual Ψ: [0,∞)→[0,∞]is given by
Ψ(r) = sup
s∈[0,∞)sr −Φ(s),
and the pair (Φ,Ψ) is called complementary pair or Legendre-Fenchel pair. For any
finite ∅ 6=B⊂V, the Orlicz-norm of f:B→Ris defined by
f
Φ,B := sup n
fg
1,B :g≥0,
Ψ(g)
1,B ≤1o.(3.1)
Notice that either Φis continuous on all of [0,∞)or there exists r0∈(0,∞)such
that Φis continuous on [0, r0)and identically +∞on [r0,∞). In particular, the dual
function Ψis also a Young function and its right-continuous inverse r7→ Ψ−1(r):=
inf{s∈[0,∞] : Ψ(s)> r}is concave. Hence, it is easy to check that for any A⊂B,
1
A
Φ,B =|A|
|B|Ψ−1|B|
|A|.(3.2)
In particular, for any f , g :B→Rthe following Orlicz-Birnbaum estimate holds (cf.
[18, Section 3.3, Proposition 1 and 4])
fg
1,B ≤2
f
Φ,B
g
Ψ,B.(3.3)
Remark 3.2.The following Legendre-Fenchel pairs are of particular importance.
a) Φp(r),Ψp(r):=rp∗
, rpfor any p, p∗∈(1,∞)with 1/p + 1/p∗= 1, the
resulting Orlicz-norm coincides with the space-averaged ℓp-norm as defined
above.
b) ΦExp(r),ΨExp(r):=(rln r−r+ 1)
1
[1,∞)(r),er−1leads to a norm with
the property that for all f:B→[0,∞)
f
Ψ,B ≤
exp(f)
1,B.(3.4)
Now we state the lower comparison result for dω
θ.
HEAT KERNEL AND INTRINSIC METRIC FOR RANDOM WALKS WITH GENERAL SPEED MEASURE 13
Theorem 3.3. Let (Φ,Ψ) be a Legendre-Fenchel pair of Young functions such that
r7−→ r
pΨ−1(rd−1)
is monotone increasing. Further, assume that
mΨ:= sup
x∈V
lim sup
n→∞
1∨µω/θ
Ψ,B(x,n)<∞.(3.5)
Then, there exists c(mΨ)>0such that the following holds. For every x∈Vthere
exists N3(ω, x)<∞such that for any y∈Vwith d(x, y )≥N3(ω, x),
dω
θ(x, y)≥c(mΨ)d(x, y)
qΨ−1d(x, y)d−1
.(3.6)
Proof. In order to simplify notation, set mω
θ(x):= 1 ∨µω(x)/θ(x)for x∈V. Since
the function t7→ 1/√tis convex, an application of the Jensen inequality yields
dω
θ(x, y)≥inf
γ∈Γx,y
lγ 1
lγ
lγ−1
X
i=0
mω
θ(zi)∨mω
θ(zi+1)!−1/2
.
Moreover,
1
lγ
lγ−1
X
i=0
mω
θ(zi)∨mω
θ(zi+1)≤2|B(x, lγ)|
lγ
1
γmω
θ
1,B(x,lγ)
(3.3)
≤4|B(x, lγ)|
lγ
1
γ
Φ,B(x,lγ)
mω
θ
Ψ,B(x,lγ)
(3.2)
= 4 Ψ−1|B(x, lγ)|
|lγ|
mω
θ
Ψ,B(x,lγ).
By combining the estimates and using (2.2) as well as the concavity of Ψ−1we
obtain that there exists c(mΨ)>0and N3(ω, x)<∞such that for any y∈Vwith
d(x, y)≥N3(ω, x),
dω
θ(x, y)≥inf
γ∈Γx,y
lγ 8 Ψ−1|B(lγ)|
|lγ|mΨ!−1/2
≥inf
γ∈Γx,y
c(mΨ)lγ
qΨ−1ld−1
γ
.
Since the function r7→ r/pΨ−1(rd−1)is assumed to be monotone increasing, the
assertion follows.
For (Φp(r),Ψp(r)) and ΦExp(r),ΨExp(r)we obtain the following result.
Corollary 3.4. Let p > (d−1)/2and suppose that
mp:= sup
x∈V
lim sup
n→∞
1∨µω/θ
p,B(x,n)<∞.(3.7)
14 SEBASTIAN ANDRES, JEAN-DOMINIQUE DEUSCHEL, AND MARTIN SLOWIK
Then, there exists c(mp)>0such that the following holds: for every x∈Vthere exists
N3(ω, x)<∞such that for any y∈Vwith d(x, y )≥N3(ω, x),
dω
θ(x, y)≥c(mp)d(x, y)1−d−1
2p.(3.8)
Moreover, if
mExp := sup
x∈V
lim sup
n→∞
exp(µω/θ)
1,B(x,n)<∞.(3.9)
Then, there exists c(mExp)>0such that the following holds: for every x∈Vthere
exists N3(ω, x)<∞such that for any y∈Vwith d(x, y )≥N3(ω, x),
dω
θ(x, y)≥c(mExp)d(x, y)
qln 1 + d(x, y)d−1
.(3.10)
Remark 3.5.Combining the result in Theorem 2.5 with Theorem 3.3 leads to a
stretched-exponential bound on the heat kernel in terms of the graph distance.
4. OPTIMALITY O F TH E LO WE R BOUN D IN THEOREM 3.3
Consider the d-dimensional Euclidean lattice (Zd, Ed)with d≥2, where Edde-
notes the set of all non-oriented nearest neighbour bonds. As pointed out in Re-
mark 2.2,(Zd, Ed)satisfies the Assumption 2.1. Further, let Pbe a probability
measure on the measurable space (Ω,F) = REd
+,B(R+)⊗Edand write Efor the
expectation with respect to P. The space shift by z∈Zdis the map τz: Ω →Ω
defined by (τzω)({x, y}):=ω({x+z, y +z})for all {x, y} ∈ Ed. Now assume that
Psatisfies the following conditions:
(i) Pis ergodic with respect to translations of Zd, i.e. P◦τ−1
x=Pfor all x∈Zd
and P[A]∈ {0,1}for any A∈ F such that τx(A) = Afor all x∈Zd.
(ii) There exist p, q ∈(1,∞]satisfying 1/p + 1/q < 2/d such that
Eω(e)p<∞and Eω(e)−q<∞(4.1)
for any e∈Ed.
Then, the spatial ergodic theorem gives that for P-a.e. ω,
lim
n→∞
µω
p
p,B(n)=Eµω(0)p<∞and lim
n→∞
νω
q
q,B(n)=Eνω(0)q<∞.
In particular, by choosing θ≡1and r=∞, Assumption 2.4 is fulfilled for P-a.e.
ωand the heat kernel estimates in Theorem 2.5 hold. Nevertheless, for general
ergodic environments we cannot control the size of the random variable N3(x),
x∈Zd, as this requires some information on the speed of convergence in the er-
godic theorem. However, if we additionally assume, for instance, that the environ-
ment satisfies a concentration inequality in form of a spectral gap inequality w.r.t.
the so-called vertical derivative, then E[N3(x)n]<∞provided a stronger moment
condition holds (depending on n), see Assumption 1.3 and Lemma 2.10 in [4].
HEAT KERNEL AND INTRINSIC METRIC FOR RANDOM WALKS WITH GENERAL SPEED MEASURE 15
In the context of the random conductance model we can now provide the follow-
ing example for which the lower bound in Theorem 3.3 or Corollary 3.4, respec-
tively, is attained up to an arbitrarily small correction in the exponent.
Theorem 4.1. Consider the VSRW, i.e. θ≡1. For any p > 1there exists an en-
vironment of ergodic random conductances {ω(e) : e∈Ed}on (Zd, Ed)satisfying
E[ω(e)p]<∞such that for any α > p and P-a.e. ωthe following hold.
(i) Suppose d= 2. There exists L0=L0(ω)<∞such that for all L≥L0there
exists x=x(ω)∈Zdwith d(0, x) = Land
dω
θ(0, x)≤c d(0, x)1−d−1
2α.
(ii) Suppose d≥2. There exists L0=L0(ω)<∞such that for all L≥L0there
exist x=x(ω), y =y(ω)∈[−2L, 2L]dwith d(x, y) = Land
dω
θ(x, y)≤c d(x, y)1−d−1
2α.
Proof. Let {Y(i, y) : i∈ {1,...,d}, y ∈Zd−1}be a family of non-negative, indepen-
dent and identically distributed random variables such that
PY(i, y)> r=r−α0+o(1) as r→ ∞
for some α0> p. For any x∈Zdwe write ˆxito denote the element of Zd−1obtained
by removing the i-th component from x. Further, set
ω({x, x ±ei}):=Y(i, ˆxi),∀x∈Zd, i ∈ {1,...,d},
where {e1,...,ed}denotes the canonical basis in Rd. Then, note that the con-
ductances are constant along the lines, but independent between different lines.
W.l.o.g. we further assume that ω(e)≥1for any e∈Ed. We refer to [3, Exam-
ple 1.11] and [13, Section 2.2] for a similar but different example for a model with
layered random conductances.
(i) Consider the nearest-neighbour path (xn:n≥0) on Nd
0defined by x0:= 0,
xn+1 :=xn+ein+1 with
i1:= argmax
i=1,...,d−1
ω({0,ei}), in+1 := argmax
i=1,...,d
ω({xn, xn+ei}), n ≥1.
In view of the definition of the chemical distance in (1.2) it suffices to show that for
any α > α0there exists L0=L0(ω)<∞such that
L
X
n=0
ω({xn, xn+1})−1/2≤c L1−d−1
2α,∀L≥L0.(4.2)
For that purpose, set Mn:= max1≤k≤nω({xk−1, xk})and un:=n1/α for n≥1.
Then, by construction Mnis the maximum of ni.i.d. random variables Z1,...,Zn
defined by
Z1:= max
i∈{1,...,d−1}ω({0,ei}), Zk:= max
i∈{1,...,d}\{ik−1}ω({xk, xk+ei}), k ≥2.
16 SEBASTIAN ANDRES, JEAN-DOMINIQUE DEUSCHEL, AND MARTIN SLOWIK
An elementary computation shows that P[Z1> uk]≤(d−1) P[ω(e)> uk]→0,
kP[Z1> uk]≥kP[ω(e)> uk]→ ∞ as k→ ∞ and
∞
X
k=1
PZ1> ukexp−kPZ1> uk≤c
∞
X
k=1
k−α0
αexp−c k1−α0
α<∞.
Thus, by [14, Theorem 3.5.2] for P-a.e. ωthere exists N0=N0(ω)<∞such that
Mn≥n1/α,∀n≥N0.(4.3)
Let (lk:k≥0) be the sequence of record times defined by
l0:= 0 and lk+1 := min j > lk:Mj> Mlk.
and denote by N(L)the number of records in the interval {0,...L}. Recall that
lim
k→∞
ln lk
k= 1,lim
L→∞
N(L)
ln L= 1,P-a.s. (4.4)
(cf. e.g. [14, Section 5.4]). Set ˆ
Mk:=Mlk. Using Abel’s summation formula the
left-hand side in (4.2) can be rewritten as
L
X
n=0
ω({xn, xn+1})−1/2≤lN(L)ˆ
M−1/2
N(L)+
N(L)−1
X
k=1
lkˆ
M−1/2
k−ˆ
M−1/2
k+1 .
By (4.3) and (4.4) the first term is of order L1−1
2α. Further, we have that lkˆ
M−1/2
k≤
(lk)1−1
2α≤ce(1−1
2α)kfor sufficiently large kand therefore
N(L)−1
X
k=1
lkˆ
M−1/2
k−ˆ
M−1/2
k+1 ≤
N(L)
X
k=1
lkˆ
M−1/2
k≤ce(1−1
2α)N(L)≤c L1−1
2α
for all Llarger than some L0=L0(ω). Thus, (4.2) is proven.
(ii) In order to show the second statement consider
eL:= argmax
e∈Ed:e−∈[−L,L]d
ω(e).
Then, by construction ω(eL)is the maximum of order Ld−1i.i.d. random variables
and again by [14, Theorem 3.5.2] there exists L0=L0(ω)such that P-a.s.
ω(eL)≥c L d−1
α,∀L≥L0.
For such Lset x:=e−
Land consider the nearest-neighbour path (xn:n∈N0)on Zd
defined by x0:=xand xn+1 :=xn+ein+1 with
in+1 := argmax
i=1,...,d
ω({xn, xn+ei}), n ≥0,
similarly as above in (i). Then, by setting y=xL, we have d(x, y) = Land
ω({xn, xn+1})≥ω(eL)≥c L d−1
α,∀n= 0,...,L−1.
In particular, (4.2) holds and (ii) follows from the definition of dω
θ.
HEAT KERNEL AND INTRINSIC METRIC FOR RANDOM WALKS WITH GENERAL SPEED MEASURE 17
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18 SEBASTIAN ANDRES, JEAN-DOMINIQUE DEUSCHEL, AND MARTIN SLOWIK
UNI VERSITY O F CAMBRID GE
Current address: Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB
E-mail address:s.andres@statslab.cam.ac.uk
TEC HN ISC HE UN IV ER SIT¨
AT BE RLIN
Current address: Strasse des 17. Juni 136, 10623 Berlin
E-mail address:deuschel@math.tu-berlin.de
TEC HN ISC HE UN IV ER SIT¨
AT BE RLIN
Current address: Strasse des 17. Juni 136, 10623 Berlin
E-mail address:slowik@math.tu-berlin.de
