Subdiffusive and Superdiffusive Exceptional Times in the Dynamical Discrete Web

0.4 MB·English

by  Dan Jenkins

Two-Sided Sub-Diffusivity in the Dynamical Discrete Web
Dan Jenkins
Courant Institute of Mathematical Sciences, NYU, New York, NY 10012
E-mail: [e-mail]
Abstract
The dynamical discrete web (DyDW) is a system of one-dimensional coalescing random
walks thatevolves inan extradynamicaltime parameter, τ. Atany deterministicτ thepaths
1
behaveascoalescingsimplesymmetricrandomwalks. Itwasshownin2009byFontes,Newman,
1
RavishankarandSchertzerthatthereexistexceptionaltimesatwhichthepathstartingatthe
0
2 origin violates the law of the iterated logarithm. To be specific, they show that there exist
c emxecaenpitniognSalτ(dt)yn≤amji+caKl √timt efosr, aτl,l ta,twwhheircehttihsethpeatrhanfdroommwthalekotrimigien,,anSd0τ,jisisKso-msuebcdoiffnsutsaivnet,.
e 0
D Thefirstgoalofthispaperistoestablishtheexistenceofexceptionaltimesforwhic√hthepath
fromtheoriginisK-subdiffusiveinbothdirections,i.e. τ suchthat|Sτ(t)|≤j+K tforallt.
0
4 We then obtain upper and lower bounds for the Hausdorff dimensions of this set, and related
1 sets, of exceptional times.

]
R 1 Introduction
P
.
h This paper examines the dynamical discrete web (DyDW), a system of coalescing random walks
t thatevolvesinacontinuousdynamicaltimeparameter. Thedynamicaldiscretewebwasintroduced
a
m by Howitt and Warren in [7]. The DyDW and related systems have been considered as models for
erosion and other hydrological phenomena (see [9],[1]). We examine “exceptional times” for the
[
DyDW. These are dynamical times at which paths from the DyDW display behavior that would

1 have probability zero for a standard random walk, or for the DyDW observed at a deterministic
v
time.
0
Now we define the dynamical discrete web, and briefly describe our main result. This paper
8
3 follows [2] closely; for a more thorough introduction to the subject see Section 1 of their paper.
3 To discuss the DyDW, we first define the discrete web (DW). The discrete web is a system
. of coalescing one-dimensional simple symmetric random walks. To construct it, we independently
2
1 assign to each point in Z2 := {(x,t) ∈ Z2 : x+t is even} a symmetric, ±1-valued Bernoulli
even
1 random variable, ξ . We then draw an arrow from (x,t) to (x+ξ ,t+1) (see Figure 1). For
(x,t) (x,t)
1 each(x,t)∈Z2 ,weletS (t)bethepaththatstartsat(x,t)andfollowsthearrowsfromthere.
v: Thediscreteweevbenisthecolle(xc,tti)onofallsuchpathsfor(x,t)∈Z2 . Asthefiguresandtheordering
even
i of (x,t) suggest, we let the path time coordinate, t, run vertically, and the space coordinate, x, run
X
horizontally. Future references to left/right, vertical/horizontal should be understood according to
r
a this convention.
The DyDW was first introduced by Howitt and Warren in [7]. It is a discrete web that evolves
in an extra dynamical time parameter, τ, by letting the arrows independently switch directions
as τ increases. To accomplish this, we assign to each (x,t) ∈ Z2 an independent, rate one
even
Poisson clock. When the clock at (x,t) rings, we reset the arrow at (x,t) by replacing it with a
new, independent arrow (that may or may not agree with the previous arrow). This corresponds
to replacing the ξ from the DW, with right-continuous τ-varying versions, ξτ . We then let
(x,t) (x,t)
1
Figure 1: A partial realization of the discrete web. Each arrow independently points left or right
with probability 1/2. In the dynamical discrete web, each arrow has an independent Poisson clock
and resets whenever it rings.
W(τ) denote the discrete web constructed from the ξτ ’s, and let Sτ (t) denote the path from
(x,t) (x,t)
W(τ) starting at (x,t). Note that at any deterministic τ, W(τ) is distributed as a discrete web.
Exceptional times for the DyDW were first studied by Fontes, Newman, Ravishankar and
Schertzer in [2]. There they show that there exist exceptional times for the DyDW at which
Sτ is subdiffusive in one direction. That is, for sufficiently large K,j:
0
√
P(∃τ ∈[0,1] s.t. Sτ(t)≤j+K t for all t≥0)>0. (1)
0
Theirworkwasmotivatedbythestudyofexceptionaltimesfordynamicalpercolation,see[6],[10].
Similarly to the DyDW, dynamical percolation consists of a lattice of Bernoulli random variables
which reset according to independent Poisson processes. For static (non-dynamical) percolation
with critical edge probabilities it is believed that no infinite cluster should exist. This is proven
for dimension two and large dimensions (see [5], for example). In [10] it was shown that critical
two-dimensionaldynamicalpercolationhasexceptionaltimeswherethisfails, i.e. whereaninfinite
cluster exists. However, no such exceptional times exist for large dimensions, see [6].
In [2] they use techniques similar to those used for dynamical percolation to prove (1) and
examine the dimensions of the corresponding sets of exceptional τ. We extend their arguments to
show the existence of exceptional times for the dynamical discrete web at which Sτ is subdiffusive
0
in both directions. To be specific, we prove:
Theorem 1. For K,j sufficiently large:
√
P(∃τ ∈[0,1] s.t. |Sτ(t)|≤j+K t for all t≥0)>0. (2)
0
An immediate consequence of this is:
Corollary 1. For K sufficiently large:
√
P(∃τ ∈[0,1] s.t. limsup|Sτ(t)/ t|≤K)>0. (3)
0
2
In the final section of the paper we examine the sets of exceptional times and study their
Hausdorff dimensions. That is, we look at the sets:
√
{τ ∈[0,∞):∃j s.t. |Sτ(t)|≤j+K t for all t≥0}, (4)
0
√
{τ ∈[0,∞):limsup|Sτ(t)/ t|≤K}, (5)
0
and derive upper and lower bounds for their Hausdorff dimensions, as functions of K. Our bounds
are analogous to, and motivated by, those from [2] for the one-sided case. As in the one-sided case,
thedimensionstendto1asK goesto∞. ForsmallK itisknownthat(4)isempty,seeProposition
5.8 of [2]. This implies (5) is also empty for small K, see Section 5. Our analysis of (5) is helped
by noting that (5) only depends on arrows with arbitrarily large time coordinate (almost surely).
This means (5) can be analysed using tail events, allowing us to improve the lower bound slightly
relative to the methods of [2]. The two sets (4) and (5) have the same dimensions, except for at
most countably many values of K (see Section 5 for details). It would seem natural that the two
dimensions would be the same for all K, but we are unable to prove this.
2 Structure of the Proof of Theorem 1
Asin[2],weshowthatsubdiffusivityoccursbyshowingthataseriesof“rectangleevents”occur.
First, we define our rectangles. Let γ > 1 and d = 2((cid:98)γk(cid:99)+1). Let R be the rectangle with
k 2 0
vertices (−d ,0), (+d ,0), (−d ,d2) and (+d ,d2). Given R we take R to be the rectangle
0 0 0 0 0 0 k k+1
of width 2d and height d2 , that is centered about the y-axis, and stacked on top of R (see
k+1 k+1 k
Figure 2). An easy computation shows that the entire stack of rectangles lies between the graphs
√ √
of −j −K t and j +K t, where j,K depend on γ. For example, we can take j = 2,K = γ,
see Proposition 2 of Section 5. Thus if Sτ stays within the stack, it will be subdiffusive in both
0
directions.
Remark1. Noticethatthisgivesaboundwithleft-rightsymmetry. Ifwewishtostudyexceptional
√ √
times where −j − K t ≤ Sτ(t) ≤ j + K t, we can skew our rectangles. This can be
L L 0 R R
accomplished by horizontally scaling the left and right halves of each rectangle by C and C ,
L R
respectively (and rounding out to the nearest point in Z2 ). For the sake of simplicity of our
even
arguments (and notation) we will largely ignore the asymmetrical case. However, it should be
noted that our results easily extend to the asymmetrical case, using the above construction.
Let t denote the time coordinate of the lower edge of R . For k ≥ 1, let l denote the upper
k k k
left vertex of R and r the upper right vertex of R . We would like to define our rectangle
k−1 k k−1
events, Bτ, as:
k
Bτ :={|Sτ(t)|≤d ∀t∈[0,t ]},
0 0 o 1
Bτ :={|Sτ (t)|≤d and |Sτ (t)|≤d ∀t∈[t ,t ]} for k ≥1.
k lk k rk k k k+1
Then on the event (cid:84) Bτ, Sτ will stay in the stack of rectangles, and thus be subdiffusive in
k≥0 k 0
both directions. This follows from the discussion above, combined with the fact that paths in the
discrete web do not cross. Thus if for some γ we can show:
(cid:92)
P(∃τ ∈[0,1] s.t. Bτ(γ) occurs)>0, (6)
k
k≥0
3
Figure2: RoughsketchofthefirstthreerectanglesandpathsforwhichtheB ’soccur. Thedarker
k
paths are the Sτ ’s and Sτ ’s. The lighter path is Sτ.
lk rk 0
then Theorem 1 will follow immediately.
Toprove(6),wewillneedtounderstandtheinteractionbetweenpairsofpathsfromtheDyDW.
This can be described as a combination of coalescing (if the paths have the same dynamical time)
and sticking (if the dynamical times differ). Let Sτ be the path from z = (x,t) ∈ Z2 at
z even
dynamical time τ, and let Sτ(cid:48) be the path from z(cid:48) = (x(cid:48),t(cid:48)) at dynamical time τ(cid:48). The paths
z(cid:48)
will evolve independently until they meet at some time t∗ ≥ Max(t,t(cid:48)). If τ = τ(cid:48), the paths
coalesce when they meet, otherwise they “stick”. To be precise, let x∗ :=Sτ(t∗)=Sτ(cid:48)(t∗) and let
z z(cid:48)
z∗ = (x∗,t∗)(∈ Z2 ). Then if the clock at z∗ has not rung in [τ,τ(cid:48)] (WLOG assume τ < τ(cid:48)), the
even
twopathswillfollowthesamearrowon[t∗,t∗+1]. Wewillsaythepathsarestickingon[t∗,t∗+1].
Thepathscontinuetostickuntiltheyreachasitewhoseclockhasrung, atwhichpointtheyfollow
independentarrows. Notethattheseindependentarrowsmayagree,butthiswillnotbeconsidered
sticking.
To prove Theorem 1, we would like to show (6). Unfortunately, we are not able to prove (6)
directly. The problem arises in the interaction between sticking and coalescing (to be specific,
(12)-(14) fail for Bτ, so we are unable to establish (16)). To get around this, we construct a larger
k
system where the relevant paths do no not coalesce. In addition to the main DyDW, W(τ), we
will need an independent, secondary DyDW, Wˆ(τ). From now on, all “arrows”, “clock rings”, etc.
should be understood to refer to W(τ) (the main DyDW), unless otherwise specified.
Given Sτ and Sτ we want to construct non-coalescing versions, Xτ and Xτ . We accomplish
this by lettilnkg Xτ r=k Sτ , and taking Xτ to be the path from r thalkt followsrkthe arrows (from
lk lk rk k
W(τ))unlessitmeetsXτ. IfXτ meetsXτ atspace-timez∗ =(x∗,t∗)∈Z2 , thenon[t∗,t∗+1]
lk rk lk even
we let Xτ follow the arrow at z∗ from Wˆ(τ) (at dynamical time τ). At time t∗+1 we repeat this,
rk
following Wˆ(τ) if the paths are together, but following W(τ) otherwise. Continuing in this manner
we get an independent pair of non-coalescing simple symmetric random walks Xτ and Xτ . Now
lk rk
4
we define new rectangle events, Cτ:
k
Cτ :=Bτ,
0 0
Cτ :={|Xτ(t)|≤d and |Xτ (t)|≤d ∀t∈[t ,t ]} for k ≥1.
k lk k rk k k k+1
Notice that Cτ implies Bτ. This is because the only difference between Xτ,Xτ and Sτ ,Sτ is the
(possible) exteknsion of Xτk beyond the initial meeting point. So if we canlkshowrk: lk rk
rk
(cid:92)
P(∃τ ∈[0,1] s.t. Cτ occurs)>0, (7)
k
k≥0
then (6), and thus Theorem 1, will follow immediately. The next two sections will be devoted to
proving (7).
3 A Decorrelation Bound
Throughoutthissectionweassumeτ,τ(cid:48) ∈[0,1],τ <τ(cid:48)andwefixarbitraryk ≥1,γ >1. Wealso
translate the paths to start at t=0. That is, we set Yτ(t):=Xτ(t +t) and Yτ(t):=Xτ (t +t)
l lk k r rk k
(k is fixed so we drop it from the notation). We will also consider diffusively rescaled versions of
thesepaths,Y˜τ(t):=Yτ(td2)/d andY˜τ(t):=Yτ(td2)/d . Therelevant“rectangleevent”isthen:
l l k k r r k k
Cτ :={|Yτ(t)|≤d and |Yτ(t)|≤d ∀t∈[0,d2]}
l k r k k
={|Y˜τ(t)|≤1 and |Y˜τ(t)|≤1 ∀t∈[0,1]}.
l r
1
Similarly to [2] we define ∆:= (take their δ =d−1). As in [2], the key ingredient for
d |τ −τ(cid:48)| k
k
the proof of (7) is a decorrelation bound for the rectangle events:
Proposition 1. There exist c,a∈(0,∞) such that:
(cid:18) 1 (cid:19)a
P(Cτ ∩Cτ(cid:48))≤P(C0)2+c(∆)a ≤P(C0)2+c ,
γk|τ −τ(cid:48)|
with a,c independent of k,τ and τ(cid:48).
Notethatthesecondinequalityisfollowsimmediatelyfromthedefinitionsof∆,d . Theremainder
k
of this section is devoted to proving the first inequality, and thus Proposition 1. The structure is
similar to the proof of Lemma 3.1 from [2], with a few necessary modifications.
Asdiscussedintheprevioussection,pathsfromtheDyDWatdifferentdynamicaltimesinteract
by sticking. This sticking leads to dependence between the web paths. Our modified paths (the
Y ’s)havetheirownversionofstickingthatisslightlymorecomplicated. ToproveProposition1we
τ
willproveboundsfortheamountofsticking, whichwillallowustoboundthedependencebetween
the Cτ’s. We begin with some notation and definitions.
We call n ∈ Z a “sticking time” if a Yτ-path and a Yτ(cid:48)-path are at the same spot and follow
the same arrow at time n. This can happen in five ways:
(i) Yτ(n)=Yτ(cid:48)(n) no ring in [τ,τ(cid:48)],
l l
(ii) Yτ(n)=Yτ(cid:48)(n)(cid:54)=Yτ(cid:48)(n) no ring in [τ,τ(cid:48)],
l r l
5
(iii) Yτ(cid:48)(n)=Yτ(n)(cid:54)=Yτ(n) no ring in [τ,τ(cid:48)],
l r l
(iv) Yτ(n)=Yτ(cid:48)(n)(cid:54)=Yτ(n),Yτ(cid:48)(n) no ring in [τ,τ(cid:48)],
r r l l
(v) Yτ(n)=Yτ(cid:48)(n)=Yτ(n)=Yτ(cid:48)(n) no Wˆ-ring in [τ,τ(cid:48)].
r r l l
We will call (i) an ll(left-left)-sticking time, (ii) an lr-sticking time, (iii) an rl-sticking time, and
(iv),(v) will both be rr-sticking times. These names refer to which pair(s) of paths are sticking at
time n.
Given s∈[0,∞) let n be the unique n∈Z such that s∈[n,n+1). We define:
s
(cid:40)
0 if n is a sticking time
g(s):= s
1 otherwise
(cid:82)t
G(t):= g(s)ds.
0
We will also need:
(cid:40)
0 if n is an ll-sticking time
g (s):= s
ll
1 otherwise
(cid:82)t
G (t):= g (s)ds,
ll 0 ll
and G ,G ,G , which are defined analogously.
lr rl rr
Notice that t−G(t) is the amount of time spent sticking up to time t. So if we make the time
change t → t−G(t) we will include only the sticking steps. Similarly if we make the time change
t→G(t) we will include only the non-sticking steps. This allows us to decompose the paths as:
Yτ(t)=Yτ(G(t))+Yτ(t−G(t)),
l ld ls
Yτ(t)=Yτ(G(t))+Yτ(t−G(t)),
r rd rs
Yτ(cid:48)(t)=Yτ(cid:48)(G(t))+Yτ(cid:48)(t−G(t)),
l ld ls
Yτ(cid:48)(t)=Yτ(cid:48)(G(t))+Yτ(cid:48)(t−G(t)), (8)
r rd rs
with Yτ(0) = Yτ(0) = −d , Yτ(0) = Yτ(0) = d , and Yτ(0) = Yτ(0) = 0 (similarly for τ(cid:48)).
ld l k−1 rd r k−1 ls rs
Recall that the Y ’s and Y ’s include only the non-sticking steps of each walk. This means that
ld rd
the τ-paths and the τ(cid:48)-paths follow different, independent arrows, and thus are independent.
TomaketheabovesplittingworkfortheY˜’stheappropriaterescalingofGisG¯(t):=G(td2)/d2.
k k
We then make the time changes t → t−G¯(t) and t → G¯(t). We would like a bound for t−G¯(t),
the amount of sticking for the rescaled paths in [0,t]. This is given by the following adaptation of
Lemma 3.4 from [2]:
Lemma 1. For any 0<β <1
(cid:16) (cid:17)
P sup (t−G¯(t))≥∆β ≤c(cid:48)(cid:48)∆1−β,
t∈[0,1]
where c(cid:48)(cid:48) ∈(0,∞) is independent of k,τ and τ(cid:48).
6
Proof. Notice that by definition:
t−G(t)≤(t−G (t))+(t−G (t))+(t−G (t))+(t−G (t)). (9)
ll lr rl rr
(a) (b) (c) (d)
LetC(t)bedefinedasin[2],i.e. suchthatt−C(t)isthestickingtimeforSτ andSτ(cid:48). Weclaim
0 0
that each of (a),(b),(c),(d) is stochastically bounded by t−C(t) (given random variables X,Y, X
is said to stochastically bound Y if P(Y > x) ≤ P(X > x) for all x ∈ R). For (a) this is obvious,
d
since t−G (t)=t−C(t) (equal in distribution). This is because the Y ’s are just translated web
ll l
paths and the DyDW is invariant under space-time translations. We now concentrate on (d); (b)
and (c) can be handled similarly.
We’dliketocomparet−G (t),theamountofstickingforYτ andYτ(cid:48),tot−C(t),theamountof
rr r r
stickingforSτ andSτ(cid:48). We’llaccomplishthisbyconstructingcoupledversionsofthetwoprocesses.
0 0
Inbothcasestherearetwopathsthatalternatebetweenidenticalstickingsectionsandindependent
non-sticking sections. To be specific, we take T =T∗ :=0 and for k ≥0 define:
0 0
T :=inf{k ≥T : The clock at Sτ(k)=Sτ(cid:48)(k) rings in [τ,τ(cid:48)]},
2k+1 2k 0 0
T :=inf{k >T : Sτ(k)=Sτ(cid:48)(k)},
2k+2 2k+1 0 0
∆ :=T −T ≥0, Γ :=T −T ≥1,
k 2k+1 2k k 2k+2 2k+1
and:
T∗ :=inf{k ≥T : k is not an rr-sticking time},
2k+1 2k
T∗ :=inf{k >T : Yτ(k)=Yτ(cid:48)(k)},
2k+2 2k+1 r r
∆∗ :=T∗ −T∗ ≥0, Γ∗ :=T∗ −T∗ ≥1.
k 2k+1 2k k 2k+2 2k+1
Thenon[T(∗),T(∗) ]wehaveSτ andSτ(cid:48)(Yτ andYτ(cid:48))stickingfor∆(∗)steps,whileon[T(∗) ,T(∗) ]
2k 2k+1 0 0 r r k 2k+1 2k+2
theymoveindependentlyuntilmeetingatT(∗) . NoticethatΓ andΓ∗ havethesamedistribution,
2k+2 k k
they are both excursion times for pairs of independent random walks. So we may take Γ =Γ∗ for
k k
our coupled versions. To compare ∆ ,∆∗, notice that:
k k
j
P(cid:16)∆(∗) ≥j(cid:17)=(cid:89)P(cid:16)∆(∗) ≥i|∆(∗) ≥i−1(cid:17)
k k k
i=1
and:
P(∆∗ ≥i|∆∗ ≥i−1)≤P(∆ ≥i|∆ ≥i−1) for all i≥1, (10)
k k k k
so:
P(∆∗ ≥j)≤P(∆ ≥j) for all j,k ≥0. (11)
k k
To see (10), consider that P(∆ ≥i|∆ ≥i−1) is just the probability of no clock ring in [τ,τ(cid:48)].
k k
For ∆∗, we have the probability that Yτ =Yτ(cid:48) (cid:54)=Yτ,Yτ(cid:48) and there is no W-ring, or Yτ =Yτ(cid:48) =
k r r l l r r
Yτ =Yτ(cid:48) and there is no Wˆ-ring. These are disjoint events and the clocks are independent of the
l l
positions of previous arrows, so this is bounded by the probability of no clock ring.
7
Combining this with the above observations, we can couple ∆ ,∆∗ and Γ ,Γ∗ such that
k k k k
∆∗ ≤ ∆ andΓ =Γ∗. Thismeansthattherr-stickingsectionsareshorterthantheSτ,Sτ(cid:48) stick-
k k k k 0 0
ingsections,whiletheindependentsectionshavethesamelength. Thisimpliest−G (t)≤t−C(t)
rr
forthecoupledversions, whichshows(d)isstochasticallyboundedbyt−C(t). Thiscanbeproven
for (b),(c) by a nearly identical coupling argument, where the portion of the left/right paths after
their first meeting is coupled with Sτ,Sτ(cid:48). So we’ve shown that (a),(b),(c),(d) are each stochasti-
0 0
cally bounded by t−C(t). Combining this with (9) we get:
(cid:32) (cid:33) (cid:32) (cid:33)
P sup (t−G¯(t))≥∆β =P sup (t−G(t))≥d2∆β
k
t∈[0,1] t∈[0,d2]
k
(cid:32) (cid:33)
∆β
≤4P sup (t−C(t))≥d2 (using (9) and above paragraph)
k 4
t∈[0,d2]
k
(cid:32) (cid:33)
∆β
=4P sup (t−C¯(t))≥
4
t∈[0,1]
(cid:18) ∆ (cid:19)1−β
≤4c˜ (by Lemma 3.4 from [2])
41/β
=c(cid:48)(cid:48)∆1−β.
This completes the proof since c˜, and thus c(cid:48)(cid:48), is independent of k,τ and τ(cid:48).
Now we define Cτ to be the rectangle event for Yτ,Yτ. That is:
d ld rd
Cτ :={|Yτ(t)|≤d and |Yτ(t)|≤d ∀t∈[0,d2]}
d ld k rd k k
={|Y˜τ(t)|≤1 and |Y˜τ(t)|≤1 ∀t∈[0,1]}.
ld rd
Given r >0 we define the r-approximations of our rectangle events as:
{Cτ +r}:={|Yτ (t)|≤(1+r)d and |Yτ (t)|≤(1+r)d ∀t∈[0,d2]}
(d) l(d) k r(d) k k
={|Y˜τ (t)|≤1+r and |Y˜τ (t)|≤1+r ∀t∈[0,1]}.
l(d) r(d)
Recall that Yτ,Yτ are independent of Yτ(cid:48),Yτ(cid:48), and therefore:
ld rd ld rd
Cτ({Cτ +r}) is independent of Cτ(cid:48)({Cτ(cid:48) +r}). (12)
d d d d
We also have:
(Yτ,Yτ)=d (Yτ,Yτ), (13)
ld rd l r
since both are just pairs of independent random walks. So:
P(Cτ)=P(Cτ)=P(C0). (14)
d
We will need the following adaptation of Lemma 3.3 from [2]:
8
Lemma 2. Given any α<1/2, there is c(cid:48) ∈(0,∞) independent of ∆,k such that:
P({Cτ +∆α}\Cτ)≤c(cid:48)∆α.
d d
Proof.
(cid:18) (cid:19) (cid:32) (cid:33)
P({Cτ +∆α}\Cτ)≤P inf Y˜τ(t)∈[−1−∆α,−1) +P sup Y˜τ(t)∈(1,1+∆α]
d d t∈[0,1] ld t∈[0,1] ld
(cid:18) (cid:19) (cid:32) (cid:33)
+P inf Y˜τ(t)∈[−1−∆α,−1) +P sup Y˜τ(t)∈(1,1+∆α] .
t∈[0,1] rd t∈[0,1] rd
Now each of the four terms on the right is bounded by c∆α. This follows exactly as in the proof
of Lemma 3.3 in [2]. To see this, note that the Y˜’s are simple symmetric random walks started at
±d /d ∈[−1,1],diffusivelyrescaledbyδ =d−1. WecanthusapproximatetheY˜’sbyBrownian
k−1 k k
motion paths (for details see [3] and [2], Lemma 3.3). The result then follows, as the maximum
(minimum) process of a Brownian motion has a bounded distribution function.
The final ingredient for the proof of Proposition 1 is a bound on the modulus of continuity of a
random walk. This is given by Lemma 3.5 from [2]:
Lemma 3. (Lemma 3.5, [2])
Let S(t) be a simple symmetric random walk and define S˜(t):=S(t/δ2)δ.
Let ω ((cid:15)):=sup |S˜(t)−S˜(s)| be the modulus of continuity of S˜. Let α,β ∈(0,∞) be
S˜ s,t∈[0,1],|s−t|<(cid:15)
such that β/2>α. For any r ≥0, there exists c (independent of ∆ and δ) such that:
(cid:18) ∆α(cid:19)
P ω (∆β)≥ ≤c∆r.
S˜ 2
This is a consequence of the Garsia-Rodemich-Rumsey inequality [4]. For a proof see [2].
We may now prove Proposition 1. The remaining steps are nearly identical to the proof of
Proposition 3.1 from [2] (see the end of Section 3). We include them for the sake of completeness.
For any 0<α<1/2, we have:
(cid:16) (cid:17) (cid:16) (cid:17)
P Cτ ∩Cτ(cid:48) ≤P {Cτ +∆α}∩{Cτ(cid:48) +∆α}
d d
+2P(Cτ \{Cτ +∆α}), (15)
d
where we used the equidistribution of (Cτ,{Cτ +∆α}) and (Cτ(cid:48),{Cτ(cid:48)+∆α}). Using (12)-(14) we
d d
get:
P({Cτ +∆α}∩{Cτ(cid:48) +∆α})=P({Cτ +∆α})P({Cτ(cid:48) +∆α})
d d d d
≤P(Cτ)2+2P({Cτ +∆α}\Cτ)
d d d
=P(C0)2+2P({Cτ +∆α}\Cτ). (16)
d d
Combined with Lemma 2 this gives:
P({Cτ +∆α}∩{Cτ(cid:48) +∆α})≤P(C0)2+2c(cid:48)∆α. (17)
d d
9
Now that we have (15) and (17) we just need cˆ,a(cid:48) such that:
P(Cτ \{Cτ +∆α})≤cˆ∆a(cid:48). (18)
d
Recall the splitting of the Yτ’s given by (8). Analogous considerations for the Y˜τ’s gives:
Y˜τ(t)=Y˜τ(G¯(t))+Y˜τ(t−G¯(t))
l ld ls
=Y˜τ(t)+[Y˜τ(G¯(t))−Y˜τ(t)]+Y˜τ(t−G¯(t)), (19)
ld ld ld ls
Y˜τ(t)=Y˜τ(t)+[Y˜τ(G¯(t))−Y˜τ(t)]+Y˜τ(t−G¯(t)). (20)
r rd rd rd rs
Notice that all the Y˜’s appearing in (19), (20) are simple symmetric random walks rescaled by
δ = d−1, as in Lemma 3. Also, we’ve taken α < 1/2, so we may choose 0 < β < 1 such that
k
β/2>α. Then:
(cid:16) (cid:17) (cid:16) (cid:17)
P(Cτ \{Cτ +∆α})≤P |Y˜τ −Y˜τ| ≥∆α +P |Y˜τ −Y˜τ| ≥∆α
d l ld ∞ r rd ∞
(cid:18) ∆α(cid:19) (cid:18) ∆α(cid:19)
≤P |Y˜τ(G¯(t))−Y˜τ(t)| ≥ +P |Y˜τ(t−G¯(t))| ≥
ld ld ∞ 2 ls ∞ 2
(cid:18) ∆α(cid:19) (cid:18) ∆α(cid:19)
+P |Y˜τ(G¯(t))−Y˜τ(t)| ≥ +P |Y˜τ(t−G¯(t))| ≥
rd rd ∞ 2 rs ∞ 2
(cid:18) ∆α(cid:19) (cid:32) (cid:33)
≤4P ω (∆β)≥ +4P sup (t−G¯(t))≥∆β
S˜ 2
t∈[0,1]
≤4c∆r+4c(cid:48)(cid:48)∆1−β,
where |·| denotes the sup norm restricted to [0,1]. The last inequality follows from Lemmas 1
∞
and 3. This completes the proof of Proposition 1.
4 Proof of Theorem 1
Now that we have Proposition 1 we are almost ready to prove Theorem 1. We’d like to show the
existence of exceptional times at which ∩ Cτ occurs. We just need one more Lemma from [2]:
k≥0 k
Lemma 4. (Lemma 4.3, [2]) There exists c∈(0,∞) such that for τ,τ(cid:48) ∈[0,1], ∀n≥0:
(cid:89)n P(Cτ ∩Cτ(cid:48)) 1
k k ≤c ,
P(C )2 |τ −τ(cid:48)|b
k
k=0
where C :=C0 and b=log(sup [P(C )−1])/logγ >0.
k k k k
This was established in [2] for a different collection of rectangle events, A . However, to make
k
their proof work for C , we just need a,c such that:
k
(cid:18) 1 (cid:19)a
P(Cτ ∩Cτ(cid:48))≤P(C )2+c ∀τ,τ(cid:48) ∈[0,1],k ≥0, (21)
k k k γk|τ −τ(cid:48)|
10