A lfo1der Condition for the Local Time
of the Brownian Sheet
Robert J. Adler
Institute of Statistics
~fimeo
Series #1221
April 1979
DEPARTMENT OF STATISTICS
Chapel Hill, North Carolina
A HOLDER CONDITION FOR THE LOCAL TIME
OF THE BROWNIAN SHEET
by
Robert J. Adler l
University of New South Wales and
University of North Carolina at Chapel Hill
Abstract
L(x,t), x
~
Brownian sheet.
E
IR l , t
E
IR
N
+
is the local time of a real valued, N-parameter,
It is shown that if B is any cube lying within the set [£,l]N
for some £ > 0, and L(x,B) is the increment of the local time over B, then as
long as the sides of B are of sufficiently small length IL(x,B) I is always
bounded above bY.~[~(B)]Y for some a.s. finite random variable ~, if
-1
Y < 1 - (2N) ..
This Holder result for the local time was previously derived
using an inequality of Tran, the proof of which is currently in doubt.
The proof
given here is effectively self-contained.
lResearch supported in part by a Queen Elizabeth II Research Fellowship and the
Office of Naval Research, Contract N00014-75-C-0809.
AMS 1970 subject classifications:
Key Words and Phrases:
Abbreviated Title:
Primary 60J65, 60J55; Secondary 60G15, 60G17.
Brownian sheets, local time, Holder conditions, Hausdorff
dimension.
Brownian sheet local time
1
1.
The problem and its background.
This paper is concerned with sample path properties of the N-parameter Wiener
process, or Brownian sheet, that involve its local time.
We let W(N), or simply
W, denote this process, so that weN) is a real valued Gaussian random field with
zero mean and covariance
N
min(s., t.) ,
II
1
i=l
where s
1
= (sl, ... ,sN) and! = (tl, ... ,t N) belong to the parameter space IR~ of
points in IR
N
with all components nonnegative.
If we use AN to denote
l
N-dimensional Lebesgue measure, and let B be a Borel set of IR , we can define an
occupation measure
~(t,
~(t, B,
B, w) for the Brownian sheet by
w)
= ''NA__ (?:
0 <
-
S. <
1 -
t., W(N) (s,w) E B) .
1
From this measure, we can define a local time function L(x,t,w) to be any function
L satisfying
~(t,B,w)
= f B L(x,t,w)dx
for almost every w.
It is now well established that such a local time exists, and, furthermore,
is jointly continuous in both its paramters.
6.4, for N
=
(Cairoli and Walsh (1975), Theorem
2, and Davydov (1978) for general N.)
For certain purposes, it is
often necessary to have more detailed knowledge of the Brownian sheet local time
other than mere joint continuity, and in Adler (1978), Lemma 4, we obtained a
Holder-type bound for it.
However, in the derivation used there, use was made
2
of some inequalities on the moments of L derived by Tran (1976).
The validity
of these inequalities has now been thrown into doubt following the discovery of
an error in Tran's work, noted in the review [8] by Pruitt.
In this paper we
present a new derivation of the aforementioned Holder condition, using a method
based on a result of Davydov (1978).
The new result, Theorem 1, can then be
used to replece Lemma 4 in Adler (1978) in the study of the Hausdorff dimension
of the level sets of the Brownian sheet presented there, as noted in the correction note [2].
To state the result we require some notation.
.
and real valued, contl.nuous,
F:~
N+l
-+ ~
1
For each x E [0,00], t,k E
~
N
+
,
we use F«(x,t), (x+h, t+k») to denote
the usual increment of F over the (N+l)-dimensional rectangle
N
[x, x+h] x
IT [t., t.+k.] obtained by taking the (N+l)-th difference of F around
i=l
l.
l. l.
the corner points of the rectangle. Similarly, for fixed, x, F(x, <!, ~+~»
denotes the increment of F (considered as a function of N variables only) over
N
N
IT [t., t.+k.]. If t, k E IR
we write <t, t+k> to denote the rectangle
i=l
l.
l. l.
+
N
IT
[t., t.+k.].
l.
l. l.
i=l
Mt)
= {s
-
E
and, for any
~
N
+
E
IT k .. Finally, for any t E IR~, set
i=l l.
t i for all i}, while ~(l) denotes the unit cube ~((l, ... ,l))
Clearly
:
si
> 0,
~
~
E
N
~«!,
t+k»
=
(1) denotes the set {t E ~(l): t. > E for all i}.
l.
In the following section we shall prove
Theorem 1.
There is a version of the local time
N+l variables, such that for any
E E
L(x,~,w),
(0,1] and every interval [a,b]
y < 1 - (2N)-1, there exist random variables nand
tive and finite for which
jointly continuous in all
s which
E
~ 1 and
+
are almost surely posi-
3
(1.1)
whenever x
E
= (k •...• k).
[a.b]. k
Ikl < n • and the cube <t.
t+k>
lies in ~ E (1).
~
~
~
For the sake of completeness we note that Walsh (1978) has also obtained a
Holder type condition for the local time of a two-dimensional Brownian sheet.
However. since his concern is with increments in the space parameter x. rather
than the time parameter, t. as in (1.1). his results do not overlap with those
presented here.
2.
Proof of Theorem 1.
The proof of Theorem 1 is built on two lemmas.
The first is essentially Lemma 2
of Adler (1978). except whereas that lemma yields a final Holder condition like
N
(2.4) below for increments over sets of the form
IT [t .• t.+k.]. the result
i=l
1
1
1
below concerns only increments over cubes. This allows some relaxation in the
conditions required for the lemma to hold.
With the exception of replacing rectan-
gles by cubes throughout. the proof of the following lemma is. verbatim. that of
the original result.
Throughout Lemma 1, k
(k •... ,k).
=
Lemma 1.
Let Ylx.t). x
U c
~(l)
(2.1)
E
be compact.
[0.1).
!
E ~(1),
Let
Suppose there are positive constants r, b. c, d such that
E\Y«(X,!). (x+h. :+~»)
~ blh\l+C Ik\N(I+d)
(2.2)
be an (N+I)-dimensional random field.
EIY(x+h. !) - Y(x.!)
~ blhl l+c
for
Ir
for
x. x+h
E
[0.1), t. t+k
Ir
x. x+ h
E [0 • 1) •
t
E U•
E
U,
4
(2.3)
EIY(x,
<
<!,
!+~»
,r
blkIN(l+d)
for
x
E
[0,1), t, t+k
E
U .
Then for every y < d/r there exists a version of Y, and almost surely positive,
finite, random variables
which
lkl <
n,
~
and
n,
such that for all x
E
[0,1), t, t+k
E
U for
we have
(2.4)
The second lemma is a consequence of Theorem 5 of Davydov (1978). To state
N-l
it, we set, for t E lR~, V to be the (N-l)-dimensional rectangle IT [O,t.],
t
i=l
1
wbile Wt(S) denotes the scaled, one-dimensional Brownian motion defined by
(2.5)
Lemma 2.
Let L(x,t) be the local time of an N-parameter Brownian sheet.
Then for each
x and t L(x,t) can be expressed as an integral of lower dimensional local times;
vjz
(2.6)
where Lt(x,') is the local time of the one-dimensional process W "
t
The importance of this lemma lies in the fact that it is comparatively simple
to obtain bounds on the moments of Lt(x,s), which, via (2.6), can be used to
obtain bounds of the form required in Lemma 1.
As a first step, we prove
5
Lemma 3.
Let
£
depending only on
t, t+k
-
-
and n
E (0,1],
£
and
~
2 be even.
Then there exists a finite constant K > 0,
n, such that if 0
E
[O,~], ~
=
(k, ... ,k) and
(1), then the following three inequalities hold:
E ~
£
(2.8)
EIL(x+h, t) _ L(x,t) In < Klhl
-
-
no
,
EIL(x, <t, ~+~» In ~ Klkl nNL1 - l /(2N)] .
(2.9)
Proof.
As the proofs of each of these inequalities are very much alike, we shall only
give a detailed proof of (2.7). We commence by noting that if for k > 0 we write
N-l
~t(k) to denote the set
IT [t., t.+k] then it follows from Lemma 2 that
1
i=l
1
Substituting this into the left hand side of (2.7) yields that the expectation
there is bounded above by
n
E
J ... J
IT
i=l
IL (i) «(x,tN), (x+h, tN+k»)
I
t
where the n-fold integral is over n copies of Vt(k) and we have left out the
n x eN-I) differentials.
However, this expression is clearly bounded above by
6
n
f ... f i~l
(2.10)
I
In lin .
[E \Ci)«Cx,t N), (x+h, tN+k»)]
Consider the expectation here.
Since we are now dealing with the local time of
a one-dimensional process, we can use known results for such processes.
In fact,
since LtCx,s) is the local time for Wt(s), it has incremental variance function
(2.11)
=s
N-l
x
n
i=l
t.
1.
Thus we can substitute directly into some inequalities of Berman ([3], p. 294-295,
[4], p. 1268-1269 or [5], p. 70, 74) to obtain that for any 0
E
[O,~],
EILt«(X,tN), (x+h, ~+k») In
2
Kl Ihl
no
f···f
n
[.n
{tN~sl~··· ~sn ~tN+k} 1.=1
where K depends only on n.
1
a(si-si_l)]
- (1+0)
dsl···ds n ,
Using (2.11) and elementary integration, it is
straightforward to check that the above expression is bounded by
where K also depends only on n.
2
But now we are almost done, for substituting
this back into (2.10) yields that the left hand side of (2.7) is bounded by
(2.12)
K Ih Ino
2
=K
3
Ik I~(l-o) f ...
Ihl no
Ikl~(1-0) ~
. 1
1.=
7
But since t. > £ for all j, the mean value theorem yields
J
It.+kl~(l-O) _ It. I~(l-o) < k x [~ (l_o)£-~(l+o)]
(2.13)
J
J
Absorbing K and the expression in brackets into one £-dependent constant,
3
(2.10)-(2.13) yield
which establishes the validity of (2.7).
To prove (2.8) we again apply Lemma 2 to obtain
L(x+h, t)
Taking moments as before yields
(2.14) EIL(x+h, t) - L(x,E) In
n
~f
f i~l [EIL~(i)(X+h,
t N) -
where, as in (2.9), we neglect to write in the differentials.
L~(i)(X,
t N) In]l/n ,
Appealing once
again to Berman's inequalities for one dimension (specifically (4.5) of [5])
establishes that, since t
N
~
£, the inner expectation is bounded above by a finite,
£-dependent, constant times Ihl
no
.
This fact is all that is required to complete
the proof of (2.8).
Finally, note that to establish (2.9) it suffices to proceed with the same
argument used to establish (2.7) appealing, on this occasion, to the inequality
(4.6) of [5].
This completes the proof of the lemma.
8
We can now turn to the proof of Theorem 1.
In fact, all we need do is
establish that L(x,t) satisfies the conditions of Lemma 1 when U =
~
£
(1).
It
is clear that inequalities (2.7)-(2.9) will correspond exactly to the conditions
(2.1)-(2.3) if we can choose appropriate r, b, c, d.
y < 1 - (2N)-1, let 0 lie in the interval
To do this, for a given
[O,~]. Then choose n (even) so large
that the following three inequalities hold:
n[l - (1+o)/(2N)] > 1,
no > 1,
y < [1 - (1+o)/(2N)] - n
Now put r
tive.
3.
= n,
b
= K,
c
= no
- 1, d
= n[l
-1
- (1+o)/(2N)] - 1.
These are all posi-
The above correspondence is thus established, as is, the proof @f Theorem 1.
Acknowledgement.
I am indebted to Professor Donald Geman for originally drawing my attention
to the problems discussed in Section 1, as well as for some extremely helpful
comments during the preparation of this work.
9
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[2]
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6~
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Submitted.
Local times and sample function properties of stationary
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[4]
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[7]
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134~
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46~
63-86.
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[8]
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[6]
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13?~
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23~
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[9]
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[10]
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3?~
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