Horizons of fractional Brownian surfaces

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Horizons of fractional Brownian surfaces

Kenneth Falconer, Jacques Lévy Véhel

To cite this version:

Kenneth Falconer, Jacques Lévy Véhel. Horizons of fractional Brownian surfaces. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2000, 456 (2001), pp.2153-2178. �10.1098/rspa.2000.0607�. �inria-00581036�

K.J. Falconer

Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland

and

J. Levy Vehel

Projet Fractales, INRIA Rocquencourt, 78153 Le Chesnay Cedex, France

Abstract

We investigate the conjecture that the horizon of an index-

fractional Brownian surface has (almost surely) the same Holder exponents as the surface itself, with corresponding relationships for fractal dimensions. We establish this formally for the usual Brownian surface (where

=

¹²

), and also for other

^;

0

^{< <}

1, assuming a hypothesis concerning maxima of index-

Brownian motion. We provide computational evidence that the conjecture is indeed true for all

.

1 Introduction and background

Fractional Brownian random surfaces have found several applications in recent years in such diverse elds as computer modeling of landscapes (Mandelbrot 1982; Peitgen &

Saupe 1988), modeling of rough surfaces in physics and chemistry and engineering and in medical imaging. The parameters for such random elds need to be chosen so as to obtain the best t to the data at hand. As an example, natural-looking landscapes typically have local Holder exponent about 0.85 almost everywhere and fractal dimension 2.15. In some situations, an additional feature has to be taken into account: the `horizon' dened by the surface may be relevant to modeling, and one needs to control its fractal characteristics (Mandelbrot 1982; Mandelbrot & Van Ness 1968). For instance, it is believed that some glasses may be well modeled by fractional Brownian surfaces. Several properties of the glass, relevant both for industrial and domestic use, depend on the roughness of its horizon, which governs its behaviour under skimming lighting. More generally, fractal electromagnetism studies how waves interact with `fractal' media (such as the sea in the case of radar imaging) (see Jaggard 1991). The horizon is then the appropriate geometrical object to consider for studying skimming waves. In imaging processes, especially medical imaging, one observes a two dimensional `horizon' of a three dimensional object, such as a lung, and tries to infer the properties of the original object (Lundahl et al 1986). Permeability elds of a porous medium provide another example (see Addison & Ndumu 1999).

It is conjectured that, for a `suciently random and homogeneous' surface, the Holder exponent of a horizon equals that of the surface, and that the `fractal dimension' of the

1

horizon is one less than that of the surface. In this paper we investigate this horizon conjecture for index-

fractional Brownian surfaces. We show that the horizon conjecture holds when

=

¹²

, and for other values of

^;

0

^<<

1 given a highly plausible assumption (the `maximum property') on the form near a maximum of index-

fractional Brownian motion in one variable. We present direct and indirect computational evidence based on very large data sets to support the conjecture.

We rst consider a function

^x

: [0

^;

1] ^d

^!

R ^{, where}

^d

is a positive integer, with graph graph

^{x f}

( t

^;x

( t )) : t

²

[0

^;

1] ^d

^g

R ^d

⁺¹

. In particular, for

^d

= 2 we think of graph

^x

as a surface above the unit square

^S

= [0

^;

1]

[0

^;

1] and we dene the horizon function

z

: [0

^;

1]

^!

R ^by

z

(

^t

) = sup

0

u

^1x

(

^t;u

)

^;

(1.1)

where t ^{= (}

^t;u

) in coordinate form. Thus graph

^z

R

²

may be thought of as the horizon of the surface graph

^x

R

³

. We are interested in the relationships between the fractality of the functions

^x

and

^z

and especially between their Holder exponents and the dimensions of the surface, graph

^x

, and its horizon, graph

^z

. Whilst our main concern in this paper is with index-

fractional Brownian surfaces, we rst gives some lemmas relating to functions of a general form. We dene the (lower) Holder exponent of

^x

at t

by

H

x ( t ) = liminf

v !t

log

^jx

( v )

^x

( t )

^j

log

^j

v t

^{j :}

^(1.2)

It is also useful to dene the lower and upper approximate Holder exponents by

H

ap x ( t ) = apliminf

v !t

log

^jx

( v ⁾

^x

⁽ t ⁾

^j

log

^j

v t

^{j ; H}

^{ap x (} t ) = aplimsup

v !t

log

^jx

( v ⁾

^x

⁽ t ⁾

^j

log

^j

v t

^{j :}

^(1.3)

Recall that the lower approximate limit is dened by apliminf

^{v !tf}

( v ^{) =}

^a

if there exists a set

^A

of Lebesgue density 1 at t such that liminf

^{v !t}

;

^{v 2}

A

^f

( v ) =

^a

, with a similar denition involving limsup for the upper approximate limit. (The set

^A

has Lebesgue density 1 at t if it is Lebesgue measurable and

lim

^!0

L

d (

^B

( t

^;

)

^\A

)

L

d (

^B

( t

^;

)) = 1 where

^L

^d is

^d

-dimensional Lebesgue measure on R ^{d ). Clearly}

H

x ( t )

^H

^ap _x ( t )

^H

^ap _x ( t )

If

^H

^ap _x ( t ) =

^H

^ap _x ( t ) we say that the approximate Holder exponent exists and write

^H

_x ^ap ( t ) for the common value. We write dim

^H;

dim

^B;

dim

^B

for Hausdor, lower box and upper box dimension respectively (see Falconer 1990) for their denitions and basic properties.

There are some basic lemmas that are useful in calculating dimensions of graphs. The rst lemma provides an upper bound for the dimension of graph

^x

, given a Holder condition on

^x

.

Lemma 1.1 Suppose that

^x

: [0

^;

1] ^d

^!

R ^satises

jx

( v ⁾

^x

⁽ t ⁾

^jcj

v t

^j

^(1.4)

for all t

^;

v

²

^[0

^;

^{1] d , where 0}

^<

^{1 and}

^c>

^{0. Then}

^H

x ( t ⁾

^{for all} t

²

^[0

^;

^{1] d , and}

dim

^H

graph

^x

dim

^B

graph

^x

dim

^B

graph

^xd

+ 1

.

2

Proof. See (Falconer 1990, Corollary 11.2).

The next lemma is useful for obtaining a lower bound for the dimension of a graph.

Lemma 1.2 ^Let

^x

: [0

^;

1] ^d

^!

R . Suppose that there is a number

^s

(1

^s

2) such that for

^L

^d -almost all t

²

[0

^;

1] ^d there exists

^c>

0 such that

L

d

f

v :

^j

v t

^jh

and

^jx

( v )

^x

( t )

^jhgch

^s for all suciently small

^h

. Then dim

^H

graph

^xs

.

Proof. We may dene a probability measure

on graph

^x

by

(

^A

) =

^L

^d

^f

t

²

[0

^;

1] ^d : ( t

^;x

( t ))

^2Ag

. Then for

-almost all ( t

^;x

( t ))

²

graph

^x

,

(

^B

(( t

^;x

⁽ t ⁾⁾

^;h

⁾⁾

⁽

^B

⁽ t

^;h

⁾

^[

^x

⁽ t ⁾

^h;x

⁽ t ^{) +}

^h

^])

^ch

^s

for all suciently small

^h

, where

^B

( t

^;h

) is the ball of radius

^h

and centre t . By Frostman's lemma dim

^H

graph

^xs

.

(There is a rather weaker condition that leads to a similar lower bound for the box- dimensions, (see Falconer 1990 Corollary 11.2).) The following useful lemma shows that a Holder condition on a surface implies a corresponding condition on its horizon, with corresponding bounds for the dimensions.

Lemma 1.3 Suppose that

^x

: [0

^;

1] ^d

^!

R ^satises

jx

( v ⁾

^x

⁽ t ⁾

^jcj

v t

^j

^(1.5)

for all t

^;

v

^2S

^{, where} 0

^<

1 and

^c>

0. Then the horizon function

^z

given by (1.1) satises a Holder condition

jz

(

^v

)

^z

(

^t

)

^{jcjv tj}

(1.6)

for all

^{t;v 2}

[0

^;

1], so in particular dim

^H

graph

^z

dim

^B

graph

^z

dim

^B

graph

^z

2

and

^H

_z (

^t

)

for all

^t2

[0

^;

1].

Proof. Fix

^t

and

^v

, and choose

^u

such that

^z

(

^t

) =

^x

(

^t;u

) (since

^u

is continuous the supremum is attained). Then

z

(

^v

)

^x

(

^v;u

)

^x

(

^t;u

)

^{cjv tj}

=

^z

(

^t

)

^{cjv tj}

^;

so

^z

(

^v

)

^z

(

^t

)

^{cjv tj}

. Combining this with the symmetric inequality gives (1.6). The Hausdor and box dimension estimates follow from Lemma 1.1, and the Holder exponent from (1.2).

Lower bounds for dimensions of horizons are much more awkward to obtain, and we proceed to consider this problem for fractional Brownian surfaces.

3

2 Fractional Brownian elds

A random eld

^X

on the

^d

-dimensional unit cube [0

^;

1] ^d is a family of random variables

fX

( t ^{) :} t

²

^[0

^;

^{1] d}

^g

dened on some probability space. The random eld

^X

is Gaussian if, for any nite set of points t

^1;:::;

t n

²

[0

^;

1] ^d and scalars

^1;:::;

n , the random vari- able

^P

_nj

⁼¹

j

^X

( t j ) has Gaussian distribution. For general denitions and properties of Gaussian random elds (see Adler 1981; Geman and Horowitz 1980; Pitt 1978; Kahane 1985). We take

^X

to be a Gaussian eld with zero mean, that is

^E

(

^X

( t )) = 0 for all

t

²

[0

^;

1] ^d , where

^E

denotes expectation. Such a Gaussian eld is completely determined (in the sense that the joint distribution of any nite set of

^X

( t j ) is determined) by the covariance functions

E

(

^X

( t

¹

)

^X

( t

²

)) ( t

^1;

t

^{2 2}

[0

^;

1] ^d )

^;

from which the variances

E

(

^jX

( t

¹

)

^X

( t

²

)

^j2

) =

^E

(

^X

( t

¹

)

²

) +

^E

(

^X

( t

²

)

²

) 2

^E

(

^X

( t

¹

)

^X

( t

²

))

may be found. The eld

^X

has stationary increments if

^E

(

^jX

( t

¹

)

^X

( t

²

)

^j2

) depends only on t

¹

t

²

and has isotropic stationary increments if it depends only on

^j

t

¹

t

^2j

. A Gaussian eld has a continuous version if there is a eld with the same nite-dimensional distributions with sample paths that are a.s. continous on [0

^;

1] ^d . We specialise to the isotropic index-

fractional Brownian eld (fBf) where 0

^{< <}

1, that is the Gaussian eld

^X

( t ) with zero mean and covariance function

E

(

^X

( t

¹

)

^X

( t

²

)) =

¹²

(

^j

t

^1j2

+

^j

t

^2j2

^j

t

¹

t

^2j2

) ( t

^1;

t

^{2 2}

[0

^;

1] ^d )

^;

(2.1) (see Adler 1981). It follows easily that

E

(

^jX

( t

¹

)

^X

( t

²

)

^j2

) =

^j

t

¹

t

^2j2

^;

(2.2) and that

^E

(

^X

( t )

²

) =

^j

t

^j2

, so in particular

^X

( 0 ) = 0 a.s. Thus

^X

has isotropic station- ary increments and is scale invariant, in the sense that for

^{h 6}

= 0 the Gaussian process

jhj

X

(

^h

t ) has the same covariance functions and thus the same distributions as

^X

( t ).

It should be emphasised that the increments are not independent in any sense unless

=

¹²

. Index-

fBf has a continuous version (see Adler 1981); indeed, given

^>

0, with probability one there is a uniform Holder inequality

jX

( t

¹

⁾

^X

⁽ t

²

⁾

^jj

t

¹

t

^2j

^(2.3)

for all

^j

t

¹

t

^2j

suciently small. We will always work with this continuous version of the process. It is not dicult to show, using Lemmas 1.1 and 1.2, that dim

^H

graph

^X

= dim

^B

graph

^X

= dim

^B

graph

^X

=

^d

+ 1

a.s. (Adler 1981; Falconer 1991).

There is an extremely useful way of regarding Gaussian elds in terms of Hilbert spaces. Let be the sample space underlying a Gaussian eld on [0

^;

1] ^d and write

^L2

() for the space of zero mean random variables

^X

on with

^E

(

^X2

)

^{< 1}

. Then

^L2

() is a real Hilbert space under the inner product given by

^E

(

^X1X2

) for

^{X1;X2 2 L2}

().

We write

^H0

for the subspace span

^fX

( t ^{) :} t

²

^[0

^;

^{1] d}

^g

^and

^H

for the closure of

^H0

in

L

2

(). With this Hilbert space approach, the covariances

^E

(

^X

( t

¹

⁾

^X

⁽ t

²

)) are just inner products. This provides a very convenient way of representing conditional variances of the

4

X

( t ). For example, if

^{A S}

, then var(

^X

( t ⁾

^jX

⁽ s ^{) :} s

^{2 A}

) is the Hilbert space distance from

^X

( t ) to the subspace

^K

span

^fX

( s ) : s

^2Ag

of

^H

, and there is a decomposition

X

( t ) =

^X1

( t ) +

^X2

( t ), where

^X1

( t )

^{2 K}

and

^X2

( t )

^{2 K?}

. To aid calculation there is a useful integral representation of the covariances of index-

fBf on [0

^;

1] ^d . It is easy to check (Geman & Horowitz 1980; Kahane 1985) that (2.1) may be written as

E

(

^X

( t )

^X

( v )) =

^kZ

Rd^j

a

^j

^d

²

(

^e

ⁱ

^ta

1)(

^e

ⁱ

^{v a}

1)d a

^;

(2.4) where

^k

depends only on

^d

and

. In particular this gives that, for t

^1;:::;

t ⁿ

²

R ^{d and}

1

;:::;

n

²

R ,

E

(

^jX

ⁿ

j

⁼¹

j

^X

( t j )

^j2

) =

^kZ

Rd^j

a

^j

^d

²

^jX

ⁿ

j

⁼¹

j

^e

i

^tj^aj

2

d a

^;

(2.5)

provided that

^P

_nj

⁼¹

j = 0. This integral form provides a very useful way of calculating conditional variances of increments of the

^X

( t ). For

^E

an ellipsoid in R ^d and

^>

1, write

E

for the homothetic (similar and similarly situated) ellipsoid with the same centre as

E

obtained by scaling

^E

by a factor

about its centre. For

^T

R ^{d we write}

^F

T for the sigma eld

(

^X

( t

¹

)

^X

( t

²

) : t

^1;

t

^{2 2 T}

) generated by the dierences of

^X

at pairs of points in

^T

.

Proposition 2.1 ^Let

^E

R ^d be an ellipsoid with semi-axis lengths 0

^{< s1 ::: s}

d

and let

^E

be dened as above for some

^>

1. If

^{V E}

and

^{T E}

_c (the complement of

^E

), and v

^2V

^and t

^2T

^{, then}

var

var(

^X

( v ⁾

^X

⁽ t ⁾

^j

⁽

^F

^T

^;F

^{V ))}

^cs

^d

¹⁺²

⁽

^s1:::s

^{d )}

^1;

^(2.6)

and hence, for all

^{1 <2}

,

P

(

^{1 <X}

( v )

^X

( t )

^2j

(

^F

T

^;F

V ))

^{c0j2 1js1}

^d=

²

(

^s1:::s

d )

¹

⁼

^2;

(2.7) where

^c

and

^c0

depend only on

^d;

and

.

Proof. Working in the Hilbert space setting, var is the square of the Hilbert space distance of

^X

( v ⁾

^X

⁽ t ) from the subspace dened by the conditioning; thus

var = inf

^E

(

^jX

( v ⁾

^X

⁽ t ^{) +}

^X

^m

j

⁼¹

j (

^X

( v j )

^X

( v ⁾⁾

+ ^m

0

X

j

⁼¹

j (

^X

( t ^{j )}

^X

⁽ t ^{)) :} v ^j

^2V;

t ^j

^2Tj2

⁾

where the inmum is over all

^1;:::;

m

^;1;:::;

m

^{0 2}

R , v

^1;:::;

v m

^2V

and t

^1;:::;

t m

^{0 2}

V

. By redening the

j

^;

j by taking appropriate linear combinations in the sums, we get that

var

inf

^E

(

^jX

j

j

^X

( v j )

^X

j

j

^X

( t j ) : v j

^2V;

t j

^{2T; X}

j

j =

^X

j

j = 1

^j2

)

^:

5

Since

^P

j

j

^P

j

j = 0, the integral representation (2.5) gives var

inf

^k

(

Z

j

P

j

j

^e

i

^vj^{a P}

j

j

^e

i

^tj^aj 2

j

a

^j

^d

⁺²

^d a

: v j

^2V;

t j

^2T;X

j

j =

^X

j

j = 1

9

=

;

:

(2.8) Dening the Fourier transform ^

^g

of a `well-behaved' function

^g

on R ^d by

^

g

( b ^{) =}

^{Z 1}

1 e

i

^bw

g

( w ^)d w

in the usual way, the inversion formula gives

g

( w ) = (2

) ^d

^{Z 1}

1 e

i

^bw

^

^g

( b )d b

^:

We take

^g

to be a

^C1

`bump function', with

^g

( w ) = 1 for w

^{2 B}

( 0

^;

1) and

^g

( w ) = 0 for w

^62B

⁽ 0

^;

^{). Let} p

²

R ^d be the centre of the ellipsoid

^E

and let

be a non-singular self-adjoint linear operator on R ^d that maps the unit ball onto the translated ellipsoid

E

p ; thus

^E

=

(

^B

( 0

^;

1)) + p and

^E

=

(

^B

( 0

^;

)) + p . With

^P

j

j =

^P

j

j = 1, it follows, using the denitions of

and

^g

, that

1 =

^X

j

j

^g

(

¹

( v j p ))

^X

j

j

^g

(

¹

( t j p ))

= (2

) ^d

^Z

(

^X

j

j

^e

i

^b

^{1(p v}j^{) X}

j

j

^e

i

^b

^{1(p t}j⁾

)^

^g

( b )d b

= (2

) ^d

^Z

(

^X

j

j

^e

i

^{a(p v}j^{) X}

j

j

^e

i

^{a(p t}j⁾

)^

^g

(

( a ))

^j

det

^j

d a

^;

writing b =

( a ) and using the self-adjointness of

. Thus by Cauchy's inequality, 1

(2

)

²

^d

^{Z je}

ⁱ

^apj2j

P

j

j

^e

i

^avj ^P

j

j

^e

i

^atjj 2

j

a

^j

^d

⁺²

^d a

^{Z g}

^ (

( a ))

^2j

a

^j

^d

⁺²

^j

det

^j2

d a

(2

)

²

^d

^{Z j}

P

j

j

^e

i

^avj ^P

j

j

^e

i

^atjj 2

j

a

^j

^d

⁺²

^d a

^Z

^

^g

( b )

^{2j 1}

( b )

^j

^d

⁺²

d b

^j

det

^j

(2

)

²

^d

^{Z j}

P

j

j

^e

i

^avj ^P

j

j

^e

i

^atjj 2

j

a

^j

^d

⁺²

^d a

^Z

^

^g

( b )

^2j

b

^j

^d

⁺²

d b

^s1

^d

²

(

^s1:::s

d )

^;

as det

=

^s1:::s

d and

^{k 1k}

=

^s11

. Since ^

^g

( b ) is continuous and rapidly decreasing, the last integral is nite, so combining this estimate with (2.8) gives (2.6). Inequality (2.7) follows on noting that, if

^W

is Gaussian with mean

and variance

²

, then

P

(

^{1 <W 2}

) = (2

)

¹

⁼

^{2 1Z}

²

1

exp( (

^s

)

⁼

2

²

)d

^s

(2

)

¹

⁼

^{2 1j2 1j:}

6

3 The maximum property for fBm

In this section we briey consider index-

fractional Brownian motion (fBm)

^X

: [0

^;

1]

^!

R , that is the index-

fBf dened by (2.1) with

^d

= 1. Then

^X

has a continuous version which a.s. attains its maximum at a unique point of (0

^;

1). We say that index-

fBm

^X

on [0

^;

1] satises the maximum property if, conditional on

^X

attaining its maximum at

t

m

²

(0

^;

1), for all

^>

0 there exists a.s.

^>

0 such that

X

(

^t

)

^X

(

^t

m )

^jt

m

^tj

⁺

for all

^t2

[0

^;

1] with

^jt

m

^tj:

(3.1) Of course, the opposite inequality, that for all

^>

0

X

(

^t

)

^X

(

^t

m )

^jt

m

^tj

for all

^t

close to

^t

m , follows from the a.s. Holder condition (2.3). Thus

^X

has the maximum property if its sample paths fall away from their maximum at a rate with power law exponent close to

. It should be emphasised that the maximum property is dened in terms of fractional Brownian motion on the real interval and not in terms of fractional Brownian elds on R

²

, although this is where our applications lie.

Proposition 3.1 Usual Brownian motion, that is index-

¹²

fBm, has the maximum prop- erty.

Proof. This fact, stated for example in (Barlow & Perkins 1984), follows from the decomposition theorem (It^o & McKean 1974; Williams 1974) which identies Brownian motion conditioned on a maximum at 0 with the 3-dimensional Bessel process (that is the radial component of Brownian motion in R

³

).

The following maximum conjecture seems extremely likely, but it does not appear to have been established rigorously.

Conjecture 3.2 ^{For all 0}

^<<

^{1, index-}

fBm has the maximum property.

This conjecture holds in an `approximate' sense: conditional on

^X

attaining its maximum at

^t

, we have a.s. that for all

^>

0

aplimsup _t

!

t (

^X

(

^t

)

^X

(

^t

m ) +

^jt

m

^tj

⁺

)

0

^;

(see Geman & Horowitz 1980, Theorem 30.4), but this is not sucient for our arguments.

4 Probabilistic estimates

From now on, we take

^d

= 2 and work with the index-

fractional Brownian eld

^X

:

^{S !}

R ^where

^S

is the unit square [0

^;

1]

[0

^;

1]. In this case we refer to graph

^X

as the index-

fractional Brownian surface. In this section we estimate some conditional variances and probabilities of

^X

preparatory to our calculations on the fractality of the horizon of

^X

. We use the vector and coordinate forms t ^{= (}

^t;u

⁾

^;

v ^{= (}

^v;w

⁾

^;

t

¹

^{= (}

^t1;u1

) etc. for points in R

²

. We introduce sigma-elds generated by the random variables

^X

(

^t;u

) on certain

7

subsets of the square

^S

. For

^t2

[0

^;

1] let

^F

t =

(

^X

(

^t;u1

)

^X

(

^t;u2

) :

^{u1;u2 2}

[0

^;

1]), and for (

^v;u

)

^2S

and

^{r >}

0 let

^F(

v;u

⁾

;r =

(

^X

(

^v;w

)

^X

(

^v;u

) :

^w2

[0

^;

1]

^{;jw ujr}

). Thus

F

t determines

^X

along a section with rst coordinate

^t

, and

^F(

v;u

⁾

;r determines

^X

along a section of length 2

^r

with midpoint (

^v;u

), in both cases to within a vertical displacement.

Proposition 4.1 ^{For all}

^{r >}

0 and all pairs of points (

^t;u

)

^;

(

^v;u

)

^2S

with 0

^{<jv tjr}

, we have, almost surely,

var(

^X

(

^v;u

)

^X

(

^t;u

)

^j

(

^F

t

^;F(

v;u

⁾

;r ))

^{cjv tj1+2}

^{r 1;}

(4.1) and for all

^{1 <2}

,

P

(

^{1 <X}

(

^v;u

)

^X

(

^t;u

)

^{2 j}

(

^F

t

^;F(

v;u

⁾

;r ))

^{c0j2 1jjv tj 1}

⁼

²

^r1

⁼

^2;

(4.2) where

^c

and

^c0

depend only on

^d

and

.

Proof. We apply Proposition 2.1, taking

^E

to be the ellipse with centre (

^u;v

) and semi- axes parallel to the coordinate axes, of lengths

^{13jt vj}

and

^r

, and taking

= 2. Then

^E

contains the interval [(

^{v;u r}

)

^;

(

^v;u

+

^r

)] and

^E

is disjoint from the line

^f

(

^t;w

) :

^{t 2}

R

^g

, so the result is immediate from Proposition 2.1

The (random) horizon function of the index-

fBf

^X

(

^t;u

) is given by

Z

(

^t

) = sup

0

u

^1X

(

^t;u

) (0

^t

1)

^:

Since

^X

is almost surely continuous, for each

^t2

[0

^;

1] we can dene the random variable

U

t by

U

t = inf

^fu2

[0

^;

1] :

^X

(

^u;t

) =

^Z

(

^t

)

^g;

(4.3) so

^U

t is

^F

t -measurable. (In fact

^X

(

^t;u

) a.s. attains a unique maximum at

^u

=

^U

t

²

(0

^;

1), with

^Z

(

^t

) =

^X

(

^t;U

t ), see Section 5.) For

^{r >}

0 and

^{t;v 2}

[0

^;

1], we dene the random variable

Z

rt (

^v

) = sup

^fX

(

^v;u

) :

^u2

[0

^;

1] and

^{ju U}

t

^jrg:

(4.4) Then

^Z

_rt (

^v

) gives the maximum of

^X

(

^v;u

) over a restricted interval of

^u

close to

^U

t .

Proposition 4.2 ^{For all}

^r>

0,

^{1 <2}

and

^{t;v 2}

[0

^;

1] with 0

^{<jv tjr}

, we have

P

(

^{1 <Z}

_rt (

^v

)

^Z

(

^t

)

²

)

^{c0j2 1jjv tj 1}

⁼

²

^r1

⁼

^2;

(4.5) where

^c0

depends only on

.

Proof. For (

^v;u

)

^2S

dene the random variable

Z

(

^v;u

) = sup

^fX

(

^v;w

) :

^{jw ujr}

and (

^v;w

)

^2Sg:

Since

^Z

(

^t

)

^X

(

^t;u

) and

^Z

(

^v;u

)

^X

(

^v;u

) are

^F

t -measurable and

^F(

v;u

⁾

;r -measurable respectively, and therefore are both

(

^F

t

^;F(

v;u

⁾

;r )-measurable, Proposition 4.1 implies that for all

^{t 2}

[0

^;

1] and (

^v;u

)

^2S

,

P

(

^{1 <Z}

(

^v;u

)

^Z

(

^t

)

^2j

(

^F

t

^;F(

v;u

⁾

;r ))

^{c0j2 1jjv tj 1}

⁼

²

^r1

⁼

^2;

8