Intersection local times of independent fractional Brownian motions as generalized white noise functionals

0.25 MB·English

by  Maria Joao Oliveira

Intersection local times of independent
0 fractional Brownian motions as generalized
1
0
white noise functionals
2

n
a
J Maria Jo˜ao Oliveira

4
Universidade Aberta, P 1269-001 Lisbon, Portugal

]
R CMAF, University of Lisbon, P 1649-003 Lisbon, Portugal
P
[email protected]
.
h
t Jos´e Lu´ıs da Silva
a
m
DME, University of Madeira, P 9000-390 Funchal, Portugal
[

CCM, University of Madeira, P 9000-390 Funchal, Portugal
1
v [email protected]
3
1
Ludwig Streit
5
0
Forschungszentrum BiBoS, Universit¨at Bielefeld, D 33501 Bielefeld, Germany
.
1
0 CCM, University of Madeira, P 9000-390 Funchal, Portugal
0
[email protected]
1
:
v
Abstract
i
X
In this work we present expansions of intersection local times of
r
a fractional Brownian motions in Rd, for any dimension d 1, with
≥
arbitrary Hurst coefficients in (0,1)d. The expansions are in terms
of Wick powers of white noises (corresponding to multiple Wiener
integrals), being well-defined in the sense of generalized white noise
functionals. As an application of our approach, a sufficient condition
on d for the existence of intersection local times in L2 is derived,
extending the results in [NOL07] to different and more general Hurst
coefficients.
Keywords: Fractional Brownian motion; White noise analysis; Local time
2000 AMS Classification: 60H40, 60G15, 60J55, 28C20, 46F25, 82D60
1
1 Introduction
In the recent years the fractional Brownian motion has become an object
of intense study, namely, due to its special properties, such as short/long
range dependence and self-similarity, yielding its proper and natural uses in
several applications in different fields (e.g. mathematical finances [MO08],
telecommunications engineering [NRT03]).
Besidesitsownspecificproperties,theintersectionpropertiesoffractional
Brownian motion paths have been studied by many authors as well, see
e.g. the works done by Gradinaru et al. [GRV03], Nualart et al. [HN07],
[HN05], Rosen [Ros87], and the references therein.
One may consider intersections of sample paths with themselves, as in
[DOS08] and references therein, or with other independent fractional Brow-
nian motions, as in [NOL07].
This work concerns the latter standpoint. Within the white noise analy-
sis framework (Section 2), a first purpose of this work is an extension of the
results presented in [AOS01] to two d-dimensional independent fractional
Brownian motions B and B with different Hurst coefficients, H and
H1 H2 1
H . Technically, this approach has the advantage that the underlying prob-
2
ability space does not depend on any Hurst coefficient under consideration.
As a consequence, one may analyze the intersection local time of any two
independent fractional Brownian motions, without any restriction on the
corresponding Hurst coefficients.
From the viewpoint of applications to physics, this absence of restrictions
on the Hurst coefficients under consideration is meaningful to widen the
modelling of polymers towards polymers molecules handling different types
of polymers.
For low dimensions, that is, either for d = 1 or for d = 2, the white noise
analysis framework allows the definition of the intersection local time of any
two independent fractional Brownian motions B in terms of an integral
Hi
over a Donsker’s δ-function
L d2tδ(B (t ) B (t )),
≡ H1 1 − H2 2
Z
intended to sum up the contributions from each pair of moments of time t ,
1
t for which the fractional Brownian motions B arrive at the same point.
2 Hi
A rigorous definition, such as, e.g., through a sequence of Gaussians ap-
2
proximating the δ-function,
x 2
(2πε)−d/2exp | | , ε > 0,
− 2ε
(cid:18) (cid:19)
will make L increasingly singular, and various “renormalizations” have to be
done as the dimension d increases. Of course, besides the dimension of the
space, the type of “renormalizations” needed depends as well on the Hurst
coefficients H (0,1)d being considered. For d > 2 with 1/max H +
i j 1,j
∈
1/max H d, the expectation diverges in the limit and must be sub-
j 2,j
≤
tracted. Depending on the values of max H , further kernel terms must be
j i,j
also subtracted (Theorem 9).
In this work we are particularly interested in the chaos decomposition
of L. We expand L in terms of Wick powers [HKPS93] of white noise, an
expansion which corresponds to that in terms of multiple Wiener integrals
when one considers the Wiener process as the fundamental random variable.
This allows us to derive the kernels for L. Due to the local structure of the
Wickpowers, thekernelfunctionsarerelativelysimpleandexhibitclearlythe
dimension dependence singularities of L (Proposition 8). For comparison, we
also calculate the regularized kernel functions corresponding to the Gaussian
δ-sequence mentioned above (Theorem 9).
As an application of this approach, in Theorem 11 we derive a sufficient
condition for the existence of the intersection local times in L2, extending the
results obtained in [NOL07] to different and more general Hurst coefficients.
2 Gaussian white noise calculus
In this section we briefly recall the concepts and results of white noise anal-
ysis used throughout this work (for a detailed explanation see e.g. [BK88],
[HKPS93], [HØUZ96], [Kuo96], [Oba94]).
2.1 Fractional Brownian motion
Thestartingpointofwhitenoiseanalysisfortheconstructionoftwoindepen-
dent d-dimensional, d 1, fractional Brownian motions is the real Gelfand
≥
triple
S (R) L2 (R) S′ (R),
2d ⊂ 2d ⊂ 2d
3
where L2 (R) := L2(R,R2d) is the real Hilbert space of all vector valued
2d
square integrable functions with respect to the Lebesgue measure on R, and
S (R), S′ (R) are the Schwartz spaces of the vector valued test functions
2d 2d
and tempered distributions, respectively. We shall denote the L2 (R)-norm
2d
by (or if there is no risk of confusion simply by ) and the dual pairing
2d
|·| |·|
between S′ (R) and S (R) by , , or simply by , , which is defined as
2d 2d h· ·i2d h· ·i
the bilinear extension of the inner product on L2 (R), i.e.,
2d
2d
g,f = dxg (x)f (x),
2d i i
h i
R
i=1 Z
X
for all g = (g ,...,g ) L2 (R) and all f = (f ,...,f ) S (R). By the
1 2d ∈ 2d 1 2d ∈ 2d
Minlos theorem, there is a unique probability measure µ on the σ-algebra
B
generated by the cylinder sets on S′ (R) with characteristic function given
2d
by
C(f) := dµ(ω~)eihω~,fi = e−21|f|2, f S2d(R).
∈
ZS2′d(R)
In this way we have defined the white noise measure space (S′ (R), ,µ).
2d B
To construct two independent d-dimensional fractional Brownian motions
we shall consider a 2d-tuple of independent Gaussian white noises
ω~ := (ω~ ,ω~ ), ω~ = (ω ,...,ω ),i = 1,2.
1 2 i i,1 i,d
Within this formalism, a version of a d-dimensional Wiener Brownian motion
is given by
B(t) := ω ,11 ,..., ω ,11 , (ω ,...,ω ) S′(R),
h 1 [0,t]i h d [0,t]i 1 d ∈ d
(cid:0) (cid:1)
where 11 denotes the indicator function of a set A and , = , . For
A 1
h· ·i h· ·i
an arbitrary d-dimensional Hurst parameter H = (H ,...,H ) (0,1)d, a
1 d
∈
version of a d-dimensional fractional Brownian motion is given by
B (t) := ω ,M 11 ,..., ω ,M 11 , (ω ,...,ω ) S′(R),
H h 1 H1 [0,t]i h d Hd [0,t]i 1 d ∈ d
(cid:0) (cid:1)
where, for a 1-dimensional Hurst parameter H (0,1) and for a generic real
∈
4
valued function f,
(1 H)K ∞ f(x) f(x+y)
2 − H lim dy − , H (0,1/2) ( )
 Γ H + 21 ε→0+Zε y23−H ∈ ∗

(M f)(x) :=  f(x(cid:0)), H =(cid:1)1 ,
H 
 2



K ∞
 H dyf(y)(y x)H−23, H (1/2,1) ( )
 Γ H − 12 Zx − ∈ ∗∗


provided the limit(cid:0)in ( ) (cid:1)exists for almost all x R and the integral in
∗ ∈
( ) exists for all x R (for more details see e.g. [Ben03] and [PT00] and
∗∗ ∈
the references therein). Independently of the case under consideration, the
normalizing constant K is given by
H
KH = Γ H + 1 1 + ∞ds (1+s)H−21 sH−21 −12 .
2 2H −
(cid:18) (cid:19)(cid:18) Z0 (cid:16) (cid:17)(cid:19)
There are several examples of functions f for which M f exists for any
H
H (0,1),namely,forf = 11 witht > 0orforf S (R). Formoredetails
[0,t] 1
∈ ∈
and proofs see e.g. [Ben03], [BHØZ08], [Mis08], [PT03], and the references
therein.
2.2 Hida distributions and characterization results
Let us now consider the complex Hilbert space (L2) := L2(S′ (R), ,µ). For
2d B
simplicity one introduces the notation
d d
n = (n , ,n ) Nd, n = n , n! = n !.
1 d i i
··· ∈
i=1 i=1
X Y
The space (L2) is canonically isomorphic to the symmetric Fock space of
symmetric square integrable functions,
∞
⊗2d
(L2) SymL2(Rk,k!dkx) ,
≃
(cid:16)Mk=0 (cid:17)
5
which leads to the chaos expansion of the elements in (L2),
F(ω~1,ω~2) = h: ω~1⊗m : ⊗ : ω~2⊗k :,fm,ki
m k
XX
d d
= : ω1⊗,imi : ⊗ : ω2⊗,jmj :,fm,k ,
m k *i=1 j=1 +
XX O O
with kernel functions fm,k in the Fock space, that is, square integrable func-
tions of the m+k arguments and symmetric in each m -, k -tuple.
i j
To proceed further we have to consider a Gelfand triple around the space
(L2). We will use the space (S)∗ of Hida distributions (or generalized Brow-
nian functionals) and the corresponding Gelfand triple (S) (L2) (S)∗.
⊂ ⊂
Here (S) is the space of white noise test functions such that its dual space
(with respect to (L2)) is the space (S)∗. Instead of reproducing the ex-
plicit construction of (S)∗ (see e.g. [HKPS93]), in Theorem 2 below we
characterize this space through its S-transform. We recall that given a
f = (f ,f ) S (R), and the Wick exponential
1 2 2d
∈
1
: exp( ω~,f ) ::= : ω~⊗m : : ω~⊗k :,f⊗m f⊗k = C(f)ehω~,fi2d,
h i m!k!h 1 ⊗ 2 1 ⊗ 2 i
m k
XX
we define the S-transform of a Φ (S)∗ by
∈
SΦ(f) := Φ,: exp( ,f ) : , f S (R). (1)
2d
hh h· i ii ∀ ∈
Here , denotes the dual pairing between (S)∗ and (S) which is defined as
hh· ·ii
the bilinear extension of the sesquilinear inner product on (L2). We observe
that the multilinear expansion of (1),
SΦ(f) := hFm,k,f1⊗m ⊗f2⊗ki,
m k
XX
extends the chaos expansion to Φ (S)∗ with distribution valued kernels
∈
Fm,k such that
Φ,ϕ = m!k! Fm,k,ϕm,k , (2)
hh ii h i
m k
XX
for every generalized test function ϕ (S) with kernel functions ϕm,k.
∈
In order to characterize the space (S)∗ through its S-transform we need
the following definition.
6
Definition 1 A function F : S (R) C is called a U-functional whenever
2d
→
1. for every f ,f S (R) the mapping R λ F(λf +f ) has an entire
1 2 2d 1 2
∈ ∋ 7−→
extension to λ C,
∈
2. there are constants K ,K > 0 such that
1 2
F(zf) K eK2|z|2kfk2, z C,f S (R)
1 2d
| | ≤ ∀ ∈ ∈
for some continuous norm on S (R).
2d
k·k
We are now ready to state the aforementioned characterization result.
Theorem 2 ([KLPSW96], [PS91]) The S-transform defines a bijection be-
tween the space (S)∗ and the space of U-functionals.
As a consequence of Theorem 2 one may derive the next two statements.
The first one concerns the convergence of sequences of Hida distributions
and the second one the Bochner integration of families of distributions of
the same type (for more details and proofs see e.g. [HKPS93], [KLPSW96],
[PS91]).
Corollary 3 Let (Φ ) be a sequence in (S)∗ such that
n n∈N
(i) for all f S (R), ((SΦ )(f)) is a Cauchy sequence in C,
∈ 2d n n∈N
(ii) there are constants K ,K > 0 such that for some continuous norm
1 2
k·k
on S (R) one has
2d
(SΦ )(zf) K eK2|z|2kfk2, z C,f S (R),n N.
n 1 2d
| | ≤ ∀ ∈ ∈ ∈
Then (Φ ) converges strongly in (S)∗ to a unique Hida distribution.
n n∈N
Corollary 4 Let (Ω, ,m) be a measure space and λ Φ be a mapping
λ
B 7→
from Ω to (S)∗. We assume that the S-transform of Φ fulfills the following
λ
two properties:
(i) the mapping λ (SΦ )(f) is measurable for every f S (R),
λ 2d
7→ ∈
(ii) the SΦ obeys a U-estimate
λ
(SΦ )(zf) C (λ)eC2(λ)|z|2kfk2, z C,f S (R),
λ 1 2d
| | ≤ ∈ ∈
for some continuous norm on S (R) and for some C L1(Ω,m),
2d 1
k·k ∈
C L∞(Ω,m).
2
∈
7
Then
dm(λ)Φ (S)∗
λ
∈
ZΩ
and
S dm(λ)Φ (f) = dm(λ) (SΦ )(f).
λ λ
(cid:18)ZΩ (cid:19) ZΩ
3 Chaos expansions
Let us now consider two independent d-dimensional fractional Brownian mo-
tions B (t) and B (t) with Hurst multiparameters H = (H ,...,H )
H1 H2 1 1,1 1,d
and H = (H ,...,H ), respectively. That is, given a 2d-tuple of indepen-
2 2,1 2,d
dent white noises (ω ,...,ω ,ω ,...,ω ),
1,1 1,d 2,1 2,d
B (t) := ω ,M 11 ,..., ω ,M 11 , i = 1,2.
Hi h i,1 Hi,1 [0,t]i h i,d Hi,d [0,t]i
Proposition 5 Fo(cid:0)r each t and s strictly positive real n(cid:1)umbers the Bochner
integral
d
1
δ(BH1(t)−BH2(s)) := 2π dλeiλ(BH1(t)−BH2(s))
(cid:18) (cid:19) ZRd
is a Hida distribution with S-transform given by
Sδ(B (t) B (s))(f)
H1 − H2
d d
1 1
= (3)
√2π √t2H1,j +s2H2,j ·
(cid:18) (cid:19) j=1
Y
e−12Pdj=1 t2H1,j+1s2H2,j(RRdx(f1,j(x)(MH1,j11[0,t])(x)−f2,j(x)(MH2,j11[0,s])(x)))2,
·
for all f = (f ,...,f ,f ,...,f ) S (R).
1,1 1,d 2,1 2,d 2d
∈
Proof. The proof of this result follows from an application of Corollary 4 to
the S-transform of the integrand function
Φ(ω~1,ω~2) := eiλ(BH1(t)−BH2(s)), ω~i = (ωi,1,...,ωi,d),i = 1,2,
with respect to the Lebesgue measure on Rd. For this purpose we begin by
observing that since the fractional Brownian motions are independent one
has
SΦ(f) = SeiλBH1(t)(f1) Se−iλBH2(s)(f2)
·
8
for every f = (f ,f ) S (R), f := (f ,...,f ), f := (f ,...,f ). Hence,
1 2 2d 1 1,1 1,d 2 2,1 2,d
∈
according e.g. to [HKPS93], for all λ = (λ ,...,λ ) Rd we obtain
1 d
∈
d
SΦ(f) = eiλj Rdx(f1,j(x)(MH1,j11[0,t])(x)−f2,j(x)(MH2,j11[0,s])(x))e−12λ2j(t2H1,j+s2H2,j),
R
j=1
Y
(4)
which clearly fulfills the measurability condition. Moreover, for all z C we
∈
find
SΦ(zf)
| |
d
= e−41λ2j(t2H1,j+s2H2,j)
·
j=1
Y
d
e−14λ2j(t2H1,j+s2H2,j)+izλj Rdx(f1,j(x)(MH1,j11[0,t])(x)−f2,j(x)(MH2,j11[0,s])(x))
· R
Yj=1(cid:12) (cid:12)
(cid:12) (cid:12)
d (cid:12) (cid:12)
e−41λ2j(t2H1,j+s2H2,j)
≤ ·
j=1
Y
d
e−41λ2j(t2H1,j+s2H2,j)+|z||λj|| Rdx(f1,j(x)(MH1,j11[0,t])(x)−f2,j(x)(MH2,j11[0,s])(x))|,
· R
j=1
Y
where, for each j = 1,...,d, the corresponding term in the second product is
bounded by
z 2 2
exp | | dx f (x)(M 11 )(x) f (x)(M 11 )(x) ,
t2H1,j +s2H2,j R 1,j H1,j [0,t] − 2,j H2,j [0,s] !
(cid:18)Z (cid:19)
(cid:0) (cid:1)
9
because
1
λ2(t2H1,j +s2H2,j)
−4 j
+ z λ dx f (x)(M 11 )(x) f (x)(M 11 )(x)
| || j| 1,j H1,j [0,t] − 2,j H2,j [0,s]
R
(cid:12)Z (cid:12)
(cid:12) z (cid:0) (cid:1)(cid:12)
(cid:12) (cid:12)
= | | dx f (x)(M 11 )(x) f (x)(M 11 )(x)
− √t2H(cid:12)1,j +s2H2,j R 1,j H1,j [0,t] − 2,j H(cid:12)2,j [0,s]
(cid:18) (cid:12)Z (cid:12)
(cid:12) 2 (cid:0) (cid:1)(cid:12)
|λj| t2H1,j +s2H2(cid:12)(cid:12),j (cid:12)(cid:12)
− 2
(cid:19)
pz 2 2
+ | | dx f (x)(M 11 )(x) f (x)(M 11 )(x) .
t2H1,j +s2H2,j R 1,j H1,j [0,t] − 2,j H2,j [0,s]
(cid:18)Z (cid:19)
(cid:0) (cid:1)
As a result,
SΦ(zf) e−41 dj=1λ2j(t2H1,j+s2H2,j)
| | ≤ P ·
e|z|2Pdj=1 t2H1,j+1s2H2,j(RRdx(f1,j(x)(MH1,j11[0,t])(x)−f2,j(x)(MH2,j11[0,s])(x)))2,
·
where, as a function of λ, the first exponential is integrable on Rd and the
second exponential is constant.
An application of the result mentioned above completes the proof. In
particular, it yields (3) by integrating (4) over λ. (cid:4)
In order to proceed further the next result shows to be very useful. It
improves the estimate obtained in [Ben03, Theorem 2.3] towards the charac-
terization results stated in Corollaries 3 and 4.
Lemma 6 ([DOS08]) Let H (0,1) and f S (R) be given. There is a
1
∈ ∈
non-negative constant C independent of f such that
H
dxf(x)(M 11 )(x) C t sup f(x) +sup f′(x) + f
H [0,t] H
(cid:12)ZR (cid:12) ≤ (cid:18)x∈R| | x∈R| | | |(cid:19)
(cid:12) (cid:12)
for all(cid:12)t > 0. (cid:12)
(cid:12) (cid:12)
In particular, the use of Lemma 6 allows to state the next result on
intersection local times L as well as on their subtracted counterparts
H1,H2
(N)
L . There, and throughout the rest of this work as well, given a H =
H1,H2
(H ,...,H ) (0,1)d we shall use the notation
1 d
∈
¯
H := max H .
j
j=1,...,d
10