Existence of generalized local times for Gaussian random fields
We consider a Gaussian centered random field that has values in the Euclidean space. We investigate the existence of local time for the random field as a generalized functional, an element of the Sobolev space constructed for our random field. We give the sufficient condition for such an existence in...
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| Цитувати: | Existence of generalized local times for Gaussian random fields / A. Rudenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 142–153. — Бібліогр.: 6 назв.— англ. |
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irk-123456789-44492009-11-11T12:00:35Z Existence of generalized local times for Gaussian random fields Rudenko, A. We consider a Gaussian centered random field that has values in the Euclidean space. We investigate the existence of local time for the random field as a generalized functional, an element of the Sobolev space constructed for our random field. We give the sufficient condition for such an existence in terms of the field covariation and apply it in a few examples: the Brownian motion with additional weight and the intersection local time of two Brownian motions. 2006 Article Existence of generalized local times for Gaussian random fields / A. Rudenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 142–153. — Бібліогр.: 6 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4449 519.21 en Інститут математики НАН України |
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English |
| description |
We consider a Gaussian centered random field that has values in the Euclidean space.
We investigate the existence of local time for the random field as a generalized functional, an element of the Sobolev space constructed for our random field. We give the
sufficient condition for such an existence in terms of the field covariation and apply it
in a few examples: the Brownian motion with additional weight and the intersection
local time of two Brownian motions. |
| format |
Article |
| author |
Rudenko, A. |
| spellingShingle |
Rudenko, A. Existence of generalized local times for Gaussian random fields |
| author_facet |
Rudenko, A. |
| author_sort |
Rudenko, A. |
| title |
Existence of generalized local times for Gaussian random fields |
| title_short |
Existence of generalized local times for Gaussian random fields |
| title_full |
Existence of generalized local times for Gaussian random fields |
| title_fullStr |
Existence of generalized local times for Gaussian random fields |
| title_full_unstemmed |
Existence of generalized local times for Gaussian random fields |
| title_sort |
existence of generalized local times for gaussian random fields |
| publisher |
Інститут математики НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/4449 |
| citation_txt |
Existence of generalized local times for Gaussian random fields / A. Rudenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 142–153. — Бібліогр.: 6 назв.— англ. |
| work_keys_str_mv |
AT rudenkoa existenceofgeneralizedlocaltimesforgaussianrandomfields |
| first_indexed |
2025-07-02T07:41:30Z |
| last_indexed |
2025-07-02T07:41:30Z |
| _version_ |
1836520142099447808 |
| fulltext |
Theory of Stochastic Processes
Vol. 12 (28), no. 1–2, 2006, pp. 142–153
UDC 519.21
ALEXEY RUDENKO
EXISTENCE OF GENERALIZED LOCAL
TIMES FOR GAUSSIAN RANDOM FIELDS
We consider a Gaussian centered random field that has values in the Euclidean space.
We investigate the existence of local time for the random field as a generalized func-
tional, an element of the Sobolev space constructed for our random field. We give the
sufficient condition for such an existence in terms of the field covariation and apply it
in a few examples: the Brownian motion with additional weight and the intersection
local time of two Brownian motions.
Introduction
This article was motivated by the following problem. Let W1, W2 be two Brownian
motions in the d-dimensional Euclidean space. Suppose that they are jointly Gaussian,
and their covariation is as follows: EW1(s)W2(t)T = min(s, t)Q (Q is a d × d matrix).
Consider the intersection local time on the time interval [0, 1] of these two processes
as a limit of the approximations Lε =
∫ 1
0
∫ 1
0
fε(W1(s) − W2(t))dsdt, where fε is an
approximation of the measure with unit mass at zero: fε(x) = 1
εd f(x
ε ), f ∈ C∞
b (Rd), f �
0,
∫
Rd f(x)dx = 1 (C∞
b (Rd) is the space of bounded infinitely differentiable functions
on R
d ). Find the conditions on d and Q such that these approximations converge in
the Sobolev spaces D2,α constructed over an abstract Wiener space associated with the
processes (for the exact definition, see below).
To answer this question, we derive the sufficient condition in a more general statement.
The main problem here is how to deal with approximations in order to find the index
of the Sobolev space, where we have the convergence. In work [1], this was done for the
self-intersection local time of the Brownian motion by finding the explicit form for the
Itô—Wiener expansion kernels of approximations. But it is enough, in fact, to find the
scalar product of those kernels in the space of square integrable functions. Moreover, it
is enough to find the asymptotic behavior of this product, as the kernel index tends to
infinity. The latter can be done using a certain integral representation of kernel scalar
products. We are able to generalize this approach and implement it for an arbitrary
Gaussian random field. In our generalization, we use the fact that the intersection local
time of two Brownian motions may be thought as the local time of a Gaussian random
field on the two-dimensional parameter space.
Let (T, B) be a measurable space with finite measure ν on it. Consider a centered
Gaussian random field ξ on (T, B), where each ξ(t) is a centered Gaussian random
vector in R
d. Denote Kij(s, t) = Eξi(s)ξj(t). We suppose that Kij are measurable
functions, and detK(t, t) > 0 ν-a.s. Additionally, we suppose that ξ is separable. In
other words, it is determined by its values in a countable set of points. Then there
exists an abstract Wiener space (B, H, μ) (see [6]) such that ξ can be constructed on the
probability space (B, F, μ) (F is the σ-algebra of Borel sets in B) as a linear functional on
2000 AMS Mathematics Subject Classification. Primary 60H07.
Key words and phrases. Local time, generalized functional, Gaussian random field.
142
EXISTENCE OF GENERALIZED LOCAL TIMES FOR GAUSSIAN RANDOM FIELDS 143
it: ξk(t)(ω) = lkt (ω), ω ∈ B, lkt ∈ H, k = 1, . . . , d, t ∈ T . Here, B is a separable Banach
space, and μ is the centered Gaussian probability measure on it. Using the structure of
an abstract Wiener space, we can introduce Sobolev spaces as in [5]. The space of square
integrable functions can be represented as a direct sum of eigenspaces of the Ornstein—
Uhlenbeck operator L: L2(B, μ) = ⊕∞
n=0 Mn, where Lx = −nx, x ∈ Mn. Consequently,
we can represent each square integrable function f ∈ L2(B, μ) as a sum of eigenvectors
f =
∑∞
n=0 fn, fn ∈ Mn. Such a decomposition is called the Itô—Wiener decomposition.
Now we have L2(Ω) = L2(B, μ), so every square integrable ξ-measurable random variable
has this decomposition. Consider P (B, μ) ⊂ L2(B, μ), being a subspace of the space of
square integrable functions which consists of the functions that have only finite non-zero
members in the associated Itô—Wiener decomposition. Introduce the norm indexed by
α on the subspace P (B, μ):
‖f‖2
2,α =
∥∥(I − L)
α
2 f
∥∥2
2
=
N∑
n=0
(1 + n)α ‖fn‖2,
where N is the maximum of indices corresponding to non-zero decomposition members,
and ‖�‖2 is norm in L2(B, μ). We denote the completion of P (B, μ) equipped with this
norm as D2,α and call it the Sobolev space with index 2, α.
We want to have the convergence of approximations for the local time in these spaces.
We consider the local time at x = 0 of ξ with regard to some finite measure ν on (T, B)
and approximations having form Lε =
∫
T
fε(ξ(t))ν(dt), where fε is the same as above.
We have to note that this approach to the local time generalization is not unique.
There exists an approach which makes use of the white noise theory (see, e.g., [2]). An-
other approach introduces generalized homogeneous functionals for the Brownian motion
[3].
In the first part, we give the sufficient condition for the existence of a square inte-
grable local time (as a limit of approximations). In the second one, we get the integral
representation for the scalar products of Itô—Wiener expansion members. In the third
part, we study the asymptotic behavior of this representation and use it to prove the
convergence of approximations in the appropriate Sobolev space. In examples, we show
how our conditions for the local time existence can be used in some cases including the
problems on the intersection and the self-intersection local time for the Brownian motion.
Existence of square integrable local time
At first, we give the sufficient condition for the existence in L2(Ω). The same condition
for the processes mentioned in work [4]. By B(s, t), we denote a symmetric matrix 2d×2d
such that
Bij(s, t) = Kij(s, s), i = 1..d, j = 1..d
Bij(s, t) = Ki−d,j(s, t), i = d + 1..2d, j = 1..d
Bij(s, t) = Ki−d,j−d(t, t), i = d + 1..2d, j = d + 1..2d.
In other words, B(s, t) is the covariance matrix of a vector (ξ(s), ξ(t)).
Theorem 1. Suppose that∫
T
∫
T
1√
detB(s, t)
ν(ds)ν(dt) < +∞,
then the limit of Lε exists in L2(Ω).
Proof. It is sufficient to prove that ELε1Lε2 → C < +∞, ε1, ε2 → 0+. Note that the
condition of Theorem 1 gives: detB(s, t) > 0 a.e. with regard to the measure ν × ν.
144 ALEXEY RUDENKO
Using the Fubini theorem and performing the change of variables ε1u = x, ε2w = y, we
get
ELε1Lε2
=
∫
T
∫
T
∫
Rd
∫
Rd
fε1(x)fε2(y)(2π)−d(det B(s, t))−
1
2
× exp
(
−1
2
(
x
y
)T
B−1(s, t)
(
x
y
))
dxdyν(ds)ν(dt)
=
∫
T
∫
T
∫
Rd
∫
Rd
f(u)f(w)(2π)−d(detB(s, t))−
1
2
× exp(−1
2
(
ε1u
ε2w
)T
B−1(s, t)
(
ε1u
ε2w
)
)dudwν(ds)ν(dt).
The function inside all integrals is monotonously converging to
f(u)f(w)(2π)−d(detB(s, t))−
1
2
when ε1, ε2 → 0+. So, by applying the theorem on monotonous convergence, we conclude
that ELε1Lε2 → ∫
T
∫
T (2π)−d(detB(s, t))−
1
2 ν(ds)ν(dt), and Theorem 1 is proved.
Covariation formula
As we stated in Introduction, we can write an Itô—Wiener expansion for the following
expression, because it is a bounded and therefore square integrable random variable:
∫
T
f(ξ(t))ν(dt) =
∞∑
n=0
an(f), f ∈ C∞
b (Rd), an(f) ∈ Mn
Note that this is the expression appeared in our local time approximations. We want to
obtain a representation for Ean(f)an(g), f, g ∈ C∞
b (Rd).
Theorem 2. For any functions f, g ∈ C∞
b (Rd),
Ean(f)an(g) =
=
∫
T
∫
T
∫
Rd
∫
Rd
∫
Rd
∫
Rd
f(x)g(y)ei(x,γ1)+i(y,γ2)qn(s, t, γ1, γ2)dxdydγ1dγ2ν(ds)ν(dt),
where
qn(s, t, x, y) = (2π)−2d (−1)n
n!
(K(s, t)x, y)ne−
1
2 (K(s,s)x,x)e−
1
2 (K(t,t)y,y)
Proof. Consider the Ornstein—Uhlenbeck semigroup {Tλ, λ ≥ 0} on our abstract Wiener
space. By definition (see [5]), it acts on any function h ∈ L2(B, μ) in the following way:
Tλh(x) =
∫
B h(e−λx+
√
1 − e−2λy)μ(dy). It also has property that its action on the same
function with the Itô—Wiener decomposion h =
∑∞
n=0 hn, hn ∈ Mn may be written as
Tλh =
∑∞
n=0 e−nλhn. Using this property, we get
E(
∫
T
f(ξ(t))ν(dt) · Tλ
∫
T
g(ξ(t))ν(dt)) =
∞∑
n=0
e−nλEan(f)an(g).
EXISTENCE OF GENERALIZED LOCAL TIMES FOR GAUSSIAN RANDOM FIELDS 145
Here, we used the fact that Ean(f)am(g) = 0, m
= n. On the other hand, by the
definition of the Ornstein—Uhlenbeck semigroup and because ξ(t)(e−λx+
√
1 − e−2λy) =
e−λξ(t)(x) +
√
1 − e−2λξ(t)(y), x, y ∈ B, we have
E(
∫
T
f(ξ(t))ν(dt) · Tλ
∫
T
g(ξ(t))ν(dt)) =
= E(
∫
T
f(ξ(t))ν(dt) · E(
∫
T
g(e−λξ(t) +
√
1 − e−2λξ̃(t))ν(dt)/ξ)) =
=
∫
T
∫
T
E(f(ξ(s))g(e−λξ(t) +
√
1 − e−2λξ̃(t)))ν(ds)ν(dt),
where ξ̃ is an independent copy of ξ. We can swap integrals and expectations, because the
expression inside all integrals is bounded. Denote Bβ(s, t) =
(
K(s, s) βK(t, s)
βK(s, t) K(t, t)
)
. The
distribution of (ξ(s), e−λξ(t) +
√
1 − e−2λξ̃(t)) is centered Gaussian with the covariation
matrix Be−λ(s, t). By Lemma 1 and our assumptions on the covariation: if λ > 0, then
detBe−λ(s, t) > 0 for all (s, t) ∈ T 2 except a set of the zero measure ν × ν. This yields
E(
∫
T
f(ξ(t))ν(dt) · Tλ
∫
T
g(ξ(t))ν(dt)) =
=
∫
T
∫
T
∫
Rd
∫
Rd
f(x)g(y)
1
(2π)d
√
detBe−λ(s, t)
·
· exp(−1
2
(
x
y
)T
B−1
e−λ(s, t)
(
x
y
)
)dxdyν(ds)ν(dt) =
=
∫
T
∫
T
∫
Rd
∫
Rd
∫
Rd
∫
Rd
f(x)g(y)ei(x,γ1)+i(y,γ2)qλ(s, t, γ1, γ2)dγ1dγ2dxdyν(ds)ν(dt),
where
qλ(s, t, x, y) = (2π)−2d exp (−1
2
(
x
y
)T
Be−λ(s, t)
(
x
y
)
) =
= (2π)−2d exp (−1
2
((K(s, s)x, x) + (K(t, t)y, y) + 2e−λ(K(s, t)x, y)))
Obviously:
qλ(s, t, x, y) =
∞∑
n=0
e−nλqn(s, t, x, y), s, t ∈ T, x, y ∈ R
d.
We substitute q in the integral with this sum. By the Lebesgue dominated convergence
theorem, we can exchange this sum with the integral by λ1, λ2. If we can show that it
can be exchanged with other integrals for all λ > 0, we get two different representations
for the starting expression in the form of a power series of e−λ. Then we match members
of these series and get the desired integral representation. Fix (s, t) ∈ T 2 such that
detK(s, s) > 0, detK(t, t) > 0, and λ > 0. Denote:
p(s, t, x, y) =
1
(2π)d
√
detB0(s, t)
exp(−1
2
(
x
y
)T
B−1
0 (s, t)
(
x
y
)
=
1
(2π)d
√
detK(s, s) detK(t, t)
exp(−1
2
((K−1(s, s)x, x) + (K−1(t, t)y, y))),
rn(s, t, x, y) =
1
p(s, t, x, y)
∫
Rd
∫
Rd
ei(x,γ1)+i(y,γ2)qn(s, t, γ1, γ2)dγ1dγ2,
146 ALEXEY RUDENKO
Rn(s, t, x, y) =
n∑
k=0
e−kλrk(s, t, ·, ·),
R(s, t, x, y) =
∞∑
k=0
e−kλrk(s, t, ·, ·)
=
1
p(s, t, x, y)
∫
Rd
∫
Rd
ei(x,γ1)+i(y,γ2)qλ(s, t, γ1, γ2)dγ1dγ2
=
√
detB0(s, t)√
detBe−λ(s, t)
exp
(
−1
2
(
x
y
)T
(B−1
e−λ − B−1
0 )(s, t)
(
x
y
))
.
We want to show that the integrals can be exchanged with the sum. Now it can be
formulated as
lim
n→∞
∫
T
∫
T
∫
Rd
∫
Rd
f(x)g(y)Rn(s, t, x, y)p(s, t, x, y)dxdyν(ds)ν(dt) =
=
∫
T
∫
T
∫
Rd
∫
Rd
f(x)g(y)R(s, t, x, y)p(s, t, x, y)dxdyν(ds)ν(dt).
Consider the space of functions on R
2d which are square integrable by a Gaussian measure
with density p(s, t, x, y) and denote it as L2(R2d, p(s, t, x, y)dxdy) with the corresponding
scalar product < ·, · >. It is easy to check that < rn(s, t, ·, ·), rm(s, t, ·, ·) >= 0 if m
= n
(because rn is a linear combination of multidimensional Hermitian polynomials of degree
2n). Therefore,
sup
n
< Rn(s, t, ·, ·), Rn(s, t, ·, ·) >=
= sup
n
n∑
k=0
e−2kλ < rk(s, t, ·, ·), rk(s, t, ·, ·) >≤
≤< R(s, t, ·, ·), R(s, t, ·, ·) > .
Using Lemmas 2 and 3 (see below), we obtain
< R(s, t, ·, ·), R(s, t, ·, ·) >=
=
∫
Rd
∫
Rd
1
(2π)d
√
det B0(s, t)(s, t)
detBe−λ(s, t)
exp(−1
2
(
x
y
)T
(2B−1
e−λ − B−1
0 )(s, t)
(
x
y
)
)dxdy =
=
√
detB−1
0 Be−λB−1
−e−λ(s, t)
√
det B0(s, t)(s, t)
detBe−λ(s, t)
=
detB0(s, t)√
detBe−λ(s, t) detB−e−λ(s, t)
≤
≤ (1 − e−λ)−2d.
Applying the inequalities given above, we get∫
Rd
∫
Rd
|f(x)g(y)Rn(s, t, x, y)|p(s, t, x, y)dxdy ≤
≤
√
< f(x)g(y), f(x)g(y) >< Rn(s, t, ·, ·), Rn(s, t, ·, ·) > ≤
≤ sup
x,y∈Rd
|f(x)g(y)|(1 − e−λ)−d.
We can see that the bound is independent of (s, t). So, by the Lebesgue dominated
convergence theorem, we proved the equality and Theorem 2.
Now we prove a few technical results. Let m, n be positive integers, and let A
be an (m + n) × (m + n) symmetric non-negative definite real matrix of the form
EXISTENCE OF GENERALIZED LOCAL TIMES FOR GAUSSIAN RANDOM FIELDS 147
A =
(
A11 A12
AT
12 A22
)
, where A11, A12, A22 are matrices with sizes m × m, m × n, n × n,
respectively, and A11, A22 are positive definite. Define Aα =
(
A11 αA12
αAT
12 A22
)
, α ∈ R.
Lemma 1. If |α| < 1, then Aα is positive definite.
Proof. Take the n + m-vector
(
u
v
)
as a concatenation of the m-vector u and the n-
vector v. Then we have to prove that (Aα
(
u
v
)
,
(
u
v
)
) = (
(
A11u + αA12v
αAT
12u + A22v
)
,
(
u
v
)
) =
(A11u, u) + 2α(A12v, u) + (A22v, v) is always positive. Let u1 = αu, v1 = v, and let us
use that A = A1 is non-negative definite. We have
0 � (A1
(
u1
v1
)
,
(
u1
v1
)
) = α2(A11u, u) + 2α(A12v, u) + (A22v, v) =
= (α2 − 1)(A11u, u) + (Aα
(
u
v
)
,
(
u
v
)
) � (Aα
(
u
v
)
,
(
u
v
)
).
So we have that Aα is non-negative definite. Moreover, if
(Aα
(
u
v
)
,
(
u
v
)
) = 0,
then the equality holds in the inequality above, and we get (A11u, u) = 0. But A11 is
positive definite, and so u = 0. Similarly using that A22 is positive definite, we get v = 0.
Hence, Aα is positive definite.
Lemma 2. If |α| < 1, then 2A−1
α − A−1
0 = A−1
0 A−αA−1
α , and the matrix on the right-
hand side is positive definite.
Proof. Note that A−1
α and A−1
0 exist by the previous lemma. Consider the obvious
relation 2A0−Aα = A−α. Multiplying it by A−1
α from the right and by A−1
0 from the left,
we get the first part of the assertion. From Lemma 1, we also have that A−1
0 , A−1
α , andA−α
are positive definite. Then A−1
0 A−αA−1
α is positive definite as a product of positive
definite symmetric matrices.
Lemma 3. If |α| < 1, then there exists a constant C(α) > 0 such that, for any matrix
A, det A0 � C(α) det Aα with C(α) = (1 − |α|)−(m+n).
Proof. We will prove that if B and C are symmetric l× l non-negative definite matrices,
then det(B + C) � detB. There exists the basis in R
l such that C is a diagonal matrix
with diagonal elements {ci � 0, i = 1, . . . , l}. Convert our matrices to this basis. By
Ci, we denote the matrix with only one non-zero element ci at the place (i, i) such that
C =
∑l
i=1 Ci. By Bi, we denote the matrix obtained from B by removing the row i
and the column i. Then we have detBi � 0, because B is non-negative definite, and
det(B+C1) = det B+c1 detB1 � detB. The matrix B+C1 is again non-negative definite
and, by the same argument, det(B + C1 + C2) � det(B + C1) � detB. Proceeding by
induction, we get det(B + C) = det(B +
∑l
i=1 Ci) � det B.
Suppose α � 0. From the statement above, we obtain
detAα = det((1 − α)A0 + αA) � det((1 − α)A0) = (1 − α)m+n det A0.
The case α < 0 is similar:
detAα = det((1 + α)A0 − αA−1) � det((1 + α)A0) = (1 + α)m+n detA0
(A−1 is non-negative definite by the same arguments as in Lemma 1).
148 ALEXEY RUDENKO
Main result
We want to know the asymptotic behavior of Ean(fε1)an(fε2) as n → ∞, where fε
approximates the delta-measure as defined in Introduction. Since
|Ean(fε1)an(fε2)|
=
∣∣∣∣
∫
T
∫
T
∫
Rd
∫
Rd
∫
Rd
∫
Rd
fε1(x)fε2(y)ei(x,γ1)+i(y,γ2)qn(s, t, γ1, γ2)dxdydγ1dγ2ν(ds)ν(dt)
∣∣∣∣
≤
∫
T
∫
T
∫
Rd
∫
Rd
∫
Rd
∫
Rd
fε1(x)fε2 (y)|qn(s, t, γ1, γ2)|dxdydγ1dγ2ν(ds)ν(dt)
=
∫
T
∫
T
∫
Rd
∫
Rd
|qn(s, t, γ1, γ2)|dγ1dγ2ν(ds)ν(dt),
we are interested in the asymptotic behavior of
∫
Rd
∫
Rd |qn(s, t, x, y)|dxdy. Fix (s, t) ∈ T 2
such that detK(s, s) > 0 and detK(t, t) > 0. Denote
G(s, t) = K−1/2(t, t)K(s, t)K−1/2(s, s)
‖G(s, t)‖ = sup
x∈Rd,|x|=1
‖G(s, t)x‖
In other words, we have the correlation matrix and its operator norm.
Theorem 3. There exists a constant C > 0 such that, for any (s, t) ∈ T 2 satisfying
detK(s, s) > 0 and detK(t, t) > 0, the following inequality holds for all integers n ≥ 0:∫
Rd
∫
Rd
|qn(s, t, x, y)|dxdy ≤ C (detK(s, s) detK(t, t))−1/2 ‖G(s, t)‖n (n + 1)d/2−1.
Proof. We have∫
Rd
∫
Rd
|qn(s, t, x, y)|dxdy =
=
∫
Rd
∫
Rd
1
(2π)2d
|(K(s, t)x, y)|n
n!
e−
1
2 (K(s,s)x,x)e−
1
2 (K(t,t)y,y)dxdy =
=
1
(2π)2d
(detK(s, s) detK(t, t))−1/2
∫
Rd
∫
Rd
|(G(s, t)x, y)|n
n!
e−
‖x‖2
2 e−
‖y‖2
2 dxdy.
Note that if n = 2k + 1, k ≥ 0, then, by the Cauchy inequality,∫
Rd
∫
Rd
|(G(s, t)x, y)|2k+1
(2k + 1)!
1
(2π)d
e−
‖x‖2
2 e−
‖y‖2
2 dxdy ≤
≤
√
2k + 2
2k + 1
√∫
Rd
∫
Rd
|(G(s, t)x, y)|2k
(2k)!
1
(2π)d
e−
‖x‖2
2 e−
‖y‖2
2 dxdy×
×
√∫
Rd
∫
Rd
|(G(s, t)x, y)|2k+2
(2k + 2)!
1
(2π)d
e−
‖x‖2
2 e−
‖y‖2
2 dxdy.
So, it is enough to consider the case n = 2k. Using Lemma 4 (see below), we can
establish a change of variables in the integral associated with the rotation around the
origin in R
2d such that (G(s, t)x, y) =
∑d
n=0 λnunvn, where u, v are the new variables,
and λ2
i , i = 1, . . . , n, are the eigenvalues of G(s, t)GT (s, t). Note that λi can be chosen
non-negative, and maxi=1,... ,n λi =
√‖G(s, t)GT (s, t)‖ = ‖G(s, t)‖. We note that
(
d∑
i=0
λiuivi)2n = (2n)!
∑
d
i=0 ki=2n
d∏
i=0
(λiuivi)ki
ki!
,
EXISTENCE OF GENERALIZED LOCAL TIMES FOR GAUSSIAN RANDOM FIELDS 149
and all members of the sum with one of ki being odd became zero after the integrating
with a Gaussian measure. All other members are non-negative, so we can increase the
sum by replacing all λi by ‖G(s, t)‖. We get∫
Rd
∫
Rd
(G(s, t)x, y)2n
(2n)!
e−
‖x‖2
2 e−
‖y‖2
2 dxdy =
∫
Rd
∫
Rd
(
∑d
i=0 λiuivi)2n
(2n)!
e−
‖u‖2
2 e−
‖v‖2
2 dudv ≤
≤ ‖G(s, t)‖2n
∫
Rd
∫
Rd
(
∑d
i=0 uivi)2n
(2n)!
e−
‖u‖2
2 e−
‖v‖2
2 dudv.
The last integral can be calculated precisely and satisfies the inequality∫
Rd
∫
Rd
(
∑d
i=0 uivi)2n
(2n)!
e−
‖u‖2
2 e−
‖v‖2
2 dudv ≤ C̃(2n + 1)d/2−1.
Hence we proved Theorem 3.
The following lemma was used in Theorem 3.
Lemma 4. For any d×d matrix A, there exists the orthogonal 2d×2d matrix U (UT U =
I) such that (Ax, y) =
∑d
n=0 λnunvn, where x, y, u, v ∈ R
d,
(
u
v
)
= U
(
x
y
)
and λ2
i , i =
1, . . . , n, are the eigenvalues of AAT .
Proof. Denote z =
(
x
y
)
∈ R
2d, x, y ∈ R
d, and Q =
(
0 AT
A 0
)
. Note that (Qz, z) =
2(Ax, y), and Q is symmetric. All eigenvalues of Q are the set {λi,−λi, i = 1, . . . , n},
where λi are from the statement of Lemma 4. This is true, because if λ2 > 0, then
the equations Qz = λz and AAT y = λ2y, λx = AT y are equivalent. So we can find
the orthogonal 2d × 2d matrix U1 such that 2(Ax, y) =
∑d
n=0 λn(ũ2
n − ṽ2
n) and
(
ũ
ṽ
)
=
U1
(
x
y
)
. This easily yields the statement of Lemma 4.
We are ready to establish the convergence of approximations. Denote
Jn =
∫
T
∫
T
‖G(s, t)‖n (detK(s, s) det K(t, t))−1/2
ν(ds)ν(dt)
Theorem 4. 1) If Jn < +∞, then limε→0+ an(fε) exists in L2(Ω). For odd n, the limit
is equal to zero.
2) If
∑∞
n=0 Jn(n + 1)α+ d
2−1 < +∞, then limε→0+ Lε exists in D2,α.
Proof. From Theorem 2, we know that
Ean(fε1)an(fε2)
=
∫
T
∫
T
∫
Rd
∫
Rd
∫
Rd
∫
Rd
fε1(x)fε2(y)ei(x,γ1)+i(y,γ2)qn(s, t, γ1, γ2)dxdydγ1dγ2ν(ds)ν(dt)
=
∫
T
∫
T
∫
Rd
∫
Rd
∫
Rd
∫
Rd
f(x)f(y)ei(ε1x,γ1)+i(ε2y,γ2)qn(s, t, γ1, γ2)dxdydγ1dγ2ν(ds)ν(dt).
From Theorem 3, we have
|Ean(fε1)an(fε2)| ≤
≤
∫
T
∫
T
∫
Rd
∫
Rd
∫
Rd
∫
Rd
f(x)f(y)|qn(s, t, γ1, γ2)|dxdydγ1dγ2ν(ds)ν(dt) ≤
≤ CJn(n + 1)d/2−1
.
150 ALEXEY RUDENKO
By the Lebesgue dominated convergence theorem
Ean(fε1)an(fε2) →
∫
T
∫
T
∫
Rd
∫
Rd
qn(s, t, x, x)dxdyν(ds)ν(dt), ε1 , ε2 → 0 + .
We conclude that the limit of an(fε) as ε → 0+ exists in L2. For odd n, the limit
is obviously zero so we proved the first part of Theorem 4. The second part easily
follows from the first part because each an(fε) satisfies (as noted above) ‖an(fε)‖2
2 ≤
CJn(n + 1)d/2−1.
Examples
All conditions we imposed on ξ in Introduction are obviously satisfied when T ⊂ R
d
and ξ is a continuous centered Gaussian random field. So, without a further notice, we
will use our results for such a kind of random fields.
Example 1. Let T = [0, 1], v(dt) = dt, ξ(t) = trW (t), where W (t) is the d-dimensional
Brownian motion. In this case, we have
Kii(s, t) = (st)r min(s, t), Kij(s, t) = 0,
detK(s, s) = s(2r+1)d,
‖G(s, t)‖ =
min(s, t)√
st
,
Jn =
∫ 1
0
∫ 1
0
(st)−
n
2 − (2r+1)d
2 (min(s, t))ndsdt.
It easy to see that the condition Jn < +∞ is satisfied for r < 1
d − 1
2 for all n � 0 (and
failed for all n in either case), and
Jn =
4
(n + 2 − (2r + 1)d)(2 − (2r + 1)d)
= O(
1
n
).
Using Theorem 4, we conclude that the limit of Lε exists in D2,α for α < 1 − d
2 .
Example 2. Let T = [0, 1], v(dt) = dt, ξ(t) =
∫ t
0 srdW (s). The integral exists if r > − 1
2 .
As in the previous example, we can easily compute Jn:
Kii(s, t) =
1
2r + 1
(min(s, t))2r+1, Kij(s, t) = 0,
detK(s, s) = (2r + 1)−ds(2r+1)d,
‖G(s, t)‖ =
(
min(s, t)√
st
)2r+1
,
Jn = (2r + 1)−d
∫ 1
0
∫ 1
0
(st)−
n(2r+1)
2 − d(2r+1)
2 (min(s, t))n(2r+1)dsdt =
= (2r + 1)−d 4
(n(2r + 1) + 2 − d(2r + 1))(2 − d(2r + 1))
= O(
1
n
).
As in the previous example, Jn < +∞ is satisfied for r < 1
d − 1
2 , and Lε converges in D2,α
for α < 1− d
2 . This example and the previous one are the generalizations of the classical
result about the existence of a local time for the Brownian motion. The additional
multiplier tr, r ∈ (− 1
2 ,− 1
2 + 1
d ) prevents the explosion of Itô—Wiener expansion kernels.
It is interesting to note that the explosion occurs near t = 0. So if we take T = [12 , 1] for
this example and the previous one, then we may drop the restriction on r (and we may
EXISTENCE OF GENERALIZED LOCAL TIMES FOR GAUSSIAN RANDOM FIELDS 151
set r = 0 to get the usual Brownian motion). The convergence still takes place under
same conditions on α.
Now we consider two special cases of the problem stated in Introduction for Q = 0
and Q = I. The second case can also be found in [1].
Example 3. Let T = [0, 1]2, v(dt) = dt1dt2, ξ(t1, t2) = W1(t1)−W2(t2), where Wi(t), i =
1, 2, are two independent d-dimensional Brownian motions. We have
Kii(s, t) = min(s1, t1) + min(s2, t2), Kij(s, t) = 0,
detK(s, s) = (s1 + s2)d,
‖G(s, t)‖ =
min(s1, t1) + min(s2, t2)√
(s1 + s2)(t1 + t2)
,
Jn =
∫ 1
0
∫ 1
0
∫ 1
0
∫ 1
0
(min(s1, t1) + min(s2, t2))n
((s1 + s2)(t1 + t2))
n
2 + d
2
ds1ds2dt1dt2.
To find the asymptotic behaviour for Jn, we divide the region of integration T 2 = [0, 1]4
into subregions T1 = {0 � s1 � t1 � 1; 0 � s2 � t2 � 1}, T2 = {0 � s1 � t1 � 1; 0 �
t2 � s2 � 1}, T3 = {0 � t1 � s1 � 1; 0 � s2 � t2 � 1}, T4 = {0 � t1 � s1 � 1; 0 � t2 �
s2 � 1}.
J1
n =
∫
T1
(min(s1, t1) + min(s2, t2))n
((s1 + s2)(t1 + t2))
n
2 + d
2
ds1ds2dt1dt2
=
∫ 1
0
∫ 1
0
∫ t2
0
∫ t1
0
(s1 + s2)
n−d
2 (t1 + t2)
−n−d
2 ds1ds2dt1dt2
=
1
(1 + n−d
2 )(2 + n−d
2 )
∫ 1
0
∫ 1
0
(t1 + t2)
−n−d
2 ((t1 + t2)
n−d
2 +2 − t
n−d
2 +2
1 − t
n−d
2 +2
1 )dt1dt2
� C̃
1
(n + 1)2
∫ 1
0
∫ 1
0
(t1 + t2)2−ddt1dt2 � ˜̃C
1
(n + 1)2
∫ 1
0
u3−ddu.
We can see that the sufficient condition for the integral to exist is d < 4 (the special
cases d = 2, n = 0; d = 3, n = 1 behave themselves in a slightly different way: logarithms
appear after the partial integration, but the corresponding integrals are still finite by
same arguments).
J2
n =
∫
T2
(min(s1, t1) + min(s2, t2))n
((s1 + s2)(t1 + t2))
n
2 + d
2
ds1ds2dt1dt2 =
=
∫ 1
0
∫ 1
0
∫ 1
s1
∫ 1
t2
(s1 + t2)n(s1 + s2)
−n−d
2 (t1 + t2)
−n−d
2 ds2dt1ds1dt2 =
=
1
(1 − n+d
2 )2
∫ 1
0
∫ 1
0
(s1 + t2)n((1 + s1)1−
n+d
2 − (s1 + t2)1−
n+d
2 )×
× ((1 + t2)1−
n+d
2 − (s1 + t2)1−
n+d
2 )ds1dt2 �
� C
1
(n + 1)2
∫ 1
0
∫ 1
0
(s1 + t2)2−dds1dt2 �
� C̃
1
(n + 1)2
∫ 1
0
u3−ddu.
We assumed above that n + d > 2. In all other cases, d = 1, n = 0, 1; d = 2, n = 0, we
can check that the integral is finite. Again we have the sufficient condition d < 4. Two
152 ALEXEY RUDENKO
last subregions can be treated similarly, and we omit them. We conclude: Jn < +∞ is
satisfied for d < 4 and all n � 0. In this case, Jn = O( 1
n2 ) , so the limit exists in D2,α
for α < 2− d
2 . As a consequence, we have the existence of the intersection local time for
two independent d-dimensional Brownian motions for d = 1, 2, 3.
Example 4. Let T = [0, 1]2, v(dt) = dt1dt2, ξ(t1, t2) = W (t1) − W (t2). We have
Kii(s, t) = min(s1, t1) + min(s2, t2) − min(s2, t1) − min(s1, t2), Kij(s, t) = 0,
detK(s, s) = (s1 + s2 − 2 min(s1, s2))d,
‖G(s, t)‖ =
min(s1, t1) + min(s2, t2) − min(s2, t1) − min(s1, t2)√
(s1 + s2 − 2 min(s1, s2))(t1 + t2 − 2 min(t1, t2))
,
Jn =
∫ 1
0
∫ 1
0
∫ 1
0
∫ 1
0
(min(s1, t1) + min(s2, t2) − min(s2, t1) − min(s1, t2))n
((s1 + s2 − 2 min(s1, s2))(t1 + t2 − 2 min(t1, t2)))
n
2 + d
2
ds1ds2dt1dt2.
As in the previous example, we have to split the region [0, 1]4 into a few subregions. We
consider only three of them, others are similar: T1 = {0 � s1 � s2 � t1 � t2 � 1},
T2 = {0 � s1 � t1 � s2 � t2 � 1} , T3 = {0 � s1 � t1 � t2 � s2 � 1}. In the first
subregion, Kii(s, t) = 0 and J1
n = 0. Consider the second subregion:
J2
n =
∫ 1
0
∫ s2
0
∫ t1
0
∫ 1
s2
(s2 − t1)n
(s2 − s1)
n+d
2 (t2 − t1)
n+d
2
dt2ds1dt1ds2 =
1
(1 − n+d
2 )2
×
∫ 1
0
∫ s2
0
(s2 − t1)n((1 − t1)1−
n+d
2 − (s2 − t1)1−
n+d
2 )(s1− n+d
2
2 − (s2 − t1)1−
n+d
2 )dt1ds2
� C
1
(n + 1)2
∫ 1
0
∫ s2
0
(s2 − t1)2−ddt1ds2 � C
1
(n + 1)2
∫ 1
0
u2−ddu.
We assumed that n+d > 2. In that case, the sufficient condition for the integral to exist
is d < 3. In the cases d = 1, n = 0, 1 and d = 2, n = 0, the integral is also finite.
J3
n =
∫ 1
0
∫ s2
0
∫ s2
s1
∫ t2
s1
(t2 − t1)
n−d
2
(s2 − s1)
n+d
2
dt1dt2ds1ds2 =
=
1
(1 + n−d
2 )(2 + n−d
2 )
∫ 1
0
∫ s2
0
(s2 − s1)2−dds1ds2 �
� C
1
(n + 1)2
∫ 1
0
u2−ddu.
The sufficient conditions for the integral to exist: d < 3 and n > 2 − d. Again we have
to treat the cases d = 1, n = 0, 1 and d = 2, n = 0 separately. But, in the last case, the
integral is infinite:
J3
0 =
∫ 1
0
∫ s2
0
∫ s2
s1
∫ t2
s1
1
(s2 − s1)(t2 − t1)
dt1dt2ds1ds2 =
=
∫ 1
0
∫ s2
0
∫ s2
s1
(
∫ t2−s1
0
1
u
du)
1
(s2 − s1)
dt2ds1ds2 = +∞.
Finally, Jn < +∞ is satisfied for d < 3 and all n � 0 except n = 0, d = 2. The
limit exists in D2,α, α < 2 − d
2 for d = 1. If d = 2, then as our calculations suggest, it is
possible to subtract the first term of the expansion and prove the convergence of the rest.
It is known as the renormalization of the self-intersection local time for the Brownian
motion in two dimensions (see [1]).
EXISTENCE OF GENERALIZED LOCAL TIMES FOR GAUSSIAN RANDOM FIELDS 153
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Brownian motion in d and renormalization, Stoch. Proc. and Appl. 56 (1995), 1-34.
2. Hisao Watanabe, The local time of self-intersections of Brownian motions as generalized Bro-
wnian functionals, Lett. in Math. Phys. 23 (1991), 1-9.
3. Dorogovtsev A.A., Bakun V.V., Random mappings and a generalized additive functional of a
Wiener process, Theor. Prob. and Appl. 48 (2003), no. 1.
4. Orey S., Gaussian sample functions and the Hausdorff dimension of level crossings, Wahr-
scheinlichkeitstheorie verw. Geb. 15 (1970), 249-256.
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E-mail : arooden@yandex.ru
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