Local time as an element of the Sobolev space
For a centered Gaussian random ?eld taking its values in R^d, we investigate the existence of a local time as a generalized functional, i.e an element of some Sobolev space. We give the sfficient condition for such an existence in terms of the field covariation and apply it in several examples: the...
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| Zitieren: | Local time as an element of the Sobolev space / A.V. Rudenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 3. — С. 65–79. — Бібліогр.:12 назв.— англ. |
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| citation_txt | Local time as an element of the Sobolev space / A.V. Rudenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 3. — С. 65–79. — Бібліогр.:12 назв.— англ. |
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| description | For a centered Gaussian random ?eld taking its values in R^d, we investigate the existence of a local time as a generalized functional, i.e an element of some Sobolev space. We give the sfficient condition for such an existence in terms of the field covariation and apply it in several examples: the self-intersection local time for a fractional Brownian motion and the intersection local time for two Brownian motions.
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Theory of Stochastic Processes
Vol. 13 (29), no. 3, 2007, pp. 65–79
UDC 519.21
ALEXEY V. RUDENKO
LOCAL TIME AS AN ELEMENT OF THE SOBOLEV SPACE
For a centered Gaussian random field taking its values in d, we investigate the
existence of a local time as a generalized functional, i.e an element of some Sobolev
space. We give the sufficient condition for such an existence in terms of the field
covariation and apply it in several examples: the self-intersection local time for a
fractional Brownian motion and the intersection local time for two Brownian motions.
Introduction
Let (T, B) be a measurable space with finite measure ν on it. Let ξ(t), t ∈ T ,
be a centered Gaussian random field with values in R
d. Consider the integrals Lε =∫
T fε(ξ(t))ν(dt), where fε approximates, in a sense, δ0 as ε → 0+ (δ0 denotes the mea-
sure with weight 1 concentrated in 0 ∈ R
d). If the limit of Lε exists, we can say that the
local time exists (in some sense) and we call this limit as the local time of ξ at 0 ∈ R
d.
We say that Lε is the approximation for the local time of ξ at 0 ∈ R
d. Our aim is
to investigate the existence of local time so the question is: When does Lε converge as
ε → 0+? The condition for L2-convergence in terms of covariation is relatively easy to
obtain (see [2],[9]). We consider the convergence in some Sobolev space D2,α defined in
the book of Watanabe [10]. The interest in this topic arises from the possibility to define
the local time as an element of the Sobolev space with negative α, while it does not exist
in the usual sense. It is possible to determine the exact smoothness of a local time in
the sense of spaces D2,α. Sometimes the local time does not exist as an element of any
Sobolev space, but if we modify the approximation by subtracting its mathematical ex-
pectation, we obtain the convergence to the so-called renormalized local time. A classical
example of renormalization is the renormalization of the self-intersection local time for
a two-dimensional Brownian motion. This result can be found, for instance, in [7]. We
introduce some kind of renormalization, and our definition includes this classical setup.
In [5], Imkeller, Perez-Abreu, and Vives found an explicit form of the Itô–Wiener
expansion for an approximation of the k-fold self-intersection local time of a Brown-
ian motion. They estimated Hermite polynomials in this expansion and obtained the
convergence of the approximation in some Sobolev space. Albeverio, Hu and Zhou [8]
employed a similar approach, but their result is opposite. They proved that the renor-
malized self-intersection local time of a planar Brownian motion is not differentiable in
the Watanabe–Meyer sense, i.e. it does not belong to the space D2,1. Another question
was considered in [11] where Nualart and Hu deal with the renormalization of the 2-fold
self-intersection local time for a fractional Brownian motion. They proved the conver-
gence of this renormalization in L2 (which is same as in D2,0) under some conditions.
2000 AMS Mathematics Subject Classification. Primary 60H07.
Key words and phrases. Local time, Itô–Wiener expansion, Sobolev spaces, Gaussian random field,
fractional Brownian motion.
Research was partially supported by the Ministry of Education and Science of Ukraine, project
GP/F13/0095.
65
66 ALEXEY V. RUDENKO
Later on, they gave a sufficient condition for the existence of the same renormalized local
time as an element of D2,α for positive α [12].
In [2], we have found a suitable integral representation for the expectation of Itô–
Wiener expansion terms for local time approximations and use it here to derive the
condition for the convergence via the covariation of a field. This condition is both neces-
sary and sufficient if the underlying Gaussian field is centered and we consider the local
time at zero. We also get a similar condition for the convergence of a renormalized local
time. In our context, the renormalization is the subtraction of several leading terms
in the Itô–Wiener expansion. We also consider some applications including the 2-fold
self-intersection local time for a fractional Brownian motion. We have to mention that
our main ideas were already present in [2].
Let us mention that there are other ways to define the local time as an element of
some generalized space of random variables when it does not exist in the usual sense. In
[1,4], the authors consider the Hida distributions instead of Sobolev spaces.
The structrure of the exposition is the following. In the first section, the necessary and
sufficient condition for the convergence is given in terms of a sequence of integrals. In the
second section, this condition is rewritten in terms of one integral. In the third section,
we introduce the renormalization using a generalization of results from the previous
sections (with an additional condition on approximations). The last section is devoted
to applications.
1. Necessary and sufficient condition for the existence of local time
Let’s describe briefly the construction of Sobolev spaces and related objects (as it
appears in [10]). To do this, we need to define a probability space more precisely and
impose some conditions on ξ. Let B be a Banach space with Gaussian probability
measure on it, which plays the role of a probability space for ξ. Denote the covariation
matrix function for ξ as K(s, t), s, t ∈ T . Suppose that each ξ(t) is linear as a function
of ω ∈ B, and σ({ξ(t), t ∈ T }) coincides with B(B), a Borel σ-algebra on B. Suppose
also that ξ is jointly measurable (as a function of both t ∈ T and ω ∈ B) and that ξ is
not degenerate ν-a.e.: ν({t| detK(t, t) = 0}) = 0. Let Hn, n = 0, 1, . . . , be a subspace of
L2(B, μ) generated by all polynomials on B with degree less or equal n. By polynomials
on B, we mean functions of the form P (l1(x), l2(x), . . . , lk(x)), where P is a polynomial
in k variables, and l1, . . . , lk ∈ B∗. By Gn, we denote an orthogonal supplement of
Hn−1 in Hn for n ∈ N. Let Pn be a projector on Gn. Then we have
∞⊕
n=0
Gn = L2(B, μ)
(polynomials are dense in L2(B, μ)). The corresponding decomposition h =
∞∑
n=0
Pnh, h ∈
L2(B, μ) is called the Itô–Wiener expansion (or the chaos decomposition [6]). We can
define a set of norms for each h ∈ Hn as
‖h‖2
2,α =
∞∑
n=0
(1 + n)α ‖h‖2
2
where ‖ · ‖2 is the norm in L2(B, μ). Then D2,α is the completion of
∞∪
n=0
Hn by the norm
‖·‖2,α.
We define an approximation for δ0 as
fε(x) =
1
εd
f(
x
ε
), f ∈ S(Rd), f � 0,
∫
Rd
f(x)dx = 1.
LOCAL TIME AS AN ELEMENT OF THE SOBOLEV SPACE 67
We denote
G(s, t) = K−1/2(t, t)K(s, t)K−1/2(s, s),
In =
∫
T
∫
T
∫
Sd
‖G(s, t)x‖n
σ(dx)
ν(ds)√
detK(s, s)
ν(dt)√
detK(t, t)
,
where Sd is a unit sphere in R
d, and σ is the uniform surface measure on Sd. The matrix
G is defined as ν × ν-a.e., thus the integral is well defined.
Theorem 1. Fix α ∈ R. The following statements are equivalent:
1. Lε → L, ε → 0+ in D2,α,
2. lim
ε→0+
‖Lε‖2,α < +∞,
3.
∞∑
n=0
I2n(2n + 1)α+d/2−1 < +∞.
Proof. We will prove the following implications 1 ⇒ 2 ⇒ 3 ⇒ 1.
1 ⇒ 2 is obvious.
2 ⇒ 3. Let’s write a formula for ‖Lε‖2,α using the Itô–Wiener expansion. By def-
inition, Lε is bounded so Lε ∈ L2(Ω) and we may write the Itô–Wiener expansion
Lε =
∞∑
n=0
an(ε). We have ‖Lε‖2,α =
∞∑
n=0
(n + 1)αE(an(ε))2. We know that (from [2])
E(an(ε))2 =
=
∫
T
∫
T
∫
Rd
∫
Rd
f(x)f(y)
∫
Rd
∫
Rd
eiε((x,u)+(y,w))qn(s, t, u, w)dudwdxdyν(ds)ν(dt),
where
qn(s, t, u, w) = (2π)−2d (−1)n
n!
(K(s, t)u, w)ne−
1
2 (K(s,s)u,u)e−
1
2 (K(t,t)w,w).
From (2), we get lim
ε→0+
E(a0(ε))2 < +∞. On the other hand,
E(a0(ε))2 =
(∫
T
∫
Rd
f(x)e−
1
2 ε(K−1(t,t)x,x) (2π)−d/2√
detK(t, t)
dxν(dt)
)2
.
We can see that E(a0(ε))2 is a monotonous function of ε with maximum at ε = 0. We
conclude that ∫
T
ν(dt)√
det K(t, t)
< +∞.
Let’s estimate E(an(ε))2:
(1) E(an(ε))2 �
∫
T
∫
T
∫
Rd
∫
Rd
f(x)f(y)
∫
Rd
∫
Rd
|qn(s, t, u, w)| dudwdxdyν(ds)ν(dt) =
=
∫
T
∫
T
∫
Rd
∫
Rd
|qn(s, t, u, w)| dudwν(ds)ν(dt) = Jn.
68 ALEXEY V. RUDENKO
We want to transform Jn.
∫
Rd
∫
Rd
|qn(s, t, u, w)| dudw =
(2π)−2d 1
n!
∫
Rd
∫
Rd
|(K(s, t)u, w)|ne−
1
2 (K(s,s)u,u)e−
1
2 (K(t,t)w,w)dudw =
=
(2π)−2d
n!
√
detK(t, t) detK(s, s)
∫
Rd
∫
Rd
|(G(s, t)u, w)|ne−
1
2 ‖u‖2− 1
2‖w‖2
dudw =
=
(2π)−2d
n!
√
det K(t, t) detK(s, s)
∫
Rd
e−
1
2 ‖u‖2
(∫
Rd
|(G(s, t)u, w)|ne−
1
2‖w‖2
dw
)
du =
=
(2π)−2d
n!
√
detK(t, t) detK(s, s)
∫
Rd
e−
1
2‖u‖2 ‖G(s, t)u‖n
(∫
Rd
|w1|n e−
1
2‖w‖2
dw
)
du =
=
(2π)−2d
n!
√
detK(t, t) detK(s, s)
∫
Rd
|w1|n e−
1
2‖w‖2
dw·
·
∫
R+
e−
1
2 r2
rn+d−1
(∫
Sd
‖G(s, t)x‖n σ(dx)
)
dr =
=
(2π)−2d
n!
√
detK(t, t) detK(s, s)
∫
Rd
|w1|n e−
1
2‖w‖2
dw·
·
∫
R+
e−
1
2 r2
rn+d−1dr
∫
Sd
‖G(s, t)x‖n σ(dx) =
=
C(n, d)√
detK(t, t) detK(s, s)
∫
Sd
‖G(s, t)x‖n
σ(dx).
As we can see, Jn = C(n, d)In, where
C(n, d) =
(2π)−2d
n!
∫
Rd
|w1|n e−
1
2‖w‖2
dw
∫
R+
e−
1
2 r2
rn+d−1dr =
=
2
n!
(2π)−(3d+1)/2
∫ ∞
0
vn
1 e−v2
1/2dv
∫ ∞
0
rn+d−1e−r2/2dr =
=
2n+(d−1)/2
n!
(2π)−(3d+1)/2
∫ ∞
0
x(n+d−2)/2e−xdx
∫ ∞
0
x(n−1)/2e−xdx =
=
2n+(d−1)/2
n!
(2π)−(3d+1)/2Γ(
n + d
2
)Γ(
n + 1
2
).
We need to find the asymptotics of C(n, d) for large n. We can use the asymptotics of
the gamma function (the Stirling formula),
Γ(x) ∼
√
2π/x
(x
e
)x
, x → +∞,
LOCAL TIME AS AN ELEMENT OF THE SOBOLEV SPACE 69
and the equality n! = Γ(n + 1). We have
C(n, d) =
2n+(d−1)/2
Γ(n + 1)
(2π)−(3d+1)/2Γ(
n + d
2
)Γ(
n + 1
2
) ∼
∼ 2(d−1)/2(2π)−(3d+1)/22n
√
8π
n + d
·
·
(
n + d
2e
)(n+d)/2(
n + 1
2e
)(n+1)/2(
e
n + 1
)n+1
=
= 2(d−1)/2(2π)−(3d+1)/22−(d+1)/2(8π)1/2e−(d+1)/2·
·
√
1
n + d
(1 +
d + 1
n + 1
)(n+1)/2(n + d)(d−1)/2 ∼
∼ (2π)−3d/2nd/2−1, n → +∞.
The integral
∫
Sd
‖G(s, t)x‖n
σ(dx) can be calculated explicitly in terms of eigenvalues
of G(s, t), but it is more convenient for us to leave it in the form of integral. Indeed, we
can use that the expression taken to the n-th power under integral, which is ‖G(s, t)x‖,
is less or equal than 1 when ‖x‖ = 1 (from the definition of G). Therefore, the condition
I0 < +∞ (this is our assumption) gives us In < +∞, n ∈ N. Using the Lebesgue theorem
of dominated convergence, we get lim
ε→0+
E(an(ε))2 = Jn for even n and lim
ε→0+
E(an(ε))2 =
0 for odd n. We can see now that
+∞∑
n=0
J2n(2n + 1)α ≤ lim
ε→0+
‖Lε‖2,α < +∞.
Using the asymptotics of C(n, d), it is easy to complete the proof of this implication.
3 ⇒ 1. Using the Hölder inequality, we get I2n+1 � M
√
I2nI2n+2, n = 0, 1, . . .
(the constant is independent of n). So if we include I2n+1 into the infinite sum, then
∞∑
n=0
In(n + 1)α+d/2−1 < +∞. Using the formula for Ean(ε1)an(ε2) (see [2]) and the
condition In < +∞, we conclude that Ean(ε1)an(ε2) is convergent to Jn for even n and
to 0 for odd n. Thus, an(ε) is convergent in L2(Ω). Define an(0) = L2 − lim
ε→0+
an(ε).
From inequality (1), we get
E(an(ε))2 � Jn = C(n, d)In
Consequently,
∞∑
n=0
(n + 1)αE(an(0))2 �
∞∑
n=0
(n + 1)αC(n, d)In < +∞,
and there exists L =
∞∑
n=0
an(0) ∈ D2,α. Now all left to do is to use a uniform bound for
tails of the Itô–Wiener expansion for approximations. We get
‖L − Lε‖2,α =
∞∑
n=0
(n + 1)αE(an(0) − an(ε))2 → 0, ε → 0 + .
Note that if statement (3) in the theorem is not fulfilled, then it is not possible to
construct the local time as an element of D2,α using approximations of this kind and
some Sobolev space with smaller α. Indeed, suppose that we have convergence to some
70 ALEXEY V. RUDENKO
L in D2,α0 with α0 < α (while there is no convergence for our fixed α). Then using same
arguments as in theorem’s proof, we can prove
‖L‖2,α =
+∞∑
n=0
E(a2n(0))2(2n + 1)α =
+∞∑
n=0
J2n(2n + 1)α = ∞.
Thus L can not be an element of D2,α.
For some values of α, it is possible to simplify the condition of the theorem.
Corollary 1. If α < − d
2 , then statements in the theorem above are equivalent to∫
T
ν(dt)√
det K(t, t)
< +∞.
Proof. Note that I0 = σ(Sd)
(∫
T
ν(dt)√
detK(t,t)
)2
. Obviously, statement (3) of the theorem
yields I0 < +∞. This proves one side of equivalence. Now suppose that I0 < +∞. As
we have seen already, In � I0, n ∈ N. Taking into account that α + d
2 − 1 < −1, we have
∞∑
n=0
I2n(2n + 1)α+d/2−1 < +∞.
2. Integral form of the condition for local time existence
We want to give a different form of the condition for the local time existence from
Theorem 1, using the following lemma.
Lemma 1. Let (E, F) be ameasurable space with measure μ on it and let f be a mea-
surable function on E with values in [0, 1]. Denote:
Jn =
∫
E
(f(x))nμ(dx),
pβ : [0, 1] → [1,∞], β ∈ R,
pβ(z) =
⎧⎨
⎩
zβ, β < 0
1 − ln z, β = 0
1, β > 0.
For any γ ∈ R. the following statements are equivalent:
1.
∞∑
n=0
Jn(n + 1)γ < +∞,
2.
∫
E
p−γ−1(1 − f(x))μ(dx) < +∞.
Proof. Note that if γ < −1, then both the sum and the integral converge when μ is
finite or diverge when μ(E) = +∞. So it is enough to consider the case γ � −1. It is
obvious that the sequence (n +1)γ may be replaced by the sequence of positive numbers
cn, n ∈ N, if cn ∼ (n + 1)γ . We use this and take cn to be coefficients near qn in the
Taylor series of p−γ−1(1 − q) at the point q = 0. We can write an explicit form for cn:
γ = −1 : p−γ−1(1 − q) = 1 − ln(1 − q) = 1 +
∞∑
n=1
1
n
qn =
∞∑
n=0
cnqn
, γ > −1 : p−γ−1(1 − q) = (1 − q)−γ−1 =
∞∑
n=0
(
n∏
k=1
(
γ
k
+ 1)
)
qn =
∞∑
n=0
cnqn.
LOCAL TIME AS AN ELEMENT OF THE SOBOLEV SPACE 71
All sums converge if |q| < 1. It can be verified that cn satisfy the needed property. The
condition f ∈ [0, 1] allows us to write
∞∑
n=0
Jncn =
∫
E
∞∑
n=0
(f(x))ncnμ(dx) =
∫
E
p−γ−1(1 − f(x))μ(dx).
Lemma 1 is proved.
Now we apply Lemma 1 to get a different form of the condition of Theorem 1. Note
that, for α < − d
2 , we have already done this above.
Corollary 2. Fix α ∈ R. The following statements are equivalent:
1. Lε → L, ε → 0+ in D2,α,
2. ∫
T
∫
T
∫
Rd
p−α−d/2(1 − ‖G(s, t)x‖)σ(dx)
ν(ds)√
detK(s, s)
ν(dt)√
detK(t, t)
< +∞.
Proof. We choose μ(dx, ds, dt) = σ(dx) ν(ds)√
det K(s,s)
ν(dt)√
detK(t,t)
to be the measure and
‖G(s, t)x‖ to be the function f from Lemma 1. Now Corollary 2 is a direct conse-
quence of Theorem 1 and Lemma 1 (we also need the fact that we can include terms
with odd numbers into sum (3) from Theorem 1, but we showed it in the proof of this
Theorem).
3. Renormalization
We want to define the renormalization for local time approximations, but before we
need to prove a more generalized form of Theorem 1. If we define local time approxima-
tions more precisely taking f to be a Gaussian density, f(x) = 1
(2π)d/2 e−
‖x‖2
2 , then we
can prove a result similar to Theorem 1 in the case where the convergence of Lε is in
the sense of some seminorm on the space of square integrable random variables on our
probability space (its values may be infinite). We define this seminorm as
S(η, {cn}) =
∞∑
n=0
cnEη2
n
where η is a square integrable random variable,
η =
∞∑
n=0
ηn
is the Itô–Wiener expansion of η and cn � 0, n = 0, 1, . . . , is an arbitrary sequence of
non-negative numbers. Norms in Sobolev spaces can be considered as partial cases of
this seminorm.
Theorem 2. The following statements are equivalent
1. Lε → L, ε → 0+ with respect to seminorm S(·, {cn}),
2. lim
ε→0+
S(Lε, {cn}) < +∞,
3.
∞∑
n=0
I2ncn(2n + 1)d/2−1 < +∞.
Proof. 1 ⇒ 2 is obvious.
72 ALEXEY V. RUDENKO
2 ⇒ 3. As in Theorem 1, we consider the Itô–Wiener expansion of Lε =
∞∑
n=0
an(ε).
Using the explicit form of an, we get
E(an(ε))2 =
=
∫
T
∫
T
∫
Rd
∫
Rd
∫
Rd
∫
Rd
f(x)f(y)eiε((x,u)+(y,w))qn(s, t, u, w)dudwdxdyν(ds)ν(dt) =
= (2π)d
∫
T
∫
T
∫
Rd
∫
Rd
qn(s, t, u, w) exp(−ε
2
(‖u‖2 + ‖w‖2))dudwν(ds)ν(dt).
We note that the function f is even and qn(s, t, u,−w) = (−1)nqn(s, t, u, w). As
a consequence, the expression is zero for odd n. For even n, the function qn is non-
negative, so the expression above is a monotonous function of ε. We can see now that
the same considerations as in Theorem 1 are valid in this case (we choose the smallest
even n such that cn > 0, then all reasonings are the same with the substitution of I0
with In).
3 ⇒ 1. Because members of the Itô–Wiener expansion with odd index are zero, the
proof is the same as that of Theorem 1.
Such a kind of seminorms is interesting because of the following considerations. If
we take cn = 0, n < 2n0; cn = (1 + n)α, n � 2n0, where α - arbitrary real number, n0
- arbitrary integer, then the convergence with respect to seminorm is equivalent to the
convergence of renormalized local time approximations Lε,n0 (first 2n0 members of the
Itô–Wiener expansion of Lε are subtracted) in D2,α. We can also apply Lemma 1 to this
case.
Corollary 3. Fix α ∈ R, n0 ∈ N, and {cn} as above. Then the following statements are
equivalent:
1. Lε,n0 → L, ε → 0+ in D2,α,
2.∫
T
∫
T
∫
Rd
‖G(s, t)x‖2n0
p−α−d/2(1 − ‖G(s, t)x‖)σ(dx)
ν(ds)√
detK(s, s)
ν(dt)√
detK(t, t)
< +∞.
Proof. We let μ(dx, ds, dt) = ‖G(s, t)x‖2n0 σ(dx) ν(ds)√
det K(s,s)
ν(dt)√
detK(t,t)
to be the measure
from Lemma 1, and let the function f be the same as that in Corollary 2. Now we apply
Theorem 2 together with Lemma 1. The only thing we need to check (like in Corollary
2) is why we can add terms with odd numbers to condition (3) for the convergence in
Theorem 2. But here, because of the form of cn, we can use the same arguments as we
did in Theorem 1.
It is interesting to observe here some immediate conclusions about this integral condi-
tion. Possible singularities which come from (detK(s, s) detK(t, t))−1/2 can be refined
with ‖G(s, t)x‖2n0 . We can expect all kinds of situations here because G(s, t), s, t ∈ T
and K(s, s), s ∈ T are in some sense independent (i.e. we can define one independently
of the other). Also we can not obtain the convergence for bigger α using renormalization
(it is obvious from the definition of renormalization, so we only confirm that) because
we can not refine the singularities of p−α−d/2(1 − ‖G(s, t)x‖) with ‖G(s, t)x‖2n0 .
4. Applications
In this section, we apply our results to some Gaussian fields. First of all, we have to
mention when the construction of the needed probability space for our Gaussian field is
LOCAL TIME AS AN ELEMENT OF THE SOBOLEV SPACE 73
possible. If T is a separable metric space and our Gaussian random field is stochastically
continuous, then we have the sequence of values {ξ(tn), n ∈ N} which generate a σ-field
of the field. Let H be a subspace of L2(Ω) generated by linear combinations of {ξ(tn)}.
It is well known (see [3],[6]) that we can construct (as a part of the abstract Wiener
space) a Banach space B and a probability Gaussian measure μ on it such that H is the
space of admissible shifts for μ (subspace of all linear functionals on B). In this setting,
(B, B(B), μ) is a replacement for the probability space, and our field will be defined
naturally on it as linear functionals on B. That is exactly the construction we need. In
all cases, we suppose that the function f from the approximation is a standart Gaussian
density and use corollaries 2,3. All our assumptions can be easily verified for processes
below. So, in proofs, we need to verify the corresponding integral conditions only.
We start from the case of fractional Brownian motion. Suppose {X(t) ∈ R
d, t ∈ [0, 1]}
is a fractional Brownian motion with the Hurst parameter H ∈ (0, 1). Coordinates are
independent and each has the covariation function rH(s, t) = s2H+t2H−|s−t|2H
2 .
Example. Let T = [0, 1]; ν(dt) = dt; ξ(t) = X(t). In this case, the local time does not
exist in any Sobolev space if d ≥ 1
H . If d < 1
H , then the limit exists in D2,α for α < 1
2H − d
2
and doesn’t exist in other cases. The value of n0 does not affect the convergence of Lε,n0
(renormalization does not work).
Example. Let T = [1/2, 1]; ν(dt) = dt; ξ(t) = X(t). The local time exists if α < 1
2H − d
2
(and doesn’t exist in other cases). Renormalization does not work.
Proof. We have
K(s, t) = rH(s, t)I
G(s, t) = rH(s, t)(rH(s, s)rH(t, t))−1/2I =
|s|2H + |t|2H − |t − s|2H
2tHsH
I,
so the integral with respect to x in the integral condition (2) from Corollary 3 can be
dropped. This condition now has form∫ 1
0
∫ 1
0
(
s2H + t2H − |s − t|2H
2sHtH
)2n0
·
·
( |s − t|2H − (sH − tH)2
2sHtH
)−α−d/2
(st)−dHdtds.
We use the polar coordinates: s = r sin φ, t = r cosφ. We can extend or reduce the
domain of integration:
{s2 + t2 < 1, s > 0, t > 0} ⊂
⊂ {s < 1, t < 1, s > 0, t > 0} ⊂
⊂ {s2 + t2 < 2, s > 0, t > 0}.
For both cases, we get similar integrals and the same conditions for their finiteness. We
consider only the case of a reduced domain (the second one can be treated in the same
way):∫ 1
0
∫ π/2
0
(
(cosφ)2H + (sin φ)2H − | cosφ − sin φ|2H
2(cosφ sin φ)H
)2n0
·
·
( | cosφ − sin φ|2H − (cosH φ − sinH φ)2
2 cosH φ sinH φ
)−α−d/2
·
· (cosφ sin φ)−dHdφr1−2dHdr < +∞
74 ALEXEY V. RUDENKO
Our integral becomes a product of two integrals. The integral with respect to r is finite,
if (and only if) dH < 1. The integral with respect to φ can be infinite only near φ = 0,
φ = π
2 and φ = π
4 , because the integrand is continuous and finite away from these points.
Note that the expression taken to power n0 is bounded. Since
|cosφ − sinφ|2H − (cosH φ − sinHφ)2
2cosHφsinHφ
=
= 1 +
| sinφ|H
2| cosφ|H +
|cosφ − sinφ|2H − | cosφ|2H
2cosHφsinHφ
∼
∼ 1 − 2H | cosφ|2H−1 sin φ
2cosHφsinHφ
→ 1
as φ → 0+, then integral with respect to φ is finite in a neighbourhood of φ = 0 if dH < 1
(because only one important multiplier under the integral is (sinφ)−dH ). The behavior
of the integral in a neighbourhood of φ = π
2 is similar. Now we investigate the behaviour
in a neighbourhood of φ = π
4 :
| cosφ − sin φ|2H − (cosH φ − sinH φ)2 ∼ 2H |φ − π
4
|2H − H22H−1(φ − π
4
)2
∼ 2H |φ − π
4
|2H , φ → π
4
.
From this relation, we get the sufficient condition for the integral to be finite 2H(−α −
d
2 ) > −1. It is also necessary, because the expression taken to power n0 can not refine
this singularity, as we mentioned earlier.
Now let T = [1/2, 1]. In this case, the points s = 0 and t = 0 lie away from the
integration domain. So, if we make same calculations as above, we get that φ = 0 and
φ = π
2 are away from the integration domain. The only one condition for the finiteness
here is α < 1
2H − d
2 .
Now we turn to the interesting case of the self-intersection for a fractional Brownian
motion. Here, we generalize the results from [5,8,11,12].
Example. Let T = [0, 1]2; ν(dt) = dt1dt2; ξ(t) = X(t1)−X(t2). The renormalized local
time exists as an element of the Sobolev space D2,α if and only if
α <
1
H
− d
2
, d <
3
2H
, n0 >
dH − 1
2(1 − H)
.
Example. Let T = [0, 1/3] × [2/3, 1]; ν(dt) = dt1dt2; ξ(t) = X(t1) − X(t2). The local
time exists in D2,α for α < 1
H − d
2 . Renormalization does not work.
Proof. The covariation in this case is given by K(s, t) = 1
2 (|s1 − t2|2H + |s2 − t1|2H −
|s1 − t1|2H − |s2 − t2|2H)I, and the correlation equals G(s, t) = g(s, t)I, where
g(s, t) =
|s1 − t2|2H + |s2 − t1|2H − |s1 − t1|2H − |s2 − t2|2H
2|s1 − s2|H |t1 − t2|H .
The integral from Corollary 3 has the form∫ 1
0
∫ 1
0
∫ 1
0
∫ 1
0
|g(s, t)|2n0 |1 − |g(s, t)||−α− d
2 |s1 − s2|−dH |t1 − t2|−dHdt1dt2ds1ds2.
By introducing the new variables x = s1 − t1, y = s2 − s1, z = t1 − t2 and extending the
integration domain, we simplify this integral to∫ 1
−1
∫ 1
−1
∫ 1
−1
∣∣∣∣D2H(x, y, z)
2|yz|H
∣∣∣∣
2n0
∣∣∣∣1 − |D2H(x, y, z)|
2|yz|H
∣∣∣∣
−α− d
2
|y|−dH |z|−dHdxdydz,
LOCAL TIME AS AN ELEMENT OF THE SOBOLEV SPACE 75
where Dγ(x, y, z) = |x + y|γ + |x + z|γ − |x|γ − |x + y + z|γ . This new integral is finite
if and only if the starting integral is finite (because we can obtain a multiple of this
integral by reducing the integration domain instead of extending it). Now we make the
change of variables x = ru, y = r cosφ, z = r sin φ and reduce the integration domain by
an additional constraint y2 + z2 < 1 (we get almost the same integral by extending this
domain, so this procedure is also two-way):∫ +∞
−∞
∫ 2π
0
∣∣∣∣D2H(u, cosφ, sin φ)
2| cosφ sin φ|H
∣∣∣∣
2n0
∣∣∣∣1 − D2H(u, cosφ, sin φ)
2| cosφ sin φ|H
∣∣∣∣
−α− d
2
·
· | cosφ|−dH | sin φ|−dH
∫ min(1/|u|,1)
0
r2−2dHdrdφdu.
Integrating with respect to r, we obtain the necessary condition for finiteness, dH < 3
2 .
To proceed, we need some inequalities describing the behavior of D2H(x, y, z).
Lemma 2.
1. For all γ ∈ (0, 1) ∪ (1, 2], there exist the positive constants C1 and C2 such that
if 3|y| < |x|, 3|z| < |x|, then
C1|yz||x|γ−2 ≤ |Dγ(x, y, z)| ≤ C2|yz||x|γ−2.
If γ = 1, then, for the same x, y, z, we have |Dγ(x, y, z)| = 0, and so we can take
C1 = C2 = 0.
2. If γ ∈ (1, 2), then
|Dγ(x, y, z)| ≤ 21−γ(||y| + |z||γ − ||y| − |z||γ).
Additionally, if y, z > 0, then the inequality changes into an equality if and only
if x = −(y + z)/2.
3. If γ ∈ (0, 1), then
|Dγ(x, y, z)| ≤ |y|γ + |z|γ − ||y| − |z||γ .
Additionally, if y, z > 0, then the inequality changes into an equality if and only
if x = −y,−z.
4. If γ = 1, then
|Dγ(x, y, z)| ≤ |y| + |z| − ||y| − |z|| .
Additionally, if y, z > 0, then the inequality changes into an equality if and only
if x ∈ [−max(y, z),−min(y, z)].
Proof. The first inequality is equivalent to
C1|uv| ≤ |Dγ(1, u, v)| ≤ C2|uv|; |u| <
1
3
, |v| <
1
3
.
Because u, v is small enough by our assumption, it is possible to use the representation
Dγ(x, y, z) = −γ(γ − 1)
∫ |y|
0
∫ |z|
0
(x + sign(y)v + sign(z)w)γ−2dvdw
which is valid for all x > max(0,−y,−z,−y − z) (here we denote sign(y) = 1 if y ≥ 0
and sign(y) = −1 if y < 0). We have
|Dγ(1, u, v)| =
∣∣∣∣∣γ(γ − 1)
∫ |u|
0
∫ |v|
0
(1 + sign(u)y + sign(v)w)γ−2dydw
∣∣∣∣∣ ≤
≤ |γ(γ − 1)uv| sup
|y|<1/3;|w|<1/3
(1 + y + w)γ−2 ≤ 32−γ |γ(γ − 1)| |uv|
76 ALEXEY V. RUDENKO
and similarly
|Dγ(1, u, v)| ≥ (
5
3
)2−γ |γ(γ − 1)| |uv|.
Note that, for γ = 1, we have |Dγ(1, u, v)| = 0 under the same assumptions on u, v.
To prove other inequalities, it is enough to consider the case 0 < y ≤ z. Indeed, if we
have both y and z negative, then we can introduce new variables x̃ = −x, ỹ = −y, z̃ = −z
and obtain the same inequality for positive variables. If, for example, y < 0 and z > 0,
then another change of variables helps: x̃ = x + y, ỹ = −y, z̃ = z. All other cases are
similar because of the symmetry with respect to the exchange y ↔ z. Moreover, if y = 0,
then all inequalities are trivial equalities, so we may suppose that y �= 0. Now we can
prove these inequalities by finding the maximum and minimum of Dγ(x, y, z) for fixed y
and z. Using the convexity of a power function, we can find the sign of the derivative of
g(x) = Dγ(x, y, z) with respect to x. For γ ∈ (1, 2), we have
g′(x) < 0, x ∈ (−∞,−(y + z)/2),
g′(x) = 0, x = −(y + z)/2,
g′(x) > 0, x ∈ (−(y + z)/2, +∞).
For γ ∈ (0, 1) (note that, in this case, the derivative may not exist at some points):
g′(x) > 0, x ∈ (−∞,−y − z),
g′(x) < 0, x ∈ (−y − z,−z),
g′(x) > 0, x ∈ (−z,−(y + z)/2),
g′(x) = 0, x = −(y + z)/2,
g′(x) < 0, x ∈ (−(y + z)/2,−y),
g′(x) > 0, x ∈ (−y, 0),
g′(x) < 0, x ∈ (0, +∞).
For γ = 1 (here the derivative also does not exist at some points):
g′(x) = 0, x < −y − z,
g′(x) < 0, x ∈ (−y − z,−z),
g′(x) = 0, x ∈ (−z,−y),
g′(x) > 0, x ∈ (−y, 0),
g′(x) = 0, x > 0.
We also know that Dγ(x, y, z) is continuous and that DH(x, y, z) → 0, |x| → ∞ (from
the first inequality). As we can see for γ ∈ [1, 2), the function has its minimum in x =
−y−z
2 and always negative. The right side of the inequality is exactly |Dγ(−y−z
2 , y, z)|, so
it is proved. For γ ∈ (0, 1] by same method, we find the maximum at x = 0; x = −y − z
and the minimum at x = −y; x = −z (values at both points of each pair are the same).
Comparing the moduli of the function values at these points, we get |Dγ(x, y, z)| ≤
|Dγ(−y, y, z)|. If y, z > 0, then the maximum is achieved only at the points x = −y−z
2
for γ ∈ (1, 2) and at x = −y; x = −z for γ ∈ (0, 1). For γ = 1, the maximum is achieved
on the interval x ∈ [−z,−y]. Lemma 2 is proved.
Now let us look at the integral
∫ +∞
−∞
∫ 2π
0
∣∣∣∣D2H(u, cosφ, sin φ)
2| cosφ sin φ|H
∣∣∣∣
2n0
∣∣∣∣1 − D2H(u, cosφ, sin φ)
2| cosφ sin φ|H
∣∣∣∣
−α− d
2
·
· | cosφ|−dH | sinφ|−dH(max(1, |u|))2dH−3dφdu
LOCAL TIME AS AN ELEMENT OF THE SOBOLEV SPACE 77
once more. We split the domain of integration into several parts by restricting ourself to
the case φ ∈ (0, π
4 ) (other parts of the domain can be treated similarly):
D1 = {|u| > M},
D2 = {|u| < M ; φ � ε,
π
4
− φ � ε},
D3 = {|u| < M, |u| � ε, |1 + u| � ε; φ < ε},
D4 = {|u|2 + | sin φ|2 < 4ε2},
D5 = {|1 + u|2 + | sin φ|2 < 4ε2},
D6 = {|u| < M, |u +
1√
2
| � ε;
π
4
− φ < ε},
D7 = {| u
cosφ
+ 1|2 + | tg φ − 1|2 < 16ε2}.
Obviously, the union of these parts covers a selected part of the domain if we choose
ε < 1
16 and M such that, for |u| > M , we have C2|u|2H−2 < 1
2 , where C2 is the constant
from the first inequality above for H �= 1
2 . We use this inequality for D1 and immediately
obtain that the finiteness of our integral is equivalent to the finiteness of two integrals∫∞
M
u2n0(2H−2)+2dH−3du and
∫ π/4
0
|sinφ|2n0(1−H)−dHdφ. Both integrals are finite if and
only if n0 > dH−1
2(1−H) . If H = 1
2 , we can choose M such that D2H(u, cosφ, sinφ) = 0 on
D1. In this case, it is easy to see that the condition dH < 1 or n0 > 0 (it is true if the
declared conditions for the local time existence hold) is sufficient for the integral over D1
to be finite.
For D2, D3, D4, D5, D6, we have that 1 − D2H(u,cosφ,sinφ)
2|cosφsinφ|H is not equal to zero on the
closure of these sets. Indeed, using the inequalities from Lemma 2, we get for H ∈ (0, 1
2 ):
|D2H(u, cosφ, sin φ)|
2| cosφ sin φ|H � | cosφ|2H + | sin φ|2H − | cosφ − sinφ|2H
2| cosφ sin φ|H =
= 1 +
(| cosφ|H − | sin φ|H)2 − | cosφ − sin φ|2H
2| cosφ sin φ|H �
= 1 − (| sin φ|H + | cosφ − sin φ|H − | cosφ|H)·
· (| cos φ|H − | sinφ|H + | cosφ − sin φ|H))
2| cosφ sin φ|H =
= 1 − (1 + | ctg φ − 1|H − | ctg φ|H)·
· (1 − | tg φ|H + |1 − tg φ|H) �
� 1 − (1 + | ctg(π/4) − 1|H − | ctg(π/4)|H)·
· (1 − | tg(π/4)|H + |1 − tg(π/4)|H) = 1.
For H ∈ (1
2 , 1),
|D2H(u, cosφ, sin φ)|
2| cosφ sin φ|H � 2−2H | cosφ + sin φ|2H − | cosφ − sin φ|2H
| cosφ sin φ|H =
= 2−2H(|tg2φ + ctg2φ + 2|H − |tg2φ + ctg2φ − 2|H) �
� 2−2H(|tg2(π/4) + ctg2(π/4) + 2|H − ∣∣tg2(π/4) + ctg2(π/4) − 2
∣∣H) = 1.
For H = 1
2 , both inequalities are true. As we can see, the equality in these inequalities
is possible only if φ = π/4, u = − 1√
2
(from Lemma 2 and the inequalities above). But
the function we consider is continuous on D2, D3, D4, D5, D6 and therefore is not equal
to zero. So, this expression taken to the power −α− d/2 can be omitted in the integral.
78 ALEXEY V. RUDENKO
For D2, the integrand is bounded, so the integral is always finite. For D3, D4, D5,
we have to deal only with the singularity of |sinφ|−dH which can be refined by the
renormalization term.
For D3, we have the bound |D2H(u, cosφ, sinφ)| ≤ C|φ|, where C is a constant de-
pending only on ε, and the condition n0 > dH−1
2(1−H) is sufficient for the integral to be finite.
Note that, for D4, D5, this inequality is not true (for 2H < 1), because the derivative of
D2H(u, cosφ, sinφ) with respect to φ at φ = 0 blows up if u = 0 or u = −1. Instead, we
have to use a change of coordinates. For D3, we can also prove that, for H = 1/2, the con-
dition n0 > dH−1
2(1−H) is necessary. It is enough to note that |D2H(u, cosφ, sin φ)| = 2 sin φ
on 0 � φ < ε; u ∈ (− cosφ,− sin φ).
For D4, let sinφ = rsinθ, u = rcosθ. We obtain the following integral:∫ 2ε
0
∫ 2π
0
||rcosθ + rsinθ|2H + |rcosθ +
√
1 − (rsinθ)2|2H − |rcosθ|2H−
− |rcosθ + rsinθ +
√
1 − (rsinθ)2|2H |2n0 |sinθ|−dH−2n0Hr1−dH−2n0Hdθdr.
Here, we already dropped multipliers under the integral bounded above and below.
The expression taken to 2n0 power can be bounded (using derivatives) by the expres-
sion Crmin(2H,1)|sinθ|. Thus, our integral is bounded by the product of two integrals∫ 2ε
0 r2n0min(2H,1)+1−dH−2n0Hdr and
∫ 2π
0 |sinθ|2n0−dH−2n0Hdθ. The conditions for their
finiteness are 2n0(min(2H, 1)− H) + 2 − dH > 0 and 2n0(1 − H) − dH + 1 > 0. Recall
that we have already the conditions dH < 3
2 and n0 > dH−1
2(1−H) as necessary and that
n0 ≥ 0. We can see that these conditions are sufficient in this case. The case of D5 is
almost similar. We have to let sinφ = rsinθ, u + 1 = rcosθ and obtain a similar bound
for the integrand.
For D6, the integrand is bounded as we have proved above. For D7, the only un-
bounded part under the sign of integral is
∣∣∣1 − D2H (u,cosφ,sinφ)
2|cosφsinφ|H
∣∣∣−α− d
2
and
2|cosφsinφ|H − D2H(u, cosφ, sinφ) =
= |cosφ|2H(2|1 + w|H − |1 + v + w|2H − |v − 1|2H + |v|2H + |v + w|2H),
where w = tgφ − 1, v = u
cosφ . We introduce w and v as new variables of integration
and note that the expression above is equivalent to |v|2H + |v + w|2H (multiplied by a
constant) when v2 + w2 → 0+. Using the polar coordinates, we obtain two integrals∫ 4ε
0 r1−2H(α+ d
2 )dr and
∫ 2π
0 (|cosθ|2H + |cosθ + sinθ|2H)−α− d
2 dθ. The second integral is
always finite and first gives us the necessary and sufficient condition α + d
2 < 1
H .
If we set T = [0, 1/3] × [2/3, 1], then we can get the same integral but on a different
domain. This domain can be treated exactly like the union of D2, D6, D7 (for example,
the singularities near s1 = s2 and t1 = t2 and, consequently, near sinφ = 0 and cosφ = 0
are outside the integration domain), and we have only the condition on α as necessary
and sufficient for the integral to be finite.
In all previous examples, we had the covariation matrix proportional to the identity
one. In the next one, the situation is different. Suppose that {W1(t), W2(t) ∈ R
d; t ∈
[0, 1]} are two Brownian motions which are not necessarily independent. Suppose that
EW1(s)W2(t) = min(s, t)Q, where Q is some d× d matrix. We consider the intersection
local time of these two processes. By {λi, i = 1, . . . , d}, we denote eigenvalues of the
self-adjoint matrix Q+QT
2 . Here, we study only the condition from Corollary 1.
Example. Let T = [0, 1]2; ν(dt) = dt1dt2; ξ(t) = W1(t1)−W2(t2). The local time exists
in D2,α for all α < − d
2 if and only if the number of eigenvalues of the matrix Q+QT
2
which are equal to 1 is smaller than 2 (#{i : γi = 1} < 2) and d < 4.
LOCAL TIME AS AN ELEMENT OF THE SOBOLEV SPACE 79
Example. Let T = [12 , 1]2; ν(dt) = dt1dt2; ξ(t) = W1(t1)−W2(t2). The local time exists
in D2,α for α < − d
2 if and only if #{i : γi = 1} < 2.
Example. Let T = [14 , 1
2 ]× [ 34 , 1]; ν(dt) = dt1dt2; ξ(t) = W1(t1)−W2(t2). The local time
always exists in D2,α for α < − d
2 .
Proof. The integral we have to study has the form∫ 1
0
∫ 1
0
d∏
i=1
(t1 + t2 − 2min(t1, t2)λi)−1/2dt1dt2.
Here, we used that
detK(t, t) = det((t1 + t2)I − min(t1, t2)(Q + QT )) =
d∏
i=1
(t1 + t2 − 2min(t1, t2)λi).
Note that t1+t2−2min(t1, t2)λi = (t1+t2)(1−λi)+|t1−t2|λi. If λi �= 1 and consequently
λi < 1, we have only one singularity near t1 = t2 = 0. Using the polar coordinates, we
conclude that the necessary condition for the integral to be finite is 2 − d
2 > 0. If, for
some i, we have λi = 1, we have to study the singularity near t1 = t2. Using the polar
coordinates, we conclude that multipliers with λi �= 1 can be omitted in the integral with
respect to φ, and we get the necessary condition 1 − k
2 > 0, where k = #{i : γi = 1}.
Two necessary conditions we obtained are also sufficient, because we do not have other
singularities in the integral. For T = [12 , 1]2 we do not have singularity near t1 = t2 = 0
and for T = [14 , 1
2 ] × [34 , 1] we also do not have singularity near t1 = t2 with obvious
consequences.
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E-mail : arooden@yandex.ru
|
| id | nasplib_isofts_kiev_ua-123456789-4508 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-07T15:45:37Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Rudenko, A.V. 2009-11-19T14:00:08Z 2009-11-19T14:00:08Z 2007 Local time as an element of the Sobolev space / A.V. Rudenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 3. — С. 65–79. — Бібліогр.:12 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4508 519.21 For a centered Gaussian random ?eld taking its values in R^d, we investigate the existence of a local time as a generalized functional, i.e an element of some Sobolev space. We give the sfficient condition for such an existence in terms of the field covariation and apply it in several examples: the self-intersection local time for a fractional Brownian motion and the intersection local time for two Brownian motions. Research was partially supported by the Ministry of Education and Science of Ukraine, project GP/F13/0095. en Інститут математики НАН України Local time as an element of the Sobolev space Article published earlier |
| spellingShingle | Local time as an element of the Sobolev space Rudenko, A.V. |
| title | Local time as an element of the Sobolev space |
| title_full | Local time as an element of the Sobolev space |
| title_fullStr | Local time as an element of the Sobolev space |
| title_full_unstemmed | Local time as an element of the Sobolev space |
| title_short | Local time as an element of the Sobolev space |
| title_sort | local time as an element of the sobolev space |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4508 |
| work_keys_str_mv | AT rudenkoav localtimeasanelementofthesobolevspace |