# Polar Functions for Anisotropic Gaussian Random Fields

Authors: Zhenlong Chen
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

LetX be an (N, d)-anisotropic Gaussian random field. Under some general conditions on X, we establish a relationship between a class of continuous functions satisfying the Lipschitz condition and a class of polar functions of X. We prove upper and lower bounds for the intersection probability for a nonpolar function and X in terms of Hausdorff measure and capacity, respectively. We also determine the Hausdorff and packing dimensions of the times set for a nonpolar function intersecting X. The class of Gaussian random fields that satisfy our conditions includes not only fractional Brownian motion and the Brownian sheet, but also such anisotropic fields as fractional Brownian sheets, solutions to stochastic heat equation driven by space-time white noise, and the operator-scaling Gaussian random field with stationary increments.


---

## Body

## 1. Introduction

Gaussian random fields have been extensively studied in probability theory and applied in a wide range of scientific areas including physics, engineering, hydrology, biology, economics, and finance. Two of the most important Gaussian random fields are, respectively, the Brownian sheet and fractional Brownian motion.On the other hand, many data sets from various areas such as image processing, hydrology, geostatistics, and spatial statistics have anisotropic nature in the sense that they have different geometric and probabilistic characteristics along different directions. Hence fractional Brownian motion, which is isotropic in the sense that the distribution of its increments depends only on the Euclidean distance of the time interval, is not adequate for modelling such phenomena. Many people have proposed to apply anisotropic Gaussian random fields as more realistic models; see [1, 2] and the references therein for more information.Typical examples of anisotropic Gaussian random fields are fractional Brownian sheets and the solution to the stochastic heat equation. It has been known that the sample path properties such as fractal dimensions of these anisotropic Gaussian random fields can be very different from those of isotropic ones such as Levy's fractional Brownian motion; see, for example, [3–7]. Recently, Xiao [2] systematically studied the analytic and geometric properties of anisotropic Gaussian random fields under certain general conditions. Biermé et al. [1] studied the hitting probabilities and the Hausdorff dimension of the inverse of anisotropic Gaussian random fields under some conditions. Their main goal is to characterize the anisotropic nature of the Gaussian random fields by a multiparameter index H
=
(
H
1
,
…
,
H
N
)
∈
(
0,1
)
N. This index is often related to the operator-self-similarity or multi-self-similarity of the Gaussian random field under study. In this paper, we further discuss the polar functions of anisotropic Gaussian random fields.We will continue to use the same setting as in Biermé et al. [1]. Let H
=
(
H
1
,
…
,
H
N
)
∈
(
0,1
)
N be a fixed vector and, for a
,
b
∈
ℝ
N with a
j
<
b
j (j
=
1
,
…
,
N), let I
=
[
a
,
b
]
:
=
∏
j
=
1
N
‍
[
a
j
,
b
j
]
⊆
ℝ
N denote a compact interval (or a rectangle). For example, we may take I
=
[
ϵ
0
,
1
]
N, where ϵ
0
∈
(
0,1
) is a fixed constant.LetX
(
t
)
=
(
X
1
(
t
)
,
…
,
X
d
(
t
)
), t
∈
ℝ
N, be a Gaussian random field on a probability space (
Ω
,
ℱ
,
ℙ
) with mean zero and whose components X
k,  k
=
1
,
…
,
d, are independent. Suppose that for each k
=
1
,
…
,
d, X
k satisfies the following general conditions.(C1)
There exist positive and finite constantsc
1.1, c
1.2, and c
1.3 such that 𝔼
[
X
k
(
t
)
]
2
≥
c
1.1 for all t
∈
I and
(1)
c
1.2
∑
ℓ
=
1
N
|
s
ℓ
-
t
ℓ
|
2
H
ℓ
≤
𝔼
[
X
k
(
s
)
-
X
k
(
t
)
]
2
≤
c
1.3
∑
ℓ
=
1
N
|
s
ℓ
-
t
ℓ
|
2
H
ℓ
,
∀
s
,
t
∈
I
.
(C2)
There exists a positive and finite constantc
1.4 such that, for all s
,
t
∈
I,
(2)
Var
⁡
(
X
k
(
t
)
∣
X
k
(
s
)
)
≥
c
1.4
∑
ℓ
=
1
N
|
s
ℓ
-
t
ℓ
|
2
H
ℓ
.HereVar
⁡
(
X
k
(
t
)
|
X
k
(
s
)
) denotes the conditional variance of X
k
(
t
) given X
k
(
s
). We will call X an (
N
,
d
)-Gaussian random field. Xiao [2] and Biermé et al. [1] gave some remarks on the above conditions. We point out that the class of Gaussian random fields that satisfy conditions (C1) and (C2) is large. It includes not only the well-known fractional Brownian motion and the Brownian sheet, but also such anisotropic random fields as fractional Brownian sheets (cf. [3, 4, 7]), solutions to stochastic heat equation driven by space-time white noise (cf. [5, 6, 8–10]), and many more.In the following, we present some notations about several classes of functions satisfying certain conditions. The relationship between them will be studied in Section3.Let𝒞
=
{
f
:
f
is
a
continuous
function
on
⁡
ℝ
N
with
values
in
ℝ
d
}. As usual, a function f
∈
𝒞 is said to be a polar function for the random field X
(
t
) if
(3)
ℙ
{
∃
t
∈
ℝ
N
such  that
X
(
t
)
=
f
(
t
)
}
=
0
.
Let 𝒫 denote the collection of the continuous functions satisfying   (3).LetK
=
(
K
1
,
…
,
K
N
)
∈
(
0,1
)
N be a fixed vector, and let ℒ
(
K
) denote the collection of all Hölder continuous functions of any order less than K
ℓ along the ℓth direction in time; that is, there exists a finite and positive constant c
1.5, depending only on I and K
ℓ
(
0
≤
ℓ
≤
N
), such that for all 0
<
δ
ℓ
<
K
ℓ
(
0
≤
ℓ
≤
N
), s
,
t
∈
I, and f
∈
𝒞,
(4)
|
f
(
s
)
-
f
(
t
)
|
≤
c
1.5
∑
ℓ
=
1
N
|
s
ℓ
-
t
ℓ
|
K
ℓ
-
δ
ℓ
.Moreover, let𝒬
(
K
) denote the collection of all functions satisfying the following condition: there exist finite and positive constants c
1.6 and c
1.7, depending only on I and K
ℓ
(
0
≤
ℓ
≤
N
), such that for all s
,
t
∈
I and f
∈
𝒞,
(5)
c
1.6
∑
ℓ
=
1
N
|
s
ℓ
-
t
ℓ
|
K
ℓ
≤
|
f
(
s
)
-
f
(
t
)
|
≤
c
1.7
∑
ℓ
=
1
N
|
s
ℓ
-
t
ℓ
|
K
ℓ
.Note that ifK
=
(
α
,
…
,
α
), then the functions in ℒ
(
K
) are called Hölder continuous of any order less than α, and the functions in 𝒬
(
K
) are called quasi-spiral with order α; see Kahane [11]. Hence ℒ
(
K
) and 𝒬
(
K
) can be regarded as a nature generalization of Hölder continuous function and quasi-spiral, respectively.In the studies of random fields, it is interesting to consider the following questions.(i)
Given a nonrandom continuous functionf
∈
𝒞, when is it nonpolar for X in the sense that ℙ
{
∃
t
∈
ℝ
N
such
that
X
(
t
)
=
f
(
t
)
}
>
0? When is it polar for X in the sense that ℙ
{
∃
t
∈
ℝ
N
such
that
X
(
t
)
=
f
(
t
)
}
=
0?
(ii)
Given a nonrandom Borel setE
∈
ℝ
N, what is the probability for the random set {
∃
t
∈
I
such
that
X
(
t
)
=
f
(
t
)
}? What is the Hausdorff and packing dimensions of the set {
t
∈
ℝ
N
:
X
(
t
)
=
f
(
t
)
}  iffis nonpolar  f  or  X?The above questions are some important questions in fractal theory of random fields and the related results have only been known for a few types of random fields. For example, Graversen [12] studied the characteristics of the polar functions for the two dimensional Brownian motions. Le Gall [13] made a further discussion for the d-dimensional Brownian motion and proposed an open problem about the existence of its no-polar continuous function satisfying the Hölder condition. Some of these results have been extended partially to fractional Brownian motion with stationary increments by Xiao [14], to the Brownian sheet with independent increments by Chen [15], and recently to the fractional Brownian sheets with anisotropy by Chen [4].In all these papers, the isotropic properties of the Brownian sheet and fractional Brownian motion have played crucial roles. Since, in general, the anisotropic random fields have neither the isotropic properties nor the properties of independent increment and stationary increments due to their general dependence structure, it is more difficult to investigate fine properties of their sample paths. The main objective of this paper is to further investigate the characteristics of the polar functions and the intersection probabilities forX satisfying conditions (C1) and (C2) by using the approach of Biermé et al. in [1] and Xiao in [2]. Our main results, in some cases, strengthen the results in the aforementioned works, and their proofs are different from the proofs for the Brownian sheet and the fractional Brownian motion. Of particular significance, we determine the exact Hausdorff and packing dimensions of the times set for a nonpolar function intersecting X. However, for the intersection probability, we can only establish an inequality in terms of Hausdorff measure and capacity, respectively; see Theorem 16. It is still an open problem to prove the best upper bound in terms of capacity. We should also point out that, compared with the isotropic case, the anisotropic nature of X induces far richer fractal structure into the properties of the nonpolar functions for X.The rest of the paper is organized as follows. In Section2, we derive a few preliminary estimates and lemmas for X that will be useful to our arguments. In Section 3, we obtain the relationship between the class of continuous functions satisfying Lipschitz condition and the class of polar functions of X. We also give upper and lower bounds for the probabilities for a nonpolar function intersecting X and determine the Hausdorff and packing dimensions of the times points for a nonpolar function intersecting X. A question proposed by Le Gall [13] about the existence of no-polar, continuous Hölder functions for the Brownian motion is also solved. Finally in Section 4, we show that our main results in Section 3 can be applied to solutions to stochastic partial differential equations.Throughout this paper we will usec to denote unspecified positive and finite constant whose precise values are not important and may be different in each appearance. More specific constants in Section i are numbered as c
i
.
1
,
c
i
.
2
,
….


## 2. Some Preliminary Estimates

Because of the complex dependence structure for the anisotropic Gaussian random fields, the proofs of the main results in Sections3 and 4 are quite involved. Therefore, we split the proofs into several lemmas to be used in Sections 3 and 4.LetE be a compact set in ℝ
N. Cov
(
X
k
(
s
)
,
X
k
(
t
)
) denotes the covariance matrix of the random vector (
X
k
(
s
)
,
X
k
(
t
)
). Then, for all s
,
t
∈
E,
(6)
det Cov
(
X
k
(
s
)
,
X
k
(
t
)
)
=
𝔼
[
X
k
2
(
s
)
]
𝔼
[
X
k
2
(
t
)
]
-
[
𝔼
(
X
k
(
s
)
X
k
(
t
)
)
]
2
,
where k
=
1
,
…
,
d.We need to estimate upper and lower bounds of the covariance determinant in (6). For the sake of completeness, we provide a simple proof by using the expression for the characteristic functions and the density functions of Gaussian random fields.Lemma 1.
LetX
=
{
X
(
t
)
,
t
∈
ℝ
N
} be an (
N
,
d
)-Gaussian random field satisfying conditions (C1) and (C2) and let E be a compact set on ℝ
N. Then there exist positive constants c
2.1 and c
2.2, such that for all s
=
(
s
1
,
…
,
s
N
),  t
=
(
t
1
,
…
,
t
N
)
∈
E,
(7)
c
2.1
∑
ℓ
=
1
N
|
s
ℓ
-
t
ℓ
|
2
H
ℓ
≤
det
⁡
⁡
Cov
(
X
k
(
s
)
,
X
k
(
t
)
)
≤
c
2.2
∑
ℓ
=
1
N
|
s
ℓ
-
t
ℓ
|
2
H
ℓ
,
1
≤
k
≤
d
.Proof.
SinceE is a compact set in ℝ
N, then there exists a positive constant c, such that E
⊂
[
-
c
,
c
]
N. In order to prove (7), it suffices to show that (7) holds for all s
,
t
∈
E with s
≠
t. We claim that for all, s
,
t
∈
E with s
≠
t,
(8)
detCov
(
X
k
(
s
)
,
X
k
(
t
)
)
=
Var
⁡
(
X
k
(
t
)
)
Var
⁡
(
X
k
(
s
)
∣
X
k
(
t
)
)
.
If detCov
(
X
k
(
s
)
,
X
k
(
t
)
)
>
0, then by using the expression for the characteristic functions and the density functions of Gaussian random fields, it turns out that
(9)
2
π
detCov
(
X
k
(
s
)
,
X
k
(
t
)
)
=
∫
ℝ
2
𝔼
exp
⁡
⁡
(
-
i
(
u
X
k
(
s
)
+
v
X
k
(
t
)
)
)
d
u
d
v
.
By applying the fact that the conditional distribution of X
k
(
s
) given X
k
(
t
) is still Gaussian with mean 𝔼
(
X
k
(
s
)
∣
X
k
(
t
)
) and variance Var
⁡
(
X
k
(
s
)
∣
X
k
(
t
)
), one can evaluate the integral in the right-hand side of (9) and thus deduce that (8) holds.
IfdetCov
(
X
k
(
s
)
,
X
k
(
t
)
)
=
0, then we can deduce that the related coefficient of X
(
s
) and X
(
t
) is equal to 0, so there exists λ
∈
ℝ such that X
(
s
)
=
λ
X
(
t
) a.s., and, in particular, a simply estimation implies that (8) still holds in this case.
We now prove the upper bound in (7). Note that (
X
k
(
s
)
,
X
k
(
t
)
) is a mean zero Gaussian vector. Since t
↦
Var
⁡
(
X
k
(
t
)
) is a positive continuous function on E, then there exists a positive constant c
2.3 such that, for all t
∈
E,
(10)
c
1.1
≤
Var
⁡
(
X
k
(
t
)
)
≤
c
2.3
.
This, together with (1), (2), and (8), implies that the upper bound in (7) holds. The lower bound in (7) follows from (2), (8), and (10). This completes the proof of Lemma 1.Similar to the argument of Testard in [16], we will provide a proof of the following lemma.Lemma 2.
LetX
=
{
X
(
t
)
,
t
∈
ℝ
N
} be an (
N
,
d
)-Gaussian random field satisfying conditions (C1) and (C2) and let E be a compact set on ℝ
N. Then there exist positive constants c
2.4 and c
2.5, such that, for all s
,
t
∈
E, we have
(11)
c
2.4
(
f
k
(
s
)
-
f
k
(
t
)
)
2
≤
𝔼
[
f
k
(
t
)
X
k
(
s
)
-
f
k
(
s
)
X
k
(
t
)
]
2
≤
c
2.5
(
f
k
(
s
)
-
f
k
(
t
)
)
2
,
where f
∈
𝒞,  k
=
1
,
…
,
d.Proof.
SinceE is a compact set on ℝ
N, then there exists a positive constant c, such that E
⊂
[
-
c
,
c
]
N. As usual, the proof is divided into proving the lower and upper bounds separately. We first prove the lower bound in (11). By (1) and (7), we have
(12)
detCov
(
X
k
(
s
)
,
X
k
(
t
)
)
=
𝔼
[
X
k
2
(
s
)
]
𝔼
[
X
k
2
(
t
)
]
-
[
𝔼
(
X
k
(
s
)
X
k
(
t
)
)
]
2
≥
c
2.1
∑
ℓ
=
1
N
|
s
ℓ
-
t
ℓ
|
2
H
ℓ
≥
c
2.1
c
1.3
𝔼
[
X
k
(
s
)
-
X
k
(
t
)
]
2
.
By taking c
2.4
=
min
⁡
{
c
1.1
,
c
2.1
/
c
1.3
}, then for all s
,
t
∈
E,
(13)
𝔼
[
X
k
2
(
s
)
]
-
c
2.4
≥
0
,
𝔼
[
X
k
2
(
t
)
]
-
c
2.4
≥
0
.
It follows from (12) and (13) that
(14)
|
𝔼
[
X
k
(
s
)
X
k
(
t
)
]
-
c
2.4
|
≤
(
𝔼
[
X
k
2
(
s
)
]
-
c
2.4
)
1
/
2
(
𝔼
[
X
k
2
(
t
)
]
-
c
2.4
)
1
/
2
.
Note that
(15)
𝔼
[
f
k
(
t
)
X
k
(
s
)
-
f
k
(
s
)
X
k
(
t
)
]
2
-
c
2.4
(
f
k
(
s
)
-
f
k
(
t
)
)
2
=
f
k
2
(
t
)
(
𝔼
[
X
k
2
(
s
)
]
-
c
2.4
)
+
f
k
2
(
s
)
(
𝔼
[
X
k
2
(
t
)
]
-
c
2.4
)
-
2
f
k
(
s
)
f
k
(
t
)
(
𝔼
[
X
k
(
s
)
X
k
(
t
)
]
-
c
2.4
)
.
Then inequalities (14) and (15) imply
(16)
𝔼
[
f
k
(
t
)
X
k
(
s
)
-
f
k
(
s
)
X
k
(
t
)
]
2
-
c
2.4
(
f
k
(
s
)
-
f
k
(
t
)
)
2
≥
f
k
2
(
t
)
(
𝔼
[
X
k
2
(
s
)
]
-
c
2.4
)
+
f
k
2
(
s
)
(
𝔼
[
X
k
2
(
t
)
]
-
c
2.4
)
-
2
|
f
k
(
s
)
|
|
f
k
(
t
)
|
(
𝔼
[
X
k
2
(
s
)
]
-
c
2.4
)
1
/
2
×
(
𝔼
[
X
k
2
(
t
)
]
-
c
2.4
)
1
/
2
=
[
|
f
k
(
t
)
|
(
𝔼
[
X
k
2
(
s
)
]
-
c
2.4
)
1
/
2
-
|
f
k
(
s
)
|
(
𝔼
[
X
k
2
(
t
)
]
-
c
2.4
)
1
/
2
]
2
≥
0
.
Now we prove the upper bound in (11). By using (1) and (7) and repeating the procedure in (12), we can derive
(17)
𝔼
[
X
k
2
(
s
)
]
𝔼
[
X
k
2
(
t
)
]
-
[
𝔼
(
X
k
(
s
)
X
k
(
t
)
)
]
2
≤
c
2.2
c
1.2
𝔼
[
X
k
(
s
)
-
X
k
(
t
)
]
2
.
By taking c
2.5
=
max
⁡
{
c
2.3
,
c
2.2
/
c
1.2
}, then for all  s
,
t
∈
E,
(18)
c
2.5
-
𝔼
[
X
k
2
(
s
)
]
≥
0
,
c
2.5
-
𝔼
[
X
k
2
(
t
)
]
≥
0
.
It follows from (17) and (18) that
(19)
|
c
2.5
-
𝔼
[
X
k
(
s
)
X
k
(
t
)
]
|
≥
(
c
2.5
-
𝔼
[
X
k
2
(
s
)
]
)
1
/
2
(
c
2.5
-
𝔼
[
X
k
2
(
t
)
]
)
1
/
2
.
Combining (15), (18), and (19), we obtain
(20)
𝔼
[
f
k
(
t
)
X
k
(
s
)
-
f
k
(
s
)
X
k
(
t
)
]
2
-
c
2.5
(
f
k
(
s
)
-
f
k
(
t
)
)
2
≤
-
f
k
2
(
t
)
(
c
2.5
-
𝔼
[
X
k
2
(
s
)
]
)
-
f
k
2
(
s
)
(
c
2.5
-
𝔼
[
X
k
2
(
t
)
]
)
+
2
|
f
k
(
s
)
|
|
f
k
(
t
)
|
(
c
2.5
-
𝔼
[
X
k
2
(
s
)
]
)
1
/
2
×
(
c
2.5
-
𝔼
[
X
k
2
(
t
)
]
)
1
/
2
=
-
[
|
f
k
(
t
)
|
(
𝔼
[
X
k
2
(
s
)
]
-
c
2.4
)
1
/
2
-
|
f
k
(
s
)
|
(
𝔼
[
X
k
2
(
t
)
]
-
c
2.4
)
1
/
2
]
2
≤
0
.
By inequalities (16) and (20), we finish the proof of Lemma 2.Letρ be a metric on ℝ
N defined by
(21)
ρ
(
s
,
t
)
=
∑
ℓ
=
1
N
|
s
ℓ
-
t
ℓ
|
H
ℓ
∧
K
ℓ
,
∀
s
,
t
∈
ℝ
N
.In the following, we will provide a slightly more general result in the proof of Proposition 4.4 by modifying the argument [8].Lemma 3.
LetX
=
{
X
(
t
)
,
t
∈
ℝ
N
} be an (
N
,
d
)-Gaussian random field satisfying conditions (C1) and (C2). Then there exist positive constants c
2.6 and δ, such that, for all r
∈
(
0
,
δ
), s
∈
E, and all f
∈
ℒ
(
K
),
(22)
ℙ
{
inf
⁡
t
∈
B
ρ
(
s
,
r
)
⁡
|
X
(
t
)
-
f
(
t
)
|
<
r
}
≤
c
2.6
r
d
,
where B
ρ
(
s
,
r
) denotes the ball of radius r centered at s in the metric ρ defined by (21).Proof.
Using the Gaussian regressions, we have(23)
𝔼
(
X
k
(
t
)
∣
X
k
(
s
)
)
=
𝔼
[
X
k
(
s
)
X
k
(
t
)
]
𝔼
[
X
k
(
s
)
]
2
X
k
(
s
)
=
^
k
(
s
,
t
)
X
k
(
s
)
.
Note that, for all t
∈
E, the Gaussian random variables X
k
(
t
)
-
k
(
s
,
t
)
X
k
(
s
)  (
s
∈
E
) and X
k
(
s
) are independent. By using the triangle inequality, we can deduce that, for all 1
≤
k
≤
d,
(24)
{
inf
⁡
t
∈
B
ρ
(
s
,
r
)
⁡
|
X
k
(
t
)
-
f
k
(
t
)
|
≥
r
}
⊇
{
inf
⁡
t
∈
B
ρ
(
s
,
r
)
⁡
|
k
(
s
,
t
)
(
X
k
(
s
)
-
f
k
(
s
)
)
|
≥
2
r
}
∩
{
inf
⁡
t
∈
B
ρ
(
s
,
r
)
⁡
|
k
(
s
,
t
)
(
X
k
(
s
)
-
f
k
(
s
)
)
|
2
Z
k
(
s
,
x
,
r
)
≤
inf
⁡
t
∈
B
ρ
(
s
,
r
)
⁡
|
k
(
s
,
t
)
(
X
k
(
s
)
-
f
k
(
s
)
)
|
}
,
where Z
k
(
s
,
r
)
=
sup
⁡
t
∈
B
ρ
(
s
,
r
)
⁡
|
(
X
k
(
t
)
-
f
k
(
t
)
)
-
k
(
s
,
t
)
(
X
k
(
s
)
-
f
k
(
s
)
)
|. Then
(25)
ℙ
{
inf
⁡
y
∈
B
ρ
(
s
,
r
)
⁡
|
X
k
(
t
)
-
f
k
(
t
)
|
<
r
}
≤
ℙ
{
inf
⁡
t
∈
B
ρ
(
s
,
r
)
⁡
|
k
(
s
,
t
)
(
X
k
(
s
)
-
f
k
(
s
)
)
|
<
2
r
}
+
ℙ
{
2
Z
k
(
s
,
r
)
>
inf
⁡
t
∈
B
ρ
(
s
,
r
)
⁡
|
k
(
s
,
t
)
(
X
k
(
s
)
-
f
k
(
s
)
)
|
}
.
By the Cauchy-Schwarz inequality, (1), and (23), we have
(26)
|
1
-
k
(
s
,
t
)
|
=
|
𝔼
[
X
k
(
s
)
(
X
k
(
s
)
-
X
k
(
t
)
)
]
|
𝔼
[
X
k
(
s
)
]
2
≤
(
c
1.3
c
1.1
∑
ℓ
=
1
N
|
s
ℓ
-
t
ℓ
|
2
H
ℓ
)
1
/
2
.
Therefore, there exists a positive constant δ such that, for all r
∈
(
0
,
δ
) and t
∈
B
ρ
(
s
,
r
), we can deduce that 1
/
2
≤
k
(
s
,
t
)
≤
3
/
2. Recall that, for the unimodality of the centered Gaussian process k
(
s
,
t
)
X
k
(
s
), we have
(27)
ℙ
{
inf
⁡
t
∈
B
ρ
(
s
,
r
)
⁡
|
k
(
s
,
t
)
(
X
k
(
s
)
-
f
k
(
s
)
)
|
<
2
r
}
≤
ℙ
{
|
X
k
(
s
)
-
f
k
(
s
)
|
<
4
r
}
≤
ℙ
{
|
X
k
(
s
)
|
<
4
r
}
≤
c
2.7
r
.
Note that Z
k
(
s
,
r
) and k
(
s
,
t
)
X
k
(
s
) are independent. It follows from (27) that
(28)
ℙ
{
2
Z
k
(
s
,
r
)
>
inf
⁡
t
∈
B
ρ
(
s
,
r
)
⁡
|
k
(
s
,
t
)
(
X
k
(
s
)
-
f
k
(
s
)
)
|
}
=
∫
0
∞
ℙ
{
|
k
(
s
,
t
)
(
X
k
(
s
)
-
f
k
(
s
)
)
|
<
4
y
∣
Z
k
(
s
,
r
)
=
y
}
×
ℙ
{
Z
k
(
s
,
r
)
∈
d
y
}
=
c
2.7
∫
0
∞
y
ℙ
{
Z
k
(
s
,
r
)
∈
d
y
}
≤
c
2.7
𝔼
[
Z
k
(
s
,
r
)
]
.
In order to estimate 𝔼
[
Z
k
(
s
,
r
)
], we denote that the Gaussian process Y
k
(
t
)
=
X
k
(
t
)
-
f
k
(
t
)  −  k
(
s
,
t
)
(
X
k
(
s
)
-
f
k
(
s
)
)  (
t
∈
B
ρ
(
s
,
r
)
) and note that Y
k
(
s
)
=
0 and the canonical metric
(29)
d
(
t
,
t
′
)
=
^
[
𝔼
(
Y
k
(
t
)
-
Y
k
(
t
′
)
)
2
]
1
/
2
,
∀
t
,
t
′
∈
B
ρ
(
s
,
r
)
.
Therefore, by the Hölder inequality and the Cauchy-Schwarz inequality, we have
(30)
d
2
(
t
,
t
′
)
≤
c
2.8
(
𝔼
(
X
k
(
t
)
-
X
k
(
t
′
)
)
2
+
(
f
k
(
t
)
-
f
k
(
t
′
)
)
2
+
(
k
(
s
,
t
)
-
k
(
s
,
t
′
)
)
2
×
(
𝔼
[
X
k
2
(
s
)
]
+
f
k
2
(
s
)
)
)
,
(
k
(
s
,
t
)
-
k
(
s
,
t
′
)
)
2
=
[
𝔼
[
X
k
(
s
)
(
X
k
(
t
)
-
X
k
(
t
′
)
)
]
]
2
[
𝔼
[
X
k
(
s
)
]
2
]
2
≤
c
1.3
c
1.1
∑
ℓ
=
1
N
|
t
ℓ
-
t
ℓ
′
|
2
H
ℓ
.
By using (1), (10), (30), and the fact that f
∈
𝒬
(
K
), we have
(31)
d
(
t
,
t
′
)
≤
c
2.9
∑
ℓ
=
1
N
|
t
ℓ
-
t
ℓ
′
|
H
ℓ
∧
K
ℓ
=
c
2.9
ρ
(
t
,
t
′
)
≤
c
2.9
r
,
∀
t
,
t
′
∈
B
ρ
(
s
,
r
)
.
Then
(32)
D
=
^
sup
⁡
t
,
t
′
∈
B
ρ
(
s
,
r
)
⁡
d
(
t
,
t
′
)
≤
c
2.9
r
,
N
d
(
B
ρ
(
s
,
r
)
,
ɛ
)
≤
c
2.10
(
r
ɛ
)
Q
,
where N
d
(
B
ρ
(
s
,
r
)
,
ɛ
) is the metric entropy number of B
ρ
(
s
,
r
) and Q
=
∑
ℓ
=
1
N
‍
(
1
/
(
H
ℓ
∧
K
ℓ
)
). It follows from Dudley’s theorem of Kahane [11] that
(33)
𝔼
[
Z
k
(
s
,
r
)
]
≤
c
2.11
∫
0
D
log
⁡
⁡
N
d
(
B
ρ
(
s
,
r
)
,
ɛ
)
d
ɛ
≤
c
2.12
r
.
Combining (25), (27), (28), and (33) and using the coordinate processes independence of X, we have
(34)
ℙ
{
inf
⁡
t
∈
B
ρ
(
s
,
r
)
⁡
|
X
(
t
)
-
f
(
t
)
|
<
r
}
≤
∏
k
=
1
d
‍
ℙ
{
inf
⁡
t
∈
B
ρ
(
s
,
r
)
⁡
|
X
k
(
t
)
-
f
k
(
t
)
|
≤
r
}
≤
c
2.6
r
d
.
This finishes the proof of Lemma 3.Lemma 4.
LetX
=
{
X
(
t
)
,
t
∈
ℝ
N
} be an (
N
,
d
)-Gaussian random field satisfying conditions (C1) and (C2) and let E be a compact set on ℝ
N. Then there exists a positive constant c
2.13 such that, for all f
∈
𝒬
(
K
), ɛ
>
0 and s
,
t
∈
E,
(35)
𝔼
(
2
π
ɛ
)
d
exp
⁡
⁡
(
-
|
X
(
s
)
-
f
(
s
)
|
2
+
|
X
(
t
)
-
f
(
t
)
|
2
2
ɛ
)
≤
c
2.13
(
∑
ℓ
=
1
N
|
s
ℓ
-
t
ℓ
|
H
ℓ
∧
K
ℓ
)
-
d
.Proof.
Note that(36)
(
2
π
ɛ
)
d
/
2
exp
⁡
⁡
(
-
|
X
(
t
)
-
f
(
t
)
|
2
2
ɛ
)
=
∫
ℝ
d
exp
⁡
⁡
(
-
ɛ
2
|
u
|
2
+
i
〈
u
,
X
(
t
)
-
f
(
t
)
〉
)
d
u
=
∏
k
=
1
d
‍
∫
ℝ
exp
⁡
⁡
(
-
ɛ
2
|
u
k
|
2
+
i
〈
u
k
,
X
k
(
t
)
-
f
k
(
t
)
〉
)
d
u
k
.
Denote by I
2 the identity matrix of order 2 and let Γ
k
(
s
,
t
)
=
ɛ
I
2
+
Cov
(
X
k
(
s
)
,
X
k
(
t
)
). Then the inverse of Γ
k
(
s
,
t
) is given by
(37)
Γ
k
-
1
(
s
,
t
)
=
1
det
⁡
Γ
k
(
s
,
t
)
×
(
ɛ
+
𝔼
[
X
k
2
(
t
)
]
-
𝔼
(
X
k
(
s
)
X
k
(
t
)
)
-
𝔼
(
X
k
(
s
)
X
k
(
t
)
)
ɛ
+
𝔼
[
X
k
2
(
s
)
]
)
,
where det
⁡
Γ
k
(
s
,
t
) denotes the determinant of Γ
k
(
s
,
t
).
By (36), Lemma 2, Fubini’s theorem, and some elementary calculations, we derive
(38)
𝔼
(
2
π
ɛ
)
d
exp
⁡
⁡
(
-
|
X
(
s
)
-
f
(
s
)
|
2
+
|
X
(
t
)
-
f
(
t
)
|
2
2
ɛ
)
=
∏
k
=
1
d
‍
∫
ℝ
∫
ℝ
exp
⁡
⁡
(
-
i
(
〈
u
k
,
f
k
(
s
)
〉
+
〈
v
k
,
f
k
(
t
)
〉
)
)
·
exp
⁡
⁡
(
-
ɛ
2
(
|
u
k
|
2
+
|
v
k
|
2
)
)
×
𝔼
exp
⁡
⁡
(
i
〈
u
k
,
X
k
(
s
)
〉
+
i
〈
v
k
,
X
k
(
t
)
〉
)
d
u
k
d
v
k
=
∏
k
=
1
d
‍
∫
ℝ
∫
ℝ
exp
⁡
⁡
(
-
i
(
〈
u
k
,
f
k
(
s
)
〉
+
〈
v
k
,
f
k
(
t
)
〉
)
)
×
exp
⁡
⁡
(
-
1
2
(
u
k
,
v
k
)
Γ
k
(
s
,
t
)
(
u
k
,
v
k
)
′
)
d
u
k
d
v
k
=
∏
k
=
1
d
‍
2
π
det
⁡
Γ
k
(
s
,
t
)
×
exp
⁡
⁡
(
-
1
2
(
f
k
(
s
)
,
f
k
(
t
)
)
Γ
k
-
1
(
s
,
t
)
(
f
k
(
s
)
,
f
k
(
t
)
)
′
)
=
∏
k
=
1
d
‍
2
π
det
⁡
Γ
k
(
s
,
t
)
exp
⁡
⁡
(
-
f
k
2
(
s
)
+
f
k
2
(
t
)
2
ɛ
det
⁡
Γ
k
(
s
,
t
)
)
·
exp
⁡
⁡
(
-
𝔼
(
f
k
(
t
)
X
k
(
s
)
-
f
k
(
s
)
X
k
(
t
)
)
2
2
det
⁡
Γ
k
(
s
,
t
)
)
≤
∏
k
=
1
d
‍
2
π
det
⁡
Γ
k
(
s
,
t
)
×
exp
⁡
(
-
c
2.4
|
f
k
(
s
)
-
f
k
(
t
)
|
2
2
det
⁡
Γ
k
(
s
,
t
)
)
.
If det
⁡
Γ
k
(
s
,
t
)
≥
|
f
k
(
s
)
-
f
k
(
t
)
|
2, then
(39)
1
det
⁡
Γ
k
(
s
,
t
)
exp
⁡
⁡
(
-
c
2.4
|
f
k
(
s
)
-
f
k
(
t
)
|
2
2
det
⁡
Γ
k
(
s
,
t
)
)
≤
1
max
⁡
⁡
{
det
⁡
Γ
k
(
s
,
t
)
,
|
f
k
(
s
)
-
f
k
(
t
)
|
}
.
For all x
≥
0, we can deduce
(40)
exp
⁡
⁡
(
-
c
2.4
2
x
)
≤
(
e
c
2.4
)
-
1
/
2
x
-
1
/
2
.
If det
⁡
Γ
k
(
s
,
t
)
<
|
f
k
(
s
)
-
f
k
(
t
)
|
2, by the inequality above and taking c
2.14
=
(
e
c
2.4
)
-
1
/
2 and x
=
|
f
k
(
s
)
-
f
k
(
t
)
|
2
/
det
⁡
Γ
k
(
s
,
t
), we have
(41)
1
det
⁡
⁡
Γ
k
(
s
,
t
)
exp
⁡
⁡
(
-
c
2.4
|
f
k
(
s
)
-
f
k
(
t
)
|
2
2
det
⁡
⁡
Γ
k
(
s
,
t
)
)
≤
c
2.14
max
⁡
⁡
{
det
⁡
Γ
k
(
s
,
t
)
,
|
f
k
(
s
)
-
f
k
(
t
)
|
}
.
It follows from Lemma 1 that, for all s
,
t
∈
E,
(42)
det
⁡
⁡
Γ
k
(
s
,
t
)
≥
det
⁡
⁡
Cov
(
X
k
(
s
)
,
X
k
(
t
)
)
≥
c
2.15
(
∑
ℓ
=
1
N
|
s
ℓ
-
t
ℓ
|
H
ℓ
)
2
.
By using (42) and the fact that f
∈
𝒬
(
K
), we have
(43)
max
⁡
⁡
{
det
⁡
Γ
k
(
s
,
t
)
,
|
f
k
(
s
)
-
f
k
(
t
)
|
}
≥
c
2.16
∑
ℓ
=
1
N
|
s
ℓ
-
t
ℓ
|
H
ℓ
∧
K
ℓ
.
Combining (36) through (43), we prove that Lemma 4 holds.For proving the lower bound in Theorem11, we will use two lemmas below, which are slightly more general results, by modifying the argument in [3, 17].Lemma 5.
Let0
<
α
<
1, b
1
>
a
1 and b
2
>
a
2 be given constants. Then for all constants p, M
>
0 and δ
>
2
α, there exists a positive and finite constant c
2.17, depending on a
1,  b
1,  a
2,  b
2,  δ,  p,  α, and M only, such that, for all 0
<
A
≤
M,
(44)
ℐ
(
A
)
=
^
∫
a
1
b
1
d
s
∫
a
2
b
2
(
A
+
|
s
-
t
|
2
α
)
-
p
d
t
≤
c
2.17
(
A
-
(
p
-
1
/
δ
)
+
1
)
.Proof.
Leta
=
min
⁡
{
a
1
,
a
2
} and b
=
max
⁡
{
b
1
,
b
2
}. By the symmetry of the integrand, we get
(45)
ℐ
(
A
)
≤
2
∫
a
b
d
s
∫
s
b
(
A
+
(
t
-
s
)
2
α
)
-
p
d
t
.
Putting u
=
t
-
s and using the fact that a
≤
s
≤
t
≤
b, we see that the above integral is bound by
(46)
2
∫
a
b
d
s
∫
0
b
-
s
(
A
+
u
2
α
)
-
p
d
u
≤
2
(
b
-
a
)
∫
0
b
-
a
(
A
+
u
2
α
)
-
p
d
u
=
b
-
a
α
∫
A
A
+
(
b
-
a
)
2
α
v
-
p
(
v
-
A
)
(
1
-
2
α
)
/
2
α
d
v
,
where we have used the substitution v
=
A
+
u
2
α.
LetB
=
^
A
+
(
b
-
a
)
2
α. If α
>
1
/
2, then for δ
>
2
α, it follows from (46) and Hölder's inequality that there exists a positive and finite constant c
2.18, which depends on a
1,  b
1,  a
2,  b
2,  δ,  p,  α, and M only, such that
(47)
ℐ
(
A
)
≤
b
-
a
α
(
∫
A
B
v
-
p
δ
d
v
)
1
/
δ
×
(
∫
A
B
(
v
-
A
)
-
(
(
2
α
-
1
)
/
2
α
)
(
δ
/
(
δ
-
1
)
)
d
v
)
(
δ
-
1
)
/
δ
≤
c
2.18
(
A
-
(
p
-
1
/
δ
)
+
1
)
,
where we have used the fact that (
(
2
α
-
1
)
/
2
α
)
(
δ
/
(
δ
-
1
)
)
<
1.
If0
<
α
≤
1
/
2, then some elementary calculations imply that, for all 0
<
A
≤
M,
(48)
ℐ
(
A
)
≤
b
-
a
α
∫
A
B
v
-
p
(
v
-
A
)
(
1
-
2
α
)
/
2
α
d
v
≤
b
-
a
α
∫
A
B
v
-
p
-
1
+
1
/
2
α
d
v
≤
c
2.19
(
A
-
(
p
-
1
/
δ
)
+
1
)
,
where c
2.19 depends on a
1,  b
1,  a
2,  b
2,  δ,  p,  α, and M only. By (47) and (48), the proof of Lemma 5 is finished.Lemma 6.
Letα,  β,  η, and b be positive constants. For A
>
0 and B
>
0, let
(49)
𝒥
(
A
,
B
)
=
^
∫
0
b
‍
d
t
(
A
+
t
α
)
β
(
B
+
t
)
η
.
Then there exist positive and finite constants c
2.20 and c
2.21, depending on α,  β,  η, and b only, such that the following hold for all reals A
,
B
>
0 satisfying A
1
/
α
≤
c
2.20
B:(i)
ifα
β
>
1, then
(50)
𝒥
(
A
,
B
)
≤
c
2.21
1
A
β
-
α
-
1
B
η
;
(ii)
ifα
β
=
1, then
(51)
𝒥
(
A
,
B
)
≤
c
2.21
1
B
η
log
⁡
⁡
(
1
+
B
A
-
1
/
α
)
;
(iii)
if0
<
α
β
<
1 and α
β
+
η
≠
1, then
(52)
𝒥
(
A
,
B
)
≤
c
2.21
(
1
B
α
β
+
η
-
1
+
1
)
.Proof.
Ifb
≤
1, by using Lemma 10 in [3], we can prove that inequalities (50), (51), and (52) hold. If b
>
1, then we can split the integral in (49) so that
(53)
𝒥
(
A
,
B
)
=
∫
0
1
d
t
(
A
+
t
α
)
β
(
B
+
t
)
η
+
∫
1
b
d
t
(
A
+
t
α
)
β
(
B
+
t
)
η
=
^
ℐ
1
+
ℐ
2
.
Let t
=
b
s. Since b
>
1 and α,  β, and η are positive constants, we get
(54)
ℐ
2
=
b
∫
1
/
b
1
d
s
(
A
+
(
b
s
)
α
)
β
(
B
+
b
s
)
η
≤
b
∫
0
1
d
s
(
A
+
s
α
)
β
(
B
+
s
)
η
.
By using (53), (54), and Lemma 10 in [3] again, we can also prove (50), (51), and (52); in this case b
>
1. Thus, the proof of Lemma 6 is finished.LetH
=
(
H
1
,
…
,
H
N
)
∈
(
0,1
)
N and K
=
(
K
1
,
…
,
K
N
)
∈
(
0,1
)
N be given vectors. For convenience, we may further assume
(55)
0
<
H
1
K
1
≤
H
2
K
2
≤
⋯
≤
H
N
K
N
<
1
.Lemma 7.
LetX
=
{
X
(
t
)
,
t
∈
ℝ
N
} be an (
N
,
d
)-Gaussian random field satisfying conditions (C1) and (C2). If ∑
ℓ
=
1
N
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
>
d, then there exists a positive and finite constant c
2.22, depending on a,  b,  H,  K,  N, and d only, such that, for all f
∈
𝒬
(
K
),
(56)
∫
I
∫
I
∏
k
=
1
d
[
|
f
k
(
s
)
-
f
k
(
t
)
|
2
{
det
⁡
⁡
Cov
(
X
k
(
s
)
,
X
k
(
t
)
)
,
}
max
⁡
⁡
{
|
f
k
(
s
)
-
f
k
(
t
)
|
2
det
⁡
⁡
Cov
⁡
(
X
k
(
s
)
,
X
k
(
t
)
)
,
|
f
k
(
s
)
-
f
k
(
t
)
|
2
{
det
⁡
⁡
Cov
(
X
k
(
s
)
,
X
k
(
t
)
)
,
}
]
-
1
/
2
d
t
d
s
≤
c
2.22
.Proof.
Note thatf
∈
𝒬
(
K
). Then, by using Lemma 1, we have
(57)
ℐ
=
^
∫
I
∫
I
∏
k
=
1
d
[
|
f
k
(
s
)
-
f
k
(
t
)
|
2
}
max
⁡
⁡
{
|
f
k
(
s
)
-
f
k
(
t
)
|
2
det
⁡
⁡
Cov
⁡
(
X
k
(
s
)
,
X
k
(
t
)
)
,
|
f
k
(
s
)
-
f
k
(
t
)
|
2
}
]
-
1
/
2
d
t
d
s
≤
c
2.23
∫
I
∫
I
[
∑
ℓ
=
1
N
|
s
ℓ
-
t
ℓ
|
H
ℓ
∧
K
ℓ
]
-
d
d
t
d
s
.
Let k
∈
{
1
,
…
,
N
} be the unique positive integer such that
(58)
∑
ℓ
=
1
k
-
1
1
H
ℓ
∧
K
ℓ
≤
d
<
∑
ℓ
=
1
k
1
H
ℓ
∧
K
ℓ
,
∑
ℓ
=
1
0
1
H
ℓ
∧
K
ℓ
=
^
0
.
Then, we choose positive constants δ
1
,
…
,
δ
N
-
1 such that δ
ℓ
>
H
ℓ
∧
K
ℓ for each 1
≤
ℓ
≤
N
-
1 and
(59)
1
δ
1
+
1
δ
2
+
⋯
+
1
δ
N
-
1
<
d
<
1
δ
1
+
1
δ
2
+
⋯
+
1
δ
N
-
1
+
1
H
N
∧
K
N
.
Applying Lemma 5 to (57) with
(60)
A
=
∑
ℓ
=
2
N
|
s
ℓ
-
t
ℓ
|
H
ℓ
∧
K
ℓ
,
p
=
d
,
we obtain that
(61)
ℐ
≤
c
2.24
+
c
2.24
∫
a
1
b
1
d
s
2
∫
a
2
b
2
d
t
2
⋯
∫
a
N
b
N
d
s
N
×
∫
a
N
b
N
[
∑
ℓ
=
2
N
|
s
ℓ
-
t
ℓ
|
H
ℓ
∧
K
ℓ
]
-
(
d
-
1
/
δ
1
)
d
t
N
.
By repeatedly using Lemma 5 to the integral in above inequality for N
-
2 steps, we have
(62)
ℐ
≤
c
2.25
+
c
2.25
∫
a
N
b
N
d
s
N
×
∫
a
N
b
N
[
|
s
N
-
t
N
|
(
H
N
∧
K
N
)
]
-
(
d
-
(
1
/
δ
1
+
⋯
+
1
/
δ
N
-
1
)
)
d
t
N
.
Since the δ
′
s satisfy (59), we have
(63)
(
(
H
N
∧
K
N
)
)
(
d
-
(
1
δ
1
+
⋯
+
1
δ
N
-
1
)
)
<
1
.
Thus, the integral in the right-hand side of (62) is finite. This completes the proof of Lemma 7.Lemma 8.
LetX
=
{
X
(
t
)
,
t
∈
ℝ
N
} be an (
N
,
d
)-Gaussian random field satisfying conditions (C1) and (C2). If ∑
ℓ
=
1
N
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
>
d, then there exist positive and finite constants δ
0 and c
2.26, such that for all f
∈
𝒬
(
K
),  δ
∈
(
0
,
δ
0
),
(64)
∫
I
∫
I
|
t
-
s
|
-
γ
∏
k
=
1
d
[
|
f
k
(
s
)
-
f
k
(
t
)
|
2
{
det
⁡
⁡
Cov
(
X
k
(
s
)
,
X
k
(
t
)
)
,
}
max
⁡
⁡
{
|
f
k
(
s
)
-
f
k
(
t
)
|
2
det
⁡
⁡
Cov
(
X
k
(
s
)
,
X
k
(
t
)
)
,
|
f
k
(
s
)
-
f
k
(
t
)
|
2
{
det
⁡
⁡
Cov
(
X
k
(
s
)
,
X
k
(
t
)
)
,
}
]
-
1
/
2
d
t
d
s
≤
c
2.26
,
where γ
=
min
⁡
1
≤
k
≤
N
⁡
{
∑
ℓ
=
1
k
(
(
H
k
∧
K
k
)
/
(
H
ℓ
∧
K
ℓ
)
)
+
N
-
k
-
(
H
k
∧
K
k
)
(
1
+
δ
)
d
} and c
2.26 depends on a,  b,  H,  K,  δ
0,  N, and d only.Proof.
For our purpose, let us note that (55) implies
(65)
∑
ℓ
=
1
k
H
k
∧
K
k
H
ℓ
∧
K
ℓ
+
N
-
k
-
(
H
k
∧
K
k
)
d
=
(
H
k
∧
K
k
)
(
(
∑
ℓ
=
1
N
1
H
ℓ
∧
K
ℓ
-
d
)
+
∑
ℓ
=
k
+
1
N
(
1
H
k
∧
K
k
-
1
H
ℓ
∧
K
ℓ
)
)
>
0
.
Then, there exist δ
0
>
0 such that for all k
∈
{
1
,
…
,
N
} and δ
∈
(
0
,
δ
0
), we have γ
>
0. By using Lemma 1 and f
∈
𝒬
(
K
), we have
(66)
𝒥
=
^
∫
I
∫
I
|
t
-
s
|
-
γ
×
∏
k
=
1
d
[
|
f
k
(
s
)
-
f
k
(
t
)
|
2
detCov
(
X
k
(
s
)
,
X
k
(
t
)
)
}
max
⁡
⁡
{
|
f
k
(
s
)
-
f
k
(
t
)
|
2
detCov
(
X
k
(
s
)
,
X
k
(
t
)
)
,
|
f
k
(
s
)
-
f
k
(
t
)
|
2
detCov
(
X
k
(
s
)
,
X
k
(
t
)
)
}
]
-
1
/
2
d
t
d
s
≤
c
2.27
∫
I
∫
I
1
(
∑
ℓ
=
1
N
|
s
ℓ
-
t
ℓ
|
)
γ
(
∑
ℓ
=
1
N
|
s
ℓ
-
t
ℓ
|
H
ℓ
∧
K
ℓ
)
d
d
s
d
t
.
By a change of variable, we have
(67)
𝒥
≤
c
2.28
∫
0
b
N
-
a
N
d
t
N
⋯
∫
0
b
1
-
a
1
1
(
∑
ℓ
=
1
N
t
ℓ
)
γ
(
∑
ℓ
=
1
N
t
ℓ
H
ℓ
∧
K
ℓ
)
d
d
t
1
.
In order to show the integral in (67) is finite, we will integrate [
d
t
1
]
,
[
d
t
2
]
,
…
,
[
d
t
k
] iteratively. We only need to consider the case when
(68)
∑
ℓ
=
1
k
-
1
1
H
ℓ
∧
K
ℓ
≤
d
<
∑
ℓ
=
1
k
1
H
ℓ
∧
K
ℓ
,
a
a
a
a
a
a
a
a
for some
1
≤
k
≤
N
.
Here and in the sequel ∑
ℓ
=
1
0
(
1
/
(
H
ℓ
∧
β
)
)
=
^
0. Then, by using (55), we can deduce that
(69)
γ
=
∑
ℓ
=
1
k
H
k
∧
K
k
H
ℓ
∧
K
ℓ
+
N
-
k
-
(
H
k
∧
K
k
)
(
1
+
δ
)
d
,
where k is the unique integer satisfying (68).
Ifk
=
1 in (68), we integrate [
d
t
1
]. Note that 0
<
(
H
1
∧
K
1
)
d
<
1 and (
H
1
∧
K
1
)
d
+
γ  =  N
-
(
H
1
∧
K
1
)
δ
d
≠
1. Then we can use (52) of Lemma 6 with A
=
∑
ℓ
=
2
N
‍
t
ℓ
H
ℓ
∧
K
ℓ and B
=
∑
ℓ
=
2
N
‍
t
ℓ to get
(70)
𝒥
≤
c
2.29
[
∫
0
b
N
-
a
N
d
t
N
⋯
∫
0
b
2
-
a
2
1
(
∑
ℓ
=
2
N
t
ℓ
)
(
H
1
∧
K
1
)
d
+
γ
-
1
d
t
2
+
1
∫
l
]
<
∞
,
since (
H
1
∧
K
1
)
d
+
γ
-
1
<
N
-
1.
Ifk
>
1 in (68), we integrate [
d
t
1
] first. Since (
H
1
∧
K
1
)
d
>
1, we can use (50) of Lemma 6 with A
=
∑
ℓ
=
2
N
‍
t
ℓ
H
ℓ
∧
K
ℓ and B
=
∑
ℓ
=
2
N
‍
t
ℓ to get
(71)
𝒥
≤
c
2.30
∫
0
b
N
-
a
N
d
t
N
⋯
∫
0
b
2
-
a
2
1
(
∑
ℓ
=
2
N
t
ℓ
H
ℓ
∧
K
ℓ
)
d
-
1
/
(
H
1
∧
K
1
)
(
∑
ℓ
=
2
N
t
ℓ
)
γ
d
t
2
.
We can repeat this procedure for integrating d
t
2
,
…
,
d
t
k
-
1.
Note that ifd
=
∑
ℓ
=
1
k
-
1
‍
(
1
/
(
H
ℓ
∧
K
ℓ
)
), then (
H
k
-
1
∧
K
k
-
1
)
(
d
-
∑
ℓ
=
1
k
-
2
‍
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
)
=
1. We need to use (51) of Lemma 6 with A
=
∑
ℓ
=
k
N
‍
t
ℓ
H
ℓ
∧
K
ℓ and B
=
∑
ℓ
=
k
N
‍
t
ℓ to integrate d
t
k
-
1 and obtain
(72)
𝒥
≤
c
2.31
∫
0
b
N
-
a
N
d
t
N
⋯
∫
0
b
k
-
a
k
1
(
∑
ℓ
=
k
N
t
ℓ
)
γ
×
log
⁡
⁡
(
1
+
1
∑
ℓ
=
k
N
t
ℓ
)
d
t
k
<
∞
,
since γ
<
N
-
k
+
1.
On the other hand, ifd
>
∑
ℓ
=
1
k
-
1
‍
(
1
/
(
H
ℓ
∧
K
ℓ
)
), then (
H
k
-
1
∧
K
k
-
1
)
(
d
-
∑
ℓ
=
1
k
-
2
‍
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
)
>
1. By using (50) of Lemma 6 with A
=
∑
ℓ
=
k
N
‍
t
ℓ
H
ℓ
∧
K
ℓ and B
=
∑
ℓ
=
k
N
‍
t
ℓ to integrate d
t
k
-
1, we can deduce
(73)
𝒥
≤
c
2.32
∫
0
b
N
-
a
N
d
t
N
⋯
∫
0
b
k
-
a
k
1
(
∑
ℓ
=
k
N
t
ℓ
H
ℓ
∧
K
ℓ
)
d
-
∑
ℓ
=
1
k
-
1
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
(
∑
ℓ
=
k
N
t
ℓ
)
γ
d
t
k
.
Note that 0
<
(
H
k
∧
K
k
)
(
d
-
∑
ℓ
=
1
k
-
1
‍
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
)
<
1 and (
H
k
∧
K
k
)
(
d
-
∑
ℓ
=
1
k
-
1
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
)
+
γ  =  N
-
k
+
1
-
(
H
k
∧
K
k
)
δ
d
≠
1 for a small enough δ
0. Applying (52) to integrate d
t
k in (73), we see that
(74)
𝒥
≤
c
2.33
×
[
∫
0
b
N
-
a
N
d
t
N
⋯

∫
0
b
k
+
1
-
a
k
+
1

1
(
∑
ℓ
=
k
+
1
N
t
ℓ
)
γ
+
(
H
k
∧
K
k
)
(
d
-
∑
ℓ
=
1
k
-
1
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
)
-
1

d
t
k
+
1
+
1
∫
0
b
N
-
a
N
d
t
N
]
<
∞
,
since γ
+
(
H
k
∧
K
k
)
(
d
-
∑
ℓ
=
1
k
-
1
‍
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
)
-
1
<
N
-
k. Combining (70) through (74) yields (64). This completes the proof of Lemma 8.


## 3. Characteristics of Polar Functions

In this section, we provide some necessary conditions and sufficient conditions for a functionf
∈
𝒞 to be polar for X. We also give the intersection probabilities for a nonpolar function and X and determine the Hausdorff and packing dimensions of the set {
t
∈
I
,
X
(
t
)
=
f
(
t
)
}.Let us note that(75)
ℙ
{
∃
t
∈
ℝ
N
,
X
(
t
)
=
f
(
t
)
}
>
0
.
If and only if there exists a rectangle I
⊂
ℝ
N, such that
(76)
ℙ
{
∃
t
∈
I
,
X
(
t
)
=
f
(
t
)
}
>
0
.
For our purpose, it suffices to consider the polar functions of X in a rectangle I
=
Π
ℓ
=
1
N
[
a
ℓ
,
b
ℓ
]
⊆
ℝ
N with a
ℓ
<
b
ℓ  (
ℓ
=
1
,
…
,
N
).Theorem 9.
LetX
=
{
X
(
t
)
,
t
∈
ℝ
N
} be an (
N
,
d
)-Gaussian random field satisfying conditions (C1) and (C2) on I. If ∑
ℓ
=
1
N
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
<
d, then ℒ
(
K
)
⊂
𝒫.Proof.
For any constants0
<
δ
<
H
ℓ
(
1
≤
ℓ
≤
N
) and any rectangle I
=
Π
ℓ
=
1
N
[
a
ℓ
,
b
ℓ
]
⊆
ℝ
N with a
ℓ
<
b
ℓ  (
ℓ
=
1
,
…
,
N
), it follows from a similar argument as in the proof of Theorem 4.2 in [2] that there is a random variable A
1 of finite moments of all orders and an event Ω
1
* of probability 1 such that, for all ω
∈
Ω
1
*,
(77)
sup
⁡
s
,
t
∈
I
⁡
|
X
(
s
,
ω
)
-
X
(
t
,
ω
)
|
∑
ℓ
=
1
N
|
s
-
t
|
H
ℓ
-
δ
≤
A
1
(
ω
)
.
For any f
∈
ℒ
(
K
), in order to prove f
∈
𝒫, it suffices to prove that, for any ɛ
>
0 and any rectangle I
⊂
ℝ
N,
(78)
ℙ
{
ω
:
∃
t
∈
I
,
X
(
t
,
ω
)
=
f
(
t
)
}
≤
ɛ
.
Fix ɛ, δ and choose η such that ∑
ℓ
=
1
N
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
<
(
1
-
η
)
d. By A
1
<
∞,  a.s., then there exist Ω
ɛ
⊂
Ω
1
* and c
3.1
>
0 such that ℙ
(
Ω
ɛ
)
>
1
-
ɛ
/
2 and for any ω
∈
Ω
ɛ, A
1
(
ω
)
≤
c
3.1. For any integer n
≥
1, divide the rectangle I into m
n
=
(
∏
ℓ
=
1
N
‍
(
b
ℓ
-
a
ℓ
)
)
n
∑
ℓ
=
1
N
(
1
/
(
H
ℓ
∧
K
ℓ
)
) subrectangles I
n
,
♭ with sides parallel to the axes and side lengths n
-
1
/
(
H
ℓ
∧
K
ℓ
)
(
ℓ
=
1
,
…
,
N
). Let τ
n
,
♭ be the lower-left vertex of I
n
,
♭.
Letω
∈
Ω
ɛ, ♭ be fixed. If there exists t
∈
I
n
,
♭ such that X
(
t
,
ω
)
=
f
(
t
), then by (77) and f
∈
ℒ
(
K
),
(79)
|
X
(
τ
n
,
♭
,
ω
)
-
f
(
τ
n
,
♭
)
|
≤
|
X
(
τ
n
,
♭
,
ω
)
-
X
(
t
,
ω
)
|
+
|
f
(
τ
n
,
♭
,
ω
)
-
f
(
t
)
|
≤
c
3.1
∑
ℓ
=
1
N
‍
|
τ
n
,
♭
,
ℓ
-
t
ℓ
|
H
ℓ
-
δ
+
c
1.5
∑
ℓ
=
1
N
‍
|
τ
n
,
♭
,
ℓ
-
t
ℓ
|
K
ℓ
-
δ
≤
c
3.2
∑
ℓ
=
1
N
‍
|
τ
n
,
♭
,
ℓ
-
t
ℓ
|
(
H
ℓ
∧
K
ℓ
)
-
δ
≤
c
3.3
n
-
1
+
γ
,
where γ
=
max
⁡
{
δ
/
(
H
ℓ
∧
K
ℓ
)
,
ℓ
=
1
,
…
,
N
}.
We can choose a positiveδ such that 0
<
γ
<
η. It follows from (79) that
(80)
ℙ
{
∃
t
∈
I
such  that
X
(
t
)
=
f
(
t
)
}
<
ɛ
2
+
ℙ
{
ω
∈
Ω
ɛ
:
∃
t
∈
I
such  that
X
(
t
,
ω
)
=
f
(
t
)
}
≤
ɛ
2
+
ℙ
{
⋃
i
=
1
m
n
{
ω
∈
Ω
ɛ
:
∃
t
∈
I
n
,
♭
such  that
X
(
t
,
ω
)
=
f
(
t
)
ω
∈
Ω
ɛ
:
∃
t
∈
I
n
,
♭
}
⋃
i
=
1
m
n
{
ω
∈
Ω
ɛ
:
∃
t
∈
I
n
,
♭
}
≤
ɛ
2
+
∑
i
=
1
m
n
ℙ
{
|
X
(
τ
n
,
♭
,
ω
)
-
f
(
τ
n
,
♭
)
|
c
3.3
n
-
1
+
γ
ω
∈
Ω
ɛ
:
|
X
(
τ
n
,
♭
,
ω
)
-
f
(
τ
n
,
♭
)
|
≤
c
3.3
n
-
1
+
γ
|
X
(
τ
n
,
♭
,
ω
)
-
f
(
τ
n
,
♭
)
|
}
≤
ɛ
2
+
c
3.4
n
∑
ℓ
=
1
N
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
-
(
1
-
γ
)
d
≤
ɛ
.
In the above, we can get the last inequality as n is big enough. This proves Theorem 9.Theorem 10.
LetX
=
{
X
(
t
)
,
t
∈
ℝ
N
} be an (
N
,
d
)-Gaussian random field satisfying conditions (C1) and (C2) and f
∈
𝒬
(
K
). If ∑
ℓ
=
1
N
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
>
d, then with probability 1,
(81)
dim
⁡
⁡
{
t
∈
I
:
X
(
t
)
=
f
(
t
)
}
≤
min
⁡
1
≤
k
≤
N
⁡
{
∑
ℓ
=
1
k
H
k
∧
K
k
H
ℓ
∧
K
ℓ
+
N
-
k
-
(
H
k
∧
K
k
)
d
}
=
∑
ℓ
=
1
k
H
k
∧
K
k
H
ℓ
∧
K
ℓ
+
N
-
k
-
(
H
k
∧
K
k
)
d
,
i
f
∑
ℓ
=
1
k
-
1
1
H
ℓ
∧
K
ℓ
≤
d
<
∑
ℓ
=
1
k
1
H
ℓ
∧
K
ℓ
.Proof.
LetF
(
ω
)
=
{
t
∈
I
:
X
(
t
,
ω
)
=
f
(
t
)
},  ω
∈
Ω. In order to prove inequality (81), it suffices to prove that, for any ɛ
>
0 and any ζ
>
∑
ℓ
=
1
k
(
(
H
k
∧
K
k
)
/
(
H
ℓ
∧
K
ℓ
)
)
+
N
-
k
-
(
H
k
∧
K
k
)
d,
(82)
ℙ
{
ω
:
dim
⁡
⁡
(
F
(
ω
)
)
≤
ζ
}
≥
1
-
ɛ
.
We choose 0
<
η
<
1 such that ζ
>
∑
ℓ
=
1
k
(
(
H
k
∧
K
k
)
/
(
H
ℓ
∧
K
ℓ
)
)
+
N
-
k
-
(
H
k
∧
K
k
)
(
1
-
η
)
d. Then for all 0
<
λ
<
η, we have
(83)
∑
ℓ
=
1
N
1
H
ℓ
∧
K
ℓ
+
∑
ℓ
=
k
+
1
N
(
1
H
k
∧
K
k
-
1
H
ℓ
∧
K
ℓ
)
-
ζ
H
k
∧
K
k
-
(
1
-
λ
)
d
<
0
.
For any integer n
≥
1, divide the interval I into m
n
=
[
∏
ℓ
=
1
N
‍
(
b
ℓ
-
a
ℓ
)
]
·
n
∑
ℓ
=
1
N
(
1
/
(
H
ℓ
∧
K
ℓ
)
) subrectangles I
n
,
♭ of side lengths n
-
1
/
(
H
ℓ
∧
K
ℓ
)  (
ℓ
=
1
,
…
,
N
). Let τ
n
,
♭ be the lower-left vertex of I
n
,
♭. It follows from (83) that
(84)
lim
⁡
n
→
∞
⁡
n
∑
ℓ
=
1
N
(
1
/
(
H
ℓ
∧
K
ℓ
)
)

+

∑
ℓ
=
k
+
1
N
(
1
/
(
H
k
∧
K
k
)

-

1
/
(
H
ℓ
∧
K
ℓ
)
)

-

ζ
/
(
H
k
∧
K
k
)

-

(
1
-
η
)
d
=
0
.
Let
(85)
I
n
,
♭
′
(
ω
)
=
{
I
n
,
♭
,
if
∃
t
∈
I
n
,
♭
,
X
(
t
,
ω
)
=
f
(
t
)
,
∅
otherwise
.
Then F
(
ω
) can be covered by  {
I
n
,
♭
′
(
ω
)
}. For every 1
≤
k
≤
N, I
n
,
♭
(
ω
) can be covered by
(86)
∏
ℓ
=
1
N
n
(
1
/
(
H
k
∧
K
k
)
-
1
/
(
H
ℓ
∧
K
ℓ
)
)
=
n
∑
ℓ
=
1
N
(
1
/
(
H
k
∧
K
k
)
-
1
/
(
H
ℓ
∧
K
ℓ
)
)
≤
n
∑
ℓ
=
k
+
1
N
(
1
/
(
H
k
∧
K
k
)
-
1
/
(
H
ℓ
∧
K
ℓ
)
)
cubes of side length n
-
1
/
(
H
k
∧
K
k
). Then, we can cover the F
(
ω
) by a sequence of cubes of side length n
-
1
/
(
H
k
∧
K
k
).
Repeating this procedure in (77) and (79) in Theorem 9, we can deduce that, for all ω
∈
Ω
ɛ, if there exists t
∈
I
n
,
♭ such that X
(
t
,
ω
)
=
f
(
t
), then
(87)
|
X
(
τ
n
,
♭
,
ω
)
-
f
(
τ
n
,
♭
)
|
≤
c
3.3
n
-
1
+
γ
,
where γ
=
max
⁡
⁡
{
δ
/
(
H
ℓ
∧
K
ℓ
)
,
ℓ
=
1
,
…
,
N
}. Hence,
(88)
lim
⁡
_
n
→
∞
⁡
∑
♭
(
ω
)
[
diam
I
n
,
♭
′
(
ω
)
]
ζ
≤
lim
⁡
_
n
→
∞
⁡
∑
♭
∈
T
n
(
ω
)
[
diam
⁡
I
n
,
♭
]
ζ
,
ω
∈
Ω
ɛ
,
where T
n
(
ω
)
=
{
♭
:
|
X
(
τ
n
,
♭
,
ω
)
-
f
(
τ
n
,
♭
)
|
≤
c
3.3
n
-
1
+
γ
}. We can choose a positive δ such that 0
<
γ
<
η. It follows from (84)~(88) and lemma of Fatou that(89)
𝔼
[
lim
⁡
_
n
→
∞
⁡
∑
♭
(
ω
)
[
diam
⁡
I
n
,
♭
′
(
ω
)
]
ζ
I
Ω
ɛ
]
≤
lim
⁡
_
n
→
∞
𝔼
[
∑
♭
∈
T
n
(
ω
)
[
diam
⁡
I
n
,
♭
′
]
ζ
I
Ω
ɛ
]
≤
c
3.5
lim
⁡
_
n
→
∞
⁡
∑
♭
=
1
m
n
[
diam
⁡
I
n
,
♭
]
ζ
×
ℙ
{
ω
∈
Ω
ɛ
:
|
X
(
τ
n
,
♭
,
ω
)
-
f
(
τ
n
,
♭
)
|
≤
c
3.3
n
-
1
+
γ
}
≤
c
3.6
×
lim
⁡
_
n
→
∞
⁡
n
∑
ℓ
=
1
N
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
+
∑
ℓ
=
k
+
1
N
(
1
/
(
H
k
∧
K
k
)
-
1
/
(
H
ℓ
∧
K
ℓ
)
)
-
ζ
/
(
H
k
∧
K
k
)
-
(
1
-
γ
)
d
=
0
.Therefore, there exists Ω
0
⊂
Ω such that ℙ
(
Ω
0
)
=
1, and for all ω
∈
Ω
0
∩
Ω
ɛ we have lim
⁡
_
n
→
∞
⁡
∑
♭
(
ω
)
‍
[
diam
⁡
I
n
,
♭
′
(
ω
)
]
ζ
=
0. Then, dim
⁡
(
F
(
ω
)
)
≤
ζ. Since ℙ
(
Ω
0
∩
Ω
ɛ
)
>
1
-
ɛ
/
2, we obtain (82).Theorem 11.
LetX
=
{
X
(
t
)
,
t
∈
ℝ
N
} be an (
N
,
d
)-Gaussian random field satisfying conditions (C1) and (C2) and f
∈
𝒬
(
K
). If ∑
ℓ
=
1
N
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
>
d, then with positive probability,(90)
dim
⁡
⁡
{
t
∈
I
:
X
(
t
)
=
f
(
t
)
}
≥
min
⁡
1
≤
k
≤
N
⁡
{
∑
ℓ
=
1
k
H
k
∧
K
k
H
ℓ
∧
K
ℓ
+
N
-
k
-
(
H
k
∧
K
k
)
d
}
=
∑
ℓ
=
1
k
H
k
∧
K
k
H
ℓ
∧
K
ℓ
+
N
-
k
-
(
H
k
∧
K
k
)
d
,
i
f
∑
ℓ
=
1
k
-
1
1
H
ℓ
∧
K
ℓ
≤
d
<
∑
ℓ
=
1
k
1
H
ℓ
∧
K
ℓ
.Proof.
Let us assume that∑
ℓ
=
1
k
-
1
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
≤
d
<
∑
ℓ
=
1
k
(
1
/
(
H
ℓ
∧
K
ℓ
)
) for some 1
≤
k
≤
N, and let δ be a positive constant such that
(91)
∑
ℓ
=
1
k
-
1
1
H
ℓ
∧
K
ℓ
<
(
1
+
δ
)
d
<
∑
ℓ
=
1
k
1
H
ℓ
∧
K
ℓ
and hence
(92)
β
=
^
min
⁡
1
≤
k
≤
N
⁡
{
∑
ℓ
=
1
k
H
k
∧
K
k
H
ℓ
∧
K
ℓ
+
N
-
k
-
(
H
k
∧
K
k
)
(
1
+
δ
)
d
}
=
∑
ℓ
=
1
k
H
k
∧
K
k
H
ℓ
∧
K
ℓ
+
N
-
k
-
(
H
k
∧
K
k
)
(
1
+
δ
)
d
.
Note that, if we can prove that there is a constant c
3.7
>
0, independent of δ and β, such that
(93)
ℙ
{
ω
:
dim
⁡
⁡
{
t
∈
I
:
X
(
t
,
ω
)
=
f
(
t
)
}
≥
β
}
≥
c
3.7
,
then the lower bound in (90) will follow by letting δ
↓
0. The proof of (93) is based on the capacity argument due to Kahane [11].
Letℳ
β
+ be the space of all nonnegative measures on ℝ
N with finite β-energy. It is known that ℳ
β
+ is a complete metric space under the metric
(94)
∥
μ
∥
β
=
∫
ℝ
N
∫
ℝ
N
μ
(
d
t
)
μ
(
d
s
)
|
t
-
s
|
β
.
We define a sequence of random positive measures μ
ɛ on the Borel sets C of I by
(95)
μ
ɛ
(
C
)
=
∫
C
(
2
π
ɛ
)
d
/
2
exp
⁡
⁡
(
-
|
X
(
t
)
-
f
(
t
)
|
2
2
ɛ
)
d
t
.
It follows from Kahane [11] or Testard [16] that if there are positive constants c
3.8 and c
3.9 such that
(96)
𝔼
(
∥
μ
ɛ
∥
)
≥
c
3.8
,
𝔼
(
∥
μ
ɛ
∥
2
)
≤
c
3.9
,
𝔼
(
∥
μ
ɛ
∥
β
)
<
∞
,
where ∥
μ
ɛ
∥
=
μ
ɛ
(
I
), then there is a subsequence of {
μ
ɛ
}, say {
μ
ε
k
}, such that μ
ε
k
→
μ in ℳ
β
+ and μ is strictly positive with probability ≥
c
3.8
2
/
(
2
c
3.9
). In this case, it follows from (95) that the measure μ has its support in F a.s. Hence, Frostman's theorem yields (93) with c
3.7
=
c
3.8
2
/
(
2
c
3.9
). It remains to verify (96).
By using Fubini's theorem and (10), for all ɛ
∈
(
0,1
), we have
(97)
𝔼
(
∥
μ
ɛ
∥
)
=
∫
I
∫
ℝ
d
exp
⁡
⁡
(
-
|
u
|
2
2
ɛ
)
×
𝔼
(
exp
⁡
⁡
(
i
〈
u
,
X
(
t
)
-
f
(
t
)
〉
)
)
d
u
d
t
=
∫
I
(
∏
k
=
1
d
∫
ℝ
exp
⁡
⁡
(
-
i
〈
u
k
,
f
k
(
t
)
〉
)
×
exp
⁡
⁡
(
-
1
2
(
|
u
k
|
2
ɛ
-
1
+
Var
⁡
(
X
k
(
t
)
)
)
×
|
u
k
|
2
-
1
2
(
ɛ
-
1
+
Var
⁡
(
X
k
(
t
)
)
)
)
d
u
k
∏
k
=
1
d
∫
ℝ
exp
⁡
⁡
(
-
i
〈
u
k
,
f
k
(
t
)
〉
)
)
d
t
=
∫
I
∏
k
=
1
d
(
2
π
ɛ
-
1
+
Var
⁡
(
X
k
(
t
)
)
)
1
/
2
×
exp
⁡
⁡
(
-
|
f
k
(
t
)
|
2
2
(
ɛ
-
1
+
Var
⁡
(
X
k
(
t
)
)
)
)
d
t
≥
∫
I
∏
k
=
1
d
(
2
π
1
+
Var
⁡
(
X
k
(
t
)
)
)
1
/
2
×
exp
⁡
⁡
(
-
|
f
k
(
t
)
|
2
2
Var
⁡
(
X
k
(
t
)
)
)
d
t
=
^
c
3.8
.
Using Lemmas 4 and 7 and Fubini’s theorem, we can deduce that
(98)
𝔼
(
∥
μ
ɛ
∥
2
)
=
∫
I
∫
I
𝔼
(
2
π
ɛ
)
d
×
exp
⁡
⁡
(
-
(
|
X
(
s
)
-
f
(
s
)
|
2
+
|
X
(
t
)
-
f
(
t
)
|
2
)
(
2
ɛ
)
-
1
)
d
t
d
s
≤
c
3.10
∫
I
∫
I
∏
k
=
1
d
[
|
f
k
(
s
)
-
f
k
(
t
)
|
2
(
X
k
(
s
)
,
X
k
(
t
)
)
}
max
⁡
⁡
{
|
f
k
(
s
)
-
f
k
(
t
)
|
2
det Cov
(
X
k
(
s
)
,
X
k
(
t
)
)
,
|
f
k
(
s
)
-
f
k
(
t
)
|
2
(
X
k
(
s
)
,
X
k
(
t
)
)
}
]
-
(
1
/
2
)
d
t
d
s
≤
c
3.9
.
Similar to (98), we have
(99)
𝔼
(
∥
μ
ɛ
∥
β
)
=
∫
I
∫
I
1
|
s
-
t
|
β
𝔼
(
2
π
ɛ
)
d
×
exp
⁡
⁡
(
-
|
X
(
s
)
-
f
(
s
)
|
2
+
|
X
(
t
)
-
f
(
t
)
|
2
2
ɛ
)
d
s
d
t
≤
c
3.11
∫
I
∫
I
|
s
-
t
|
-
β
×
∏
k
=
1
d
[
|
f
k
(
s
)
-
f
k
(
t
)
|
2
(
X
k
(
s
)
,
X
k
(
t
)
)
}
max
⁡
⁡
{
|
f
k
(
s
)
-
f
k
(
t
)
|
2
det
⁡
⁡
Cov
(
X
k
(
s
)
,
X
k
(
t
)
)
,
|
f
k
(
s
)
-
f
k
(
t
)
|
2
(
X
k
(
s
)
,
X
k
(
t
)
)
}
]
-
(
1
/
2
)
d
t
d
s
≤
c
3.12
,
where the last inequality follows from Lemma 8. This completes the proof of Theorem 11.By using Theorems10 and 11, we can derive the following corollaries.Corollary 12.
If∑
ℓ
=
1
N
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
>
d and f
∈
𝒬
(
K
), then with positive probability,
(100)
dim
⁡
⁡
{
t
∈
I
:
X
(
t
)
=
f
(
t
)
}
=
min
⁡
1
≤
k
≤
N
⁡
{
∑
ℓ
=
1
k
H
k
∧
K
k
H
ℓ
∧
K
ℓ
+
N
-
k
-
(
H
k
∧
K
k
)
d
}
=
∑
ℓ
=
1
k
H
k
∧
K
k
H
ℓ
∧
K
ℓ
+
N
-
k
-
(
H
k
∧
K
k
)
d
,
i
f
∑
ℓ
=
1
k
-
1
1
H
ℓ
∧
K
ℓ
≤
d
<
∑
ℓ
=
1
k
1
H
ℓ
∧
K
ℓ
.
In particular, we have the following.Corollary 13.
If∑
ℓ
=
1
N
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
>
d, then 𝒬
(
K
)
⊂
𝒞
∖
𝒫.The following corollary presents the Hausdorff dimension about the fixed points ofX
(
t
),  t
∈
I.Corollary 14.
LetF
=
^
{
t
∈
I
,
X
(
t
)
=
t
} and N
=
d. Then, with positive probability,
(101)
dim
⁡
⁡
F
=
min
⁡
1
≤
k
≤
N
⁡
{
∑
ℓ
=
1
k
(
H
k
H
ℓ
-
1
)
+
(
1
-
H
k
)
d
}
=
∑
ℓ
=
1
k
(
H
k
H
ℓ
-
1
)
+
(
1
-
H
k
)
d
,
i
f
∑
ℓ
=
1
k
-
1
1
H
ℓ
≤
d
<
∑
ℓ
=
1
k
1
H
ℓ
.The following corollary also solves the question proposed by Le Gall [13] about the existence of nonpolar, continuous functions satisfying the Hölder condition for the Brownian motion.Corollary 15.
LetB
=
{
B
(
t
)
,
t
∈
ℝ
+
} be (
1
,
d
) Brownian motion. Then for any 0
<
β
<
1
/
d, there exists a function f satisfying the Hölder condition with index β such that f
∈
𝒞
∖
𝒫.The following theorem provides the intersection probability for a nonpolar function andX in terms of Hausdorff measure and capacity, respectively.Theorem 16.
LetX
=
{
X
(
t
)
,
t
∈
ℝ
N
} be an (
N
,
d
)-Gaussian random field satisfying conditions (C1) and (C2). If E
⊆
I is a compact set on ℝ
N and f
∈
𝒬
(
K
), then there exist positive constants c
3.13 and c
3.14 depending on E, K, and H only, such that
(102)
c
3.13
𝒞
d
ρ
(
E
)
≤
ℙ
{
∃
t
∈
E
such that
X
(
t
)
=
f
(
t
)
}
≤
c
3.14
ℋ
d
ρ
(
E
)
,
where the metric ρ is defined in (21), 𝒞
d
ρ
(
E
) is the capacity of E on ℝ
N generated by the kernel function ρ
-
d
(
s
,
t
), and ℋ
d
ρ
(
E
) is defined as the d-dimensional Hausdorff measure of E in the metric space (
ℝ
N
,
ρ
).Proof.
We first prove the lower bound in (102). When 𝒞
d
ρ
(
E
)
=
0, the lower bound in (102) holds automatically. On the other hand, when 𝒞
d
ρ
(
E
)
>
0, by the definition of 𝒞
d
ρ
(
E
), then there exists a finite positive measure σ supported on E, such that
(103)
ℐ
d
ρ
(
σ
)
=
^
∫
E
∫
E
σ
(
d
t
)
σ
(
d
s
)
[
ρ
(
s
,
t
)
]
d
<
1
𝒞
d
ρ
(
E
)
.
For all ɛ
>
0, we define a family of random measures μ
ɛ
=
^
μ
ɛ
(
x
,
·
) on the Borel sets C of E by
(104)
μ
ɛ
(
C
)
=
∫
C
(
2
π
ɛ
)
d
/
2
exp
⁡
⁡
(
-
|
X
(
s
)
-
f
(
s
)
-
x
|
2
2
ɛ
)
σ
(
d
s
)
=
∫
C
∫
ℝ
d
exp
⁡
⁡
(
-
ɛ
|
ξ
|
2
2
+
i
〈
ξ
,
X
(
s
)
-
f
(
s
)
-
x
〉
ɛ
|
ξ
|
2
2
)
d
ξ
σ
(
d
s
)
.
We claim that there are positive constants c
3.15 and c
3.16, such that
(105)
𝔼
(
∥
μ
ɛ
∥
)
≥
c
3.15
,
𝔼
(
∥
μ
ɛ
∥
2
)
≤
c
3.16
ℐ
d
ρ
(
σ
)
,
where ∥
μ
ɛ
∥
=
μ
ɛ
(
E
).
For anyɛ
∈
(
0,1
), by Fubini’s theorem and (10), we have
(106)
𝔼
(
∥
μ
ɛ
∥
)
=
∫
E
∫
ℝ
d
exp
⁡
⁡
(
-
ɛ
2
|
ξ
|
2
)
×
𝔼
(
exp
⁡
⁡
(
i
〈
ξ
,
X
(
s
)
-
f
(
s
)
-
x
〉
)
)
d
ξ
σ
(
d
s
)
=
∫
E
∏
k
=
1
d
(
2
π
ɛ
+
Var
⁡
(
X
k
(
s
)
-
f
k
(
s
)
)
)
1
/
2
×
exp
⁡
⁡
(
-
x
k
2
2
(
ɛ
+
Var
⁡
(
X
k
(
s
)
-
f
k
(
s
)
)
)
)
σ
(
d
s
)
≥
∫
E
∏
k
=
1
d
(
2
π
1
+
Var
⁡
(
X
k
(
s
)
)
)
1
/
2
×
exp
⁡
⁡
(
-
x
k
2
2
Var
⁡
(
X
k
(
s
)
)
)
σ
(
d
s
)
=
^
c
3.15
.
Let M
k
ɛ
(
s
,
t
)
=
ɛ
I
2
+
Cov
(
X
k
(
s
)
-
f
k
(
s
)
,
X
k
(
t
)
-
f
k
(
t
)
). It follows from Lemmas 1 and 2 that
(107)
det
⁡
⁡
M
k
ɛ
(
s
,
t
)
≥
Cov
(
X
k
(
s
)
-
f
k
(
s
)
,
X
k
(
t
)
-
f
k
(
t
)
)
=
det
⁡
⁡
Cov
(
X
k
(
s
)
,
X
k
(
t
)
)
+
𝔼
[
f
k
(
t
)
X
k
(
s
)
-
f
k
(
s
)
X
k
(
t
)
]
2
≥
c
3.17
∑
ℓ
=
1
N
|
s
ℓ
-
t
ℓ
|
2
(
H
ℓ
∧
K
ℓ
)
=
^
c
3.18
ρ
2
(
s
,
t
)
.
By Fubini’s theorem, (103), and (107), we have
(108)
𝔼
(
∥
μ
ɛ
∥
2
)
=
∫
E
∫
E
𝔼
(
2
π
ɛ
)
d
×
exp
⁡
⁡
(
-
(
|
X
(
s
)
-
f
(
s
)
-
x
|
2
+
|
X
(
t
)
-
f
(
t
)
-
x
|
2
)
(
2
ɛ
)
-
1
)
×
σ
(
d
s
)
σ
(
d
t
)
=
∫
E
∫
E
∏
k
=
1
d
2
π
(
det
⁡
⁡
M
k
ɛ
(
s
,
t
)
)
1
/
2
×
exp
⁡
⁡
(
-
1
2
(
x
k
,
x
k
)
(
M
k
ɛ
(
s
,
t
)
)
-
1
(
x
k
,
x
k
)
′
)
×
σ
(
d
s
)
σ
(
d
t
)
≤
∫
E
∫
E
c
3.16
ρ
d
(
s
,
t
)
σ
(
d
s
)
σ
(
d
t
)
<
c
3.16
ℐ
d
ρ
(
σ
)
<
c
3.16
𝒞
d
ρ
(
E
)
.
By modifying an argument from Kahane [11] or Testard [16], we can verify that there is a subsequence of {
μ
ɛ
}, say {
μ
ε
n
}, such that μ
ε
n
→
μ and μ is strictly positive with probability that at least c
3.15
2
/
(
c
3.16
ℐ
d
ρ
(
σ
)
). It follows from (104) and the continuity of X that μ has its support on {
t
∈
E
:
X
(
t
)
-
f
(
t
)
=
x
} a.s. Then, we apply the Paley-Zygmund inequality, (103), and (105), to deduce that
(109)
ℙ
{
t
∈
E
:
X
(
t
)
=
f
(
t
)
}
≥
[
𝔼
(
∥
μ
ɛ
∥
)
]
2
𝔼
(
∥
μ
ɛ
∥
2
)
≥
c
3.15
2
c
3.16
I
d
ρ
(
σ
)
≥
c
3.15
2
c
3.16
𝒞
d
ρ
(
E
)
.
Next we prove that the upper of (102) holds. When ℋ
d
ρ
(
E
)
=
∞, the result follows immediately; when 0
<
ℋ
d
ρ
(
E
)
<
∞, we can choose and fix an arbitrary constant γ
>
ℋ
d
ρ
(
E
). By using the definition of ℋ
d
ρ
(
E
) and modifying an argument from Theorem 32 in Rogers [18], there is a sequence of balls B
ρ
(
s
n
,
r
n
)
(
0
<
r
n
<
δ,n
=
1,2
,
…
) in the metric space (
ℝ
N
,
ρ
) such that
(110)
E
⊆
⋃
n
=
1
∞
B
ρ
(
s
n
,
r
n
)
,
∑
n
=
1
∞
r
n
d
<
γ
.
By (110) and Lemma 3, we have
(111)
ℙ
{
t
∈
E
:
X
(
t
)
=
f
(
t
)
}
≤
∑
n
=
1
∞
ℙ
{
t
∈
B
ρ
(
s
n
,
r
n
)
:
X
(
t
)
=
f
(
t
)
}
≤
c
3.19
∑
n
=
1
∞
r
n
d
<
c
3.19
γ
.
This implies that the upper of (102) holds in this case.
Whenℋ
d
ρ
(
E
)
=
0, by using Theorem 32 in Rogers [18] again, we can deduce that there exist sequences of open balls B
ρ
(
s
n
,
r
n
)
(
0
<
r
n
<
δ,  n
=
1,2
,
…
) in the metric space (
ℝ
N
,
ρ
) such that
(112)
E
⊆
lim
⁡
sup
⁡
n
→
∞
⁡
B
ρ
(
s
n
,
r
n
)
,
∑
n
=
1
∞
r
n
d
<
∞
.
Let
(113)
D
n
=
{
t
∈
B
ρ
(
s
n
,
r
n
)
:
X
(
t
)
=
f
(
t
)
}
,
and then by Lemma 3 and (112) we have
(114)
∑
n
=
1
∞
ℙ
(
D
n
)
≤
c
3.20
∑
n
=
1
∞
r
n
d
<
∞
.
Therefore the Borel-Cantelli Lemma implies
(115)
ℙ
{
lim
⁡
sup
⁡
n
→
∞
⁡
D
n
}
=
0
.
On the other hand, by (112) we have
(116)
{
t
∈
E
:
X
(
t
)
=
f
(
t
)
}
⊆
lim
⁡
sup
⁡
n
→
∞
⁡
D
n
.
Then (115) and (116) imply the upper bound of (102) when ℋ
d
ρ
(
E
)
=
0. Thus, the proof of Theorem 16 is finished.Finally, we discuss the packing dimension for the(
N
,
d
)-anisotropic Gaussian random fields.For anyɛ
>
0 and any bounded set E
⊂
ℝ
N, we use M
(
ɛ
,
E
) to denote the smallest number of cubes of side lengths ɛ that are needed to cover E. Then the upper box-counting dimension of E is defined as
(117)
Δ
(
E
)
=
lim
⁡
sup
⁡
ɛ
→
0
⁡
log
⁡
⁡
M
(
ɛ
,
E
)
-
log
⁡
⁡
ɛ
.
The packing dimension of E is defined as
(118)
Dim
(
E
)
=
inf
⁡
⁡
{
sup
⁡
⁡
Δ
(
E
n
)
,
E
⊂
⋃
n
=
1
∞
E
n
}
.
It is proved in Tricot Jr. [19] that, for any bounded set E
⊂
ℝ
N,
(119)
0
≤
dim
⁡
⁡
(
E
)
≤
Dim
(
E
)
≤
Δ
(
E
)
≤
N
.Theorem 17.
LetX
=
{
X
(
t
)
,
t
∈
ℝ
N
} be an (
N
,
d
)-Gaussian random field satisfying conditions (C1) and (C2). If ∑
ℓ
=
1
N
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
>
d, then for any f
∈
𝒬
(
K
), with positive probability
(120)
Dim
⁡
{
t
∈
I
,
X
(
t
)
=
f
(
t
)
}
=
min
⁡
1
≤
k
≤
N
⁡
{
∑
ℓ
=
1
k
‍
H
k
∧
K
k
H
ℓ
∧
K
ℓ
+
N
-
k
-
(
H
k
∧
K
k
)
d
}
=
∑
ℓ
=
1
k
H
k
∧
K
k
H
ℓ
∧
K
ℓ
+
N
-
k
-
(
H
k
∧
K
k
)
d
,
i
f
∑
ℓ
=
1
k
-
1
1
H
ℓ
∧
K
ℓ
≤
d
<
∑
ℓ
=
1
k
1
H
ℓ
∧
K
ℓ
.Proof.
The lower bound of (120) follows from (90) and (119). In order to prove the upper bound in (120), let us assume that ∑
ℓ
=
1
k
-
1
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
≤
d
<
∑
ℓ
=
1
k
(
1
/
(
H
ℓ
∧
K
ℓ
)
) for some 1
≤
k
≤
N. Then, by (119) we only need to prove that
(121)
Δ
(
F
)
≤
∑
ℓ
=
1
k
H
k
∧
K
k
H
ℓ
∧
K
ℓ
+
N
-
k
-
(
H
k
∧
K
k
)
d
a.s.
,
where F
=
{
t
∈
I
:
X
(
t
)
=
f
(
t
)
}.
For any integern
≥
1, divide the I into m
n
=
[
∏
ℓ
=
1
N
‍
(
b
ℓ
-
a
ℓ
)
]
·
n
∑
ℓ
=
1
N
(
1
/
(
H
ℓ
∧
K
ℓ
)
) subrectangles I
n
,
♭ with sides parallel to the axes and side lengths n
-
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
(
1
≤
ℓ
≤
N
). Then I can be covered by {
I
n
,
♭
} and each I
n
,
♭ is equivalent to a ball of radius n
-
1 under the metric ρ. It follows from Lemma 3 that
(122)
ℙ
{
∃
t
∈
I
n
,
♭
:
X
(
t
)
=
f
(
t
)
}
≤
ℙ
{
inf
⁡
t
∈
I
n
,
♭
⁡
|
X
(
t
)
-
f
(
t
)
|
<
1
n
}
≤
c
3.21
n
-
d
.
For every ♭, let
(123)
I
n
,
♭
′
=
{
I
n
,
♭
,
if
∃
t
∈
I
n
,
♭
,
X
(
t
)
=
f
(
t
)
∅
,
otherwise.
Then F can be covered by {
I
n
,
♭
′
}. For every 1
≤
k
≤
N, I
n
,
♭
′ can be covered by
(124)
∏
ℓ
=
1
N
n
(
1
/
(
H
k
∧
K
k
)
-
1
/
(
H
ℓ
∧
K
ℓ
)
)
≤
n
∑
ℓ
=
k
+
1
N
(
1
/
(
H
k
∧
K
k
)
-
1
/
(
H
ℓ
∧
K
ℓ
)
)
cubes of side length n
-
1
/
(
H
k
∧
K
k
). Thus, we can cover the F by a sequence of cubes of side length n
-
1
/
(
H
k
∧
K
k
). Denote the number of such cubes by M
n
,
k. Using (122) and (124), we have
(125)
𝔼
[
M
(
n
-
(
1
/
(
H
k
∧
K
k
)
)
,
F
)
]
≤
𝔼
[
M
n
,
k
]
≤
c
3.22
n
∑
ℓ
=
1
N
(
1
/
(
H
ℓ
∧
K
ℓ
)
)
+
∑
ℓ
=
k
+
1
N
(
1
/
(
H
k
∧
K
k
)
-
1
/
(
H
ℓ
∧
K
ℓ
)
)
·
n
-
d
=
c
3.22
n
∑
ℓ
=
1
k
(
1
/
(
H
ℓ
∧
K
ℓ
)
+
(
N
-
K
)
/
(
H
k
∧
K
k
)
-
d
)
.
Now let 0
<
δ
<
1 be fixed and let η be the constant defined by
(126)
η
=
∑
ℓ
=
1
k
1
H
ℓ
∧
K
ℓ
+
N
-
k
H
k
∧
K
k
-
(
1
-
δ
)
d
.
We consider the sequence of integers n
i
=
2
i
(
i
≥
1
). By using (125) and Markov inequality, we have
(127)
∑
i
=
1
∞
ℙ
{
M
(
n
i
-
1
/
(
H
k
∧
K
k
)
,
F
)
>
c
n
i
η
}
≤
c
3.23
∑
i
=
1
∞
2
-
δ
d
i
<
∞
.
Then it follows from the Borel-Cantelli lemma that a.s.                             (128)
M
(
n
i
-
1
/
(
H
k
∧
K
k
)
,
F
)
≤
c
n
i
η
for  all
i
large  enough
.
For any 0
<
ɛ
<
1, we can choose some positive integer i such that 2
-
i
-
1
<
ɛ
≤
2
-
i. Then, this, together with (128), implies that a.s.                            (129)
Δ
(
F
)
=
lim
⁡
¯
ɛ
→
0
log
⁡
⁡
[
M
(
ɛ
1
/
(
H
k
∧
K
k
)
,
F
)
]
-
log
⁡
⁡
ɛ
1
/
(
H
k
∧
K
k
)
≤
lim
⁡
¯
i
→
∞
log
⁡
[
M
(
n
i
+
1
-
1
/
(
H
k
∧
K
k
)
,
F
)
]
-
log
⁡
⁡
n
i
-
1
/
(
H
k
∧
K
k
)
≤
∑
ℓ
=
1
k
H
k
∧
K
k
H
ℓ
∧
K
ℓ
+
N
-
k
-
(
1
-
δ
)
(
H
k
∧
K
k
)
d
.
Letting δ
↘
0 along rational numbers and optimizing over k
=
1
,
…
,
N, we can deduce that (121) holds.


## 4. Applications to SPDEs

These results in this paper are applicable to solutions of SPDEs such as the linear string process considered by Mueller and Tribe [6], linear hyperbolic SPDEs considered by Dalang and Nualart [10], and nonlinear stochastic heat equations considered by Dalang et al. [8]. In this section, we only consider the Hausdorff and packing dimensions of the set {
t
∈
I
:
X
(
t
)
=
f
(
t
)
} for nonlinear stochastic heat equations in [8].LetW
˙
=
(
W
˙
1
,
…
,
W
˙
d
) be a space-time white noise in ℝ
d. That is, the components W
˙
1
(
x
,
s
)
,
…
,
W
˙
d
(
x
,
s
) are independent space-time white noises, which are generalized Gaussian processes with covariance given by
(130)
𝔼
[
W
˙
i
(
x
,
s
)
W
˙
j
(
y
,
t
)
]
=
δ
(
x
-
y
)
δ
(
s
-
t
)
,
(
i
=
1
,
…
,
d
)
,
where δ
(
·
) is the Dirac delta function. For all 1
≤
j
≤
d, let b
j
:
ℝ
d
↦
ℝ be globally Lipschitz and bounded functions, and let σ
=
^
(
σ
i
j
) be a deterministic d
×
d invertible matrix.Consider the system of SPDEs(131)
∂
u
i
(
s
,
x
)
∂
s
=
∂
2
u
i
(
s
,
x
)
∂
x
2
+
∑
j
=
1
d
σ
i
,
j
W
˙
j
(
s
,
x
)
+
b
i
(
u
(
s
,
x
)
)
,
for 1
≤
i
≤
d,  s
∈
[
0
,
T
] and x
∈
[
0,1
], with the initial conditions u
(
0
,
x
)
=
0 for all x
∈
[
0,1
], and the Neumann boundary conditions
(132)
∂
u
i
(
s
,
0
)
∂
x
=
∂
u
i
(
s
,
1
)
∂
x
=
0
,
0
≤
s
≤
T
,
where u
(
s
,
x
)
=
(
u
1
(
s
,
x
)
,
…
,
u
d
(
s
,
x
)
). Equation (131) can be interpreted rigorously as in Dalang et al. [8].A random fieldu
=
{
u
(
s
,
x
)
,
s
∈
[
0
,
T
]
,
x
∈
[
0,1
]
} is a solution of (131) if u is adapted to (
ℱ
s
) and if for every i
∈
{
1
,
…
,
d
},  s
∈
[
0
,
T
] and x
∈
[
0,1
],
(133)
u
i
(
s
,
x
)
=
∫
0
s
∫
0
1
G
s
-
r
(
x
,
v
)
∑
j
=
1
d
σ
i
,
j
W
j
(
d
r
,
d
v
)
+
∫
0
s
∫
0
1
G
s
-
r
(
x
,
v
)
b
i
(
u
(
s
,
x
)
)
d
r
d
v
,
where G
s
(
x
,
y
) is the Green kernel for the heat equation with Neumann boundary conditions (see Walsh [20]).For the linear form of (131) (i.e., b
≡
0 and σ
≡
I
d (the d
×
d identity matrix)), Mueller and Tribe [6] found necessary and sufficient conditions (in terms of the dimension d) for its solution u to hit points or to have double points of various types. Wu and Xiao [21] further studied the fractal properties of the sample paths of u and obtained the Hausdorff dimensions of the level sets and the set of double times of u. Recently, Chen [5] studied the fractal properties of the algebraic sum of the image sets for u and obtained the Hausdorff and packing dimensions of the algebraic sum of the image sets of the string. More generally, Dalang et al. [8] studied hitting probabilities for the nonlinear equation (131). They also determined the Hausdorff dimensions of the range and level sets of these processes.In the following, we show the Hausdorff dimension and the packing dimension of the intersecting sets{
t
∈
I
:
X
(
t
)
=
f
(
t
)
} of the nonpolar functions for nonlinear stochastic heat equations in [8]. As shown by [8, Proposition 4.1], it is sufficient to consider these problems for the solution of (131) in the following drift-free case (i.e., b
≡
0):
(134)
∂
u
∂
s
(
s
,
x
)
=
∂
2
u
∂
x
2
(
s
,
x
)
+
σ
W
˙
.
The solution of (134) is the mean zero Gaussian random field u
=
{
u
(
s
,
x
)
,
s
∈
[
0
,
T
]
,
x
∈
[
0,1
]
} with values in ℝ
d defined by
(135)
u
(
s
,
x
)
=
∫
0
s
∫
0
1
G
s
-
r
(
x
,
y
)
σ
W
(
d
r
,
d
y
)
a
a
a
a
a
a
a
a
a
a
a
a
a
a
s
∈
[
0
,
T
]
,
x
∈
[
0,1
]
.
Moreover, since the matrix σ is invertible, a change of variables shows (see proof of Proposition 4.1 in [8]) that v
=
^
σ
-
1
u solves the following uncoupled system of SPDE:
(136)
∂
v
∂
s
(
s
,
x
)
=
∂
2
v
∂
x
2
(
s
,
x
)
+
σ
W
˙
.
Note that f
∈
𝒬
(
K
) if and only if σ
-
1
f
∈
𝒬
(
K
); that is, they belong or do not belong to the same functional class 𝒬
(
K
). Thus, both processes u and v have the same intersection probability, Hausdorff dimension and packing dimension properties. Therefore, without loss of generality, we will assume that σ
=
I
d in (134).The following is a consequence of Lemmas 4.2 and 4.3 of Dalang et al. [8] or Lemma 4.1, in Biermé et al. [1], which indicates that the Gaussian random field u satisfies conditions (C1) and (C2) with H
1
=
1
/
4 and H
2
=
1
/
2.Lemma 18.
Letu
=
{
u
(
s
,
x
)
,
s
∈
[
0
,
T
]
,
x
∈
[
0,1
]
} be the solution of (134). Then for any compact set E
⊆
(
0
,
T
]
×
[
0,1
], there exist positive and finite constants c
4.1
,
…
,
c
4.5 such that the following hold. (i)
For all(
s
,
x
)
∈
E, c
4.1
≤
𝔼
[
u
(
s
,
x
)
]
2
≤
c
4.2 and for all (
s
,
x
)
,
(
t
,
y
)
∈
E,
(137)
c
4.3
(
|
s
-
t
|
1
/
2
+
|
x
-
y
|
)
≤
𝔼
[
u
(
s
,
x
)
-
u
(
t
,
y
)
]
2
≤
c
4.4
(
|
s
-
t
|
1
/
2
+
|
x
-
y
|
)
.
(ii)
For all(
s
,
x
)
,
(
t
,
y
)
∈
E,
(138)
Var
⁡
(
u
(
s
,
x
)
∣
u
(
t
,
y
)
)
≥
c
4.5
(
|
s
-
t
|
1
/
2
+
|
x
-
y
|
)
.Proof.
SinceE is a compact set on (
0
,
T
]
×
[
0,1
], then there exists a positive constant c, such that E
⊂
[
c
,
T
]
×
[
0,1
]. By (135), we have
(139)
𝔼
[
u
(
s
,
x
)
]
2
=
∫
0
s
d
r
∫
0
1
(
G
s
-
r
(
x
,
y
)
)
2
d
y
.
Note that 𝔼
[
u
(
s
,
x
)
]
2 is a continuous function in (
s
,
x
) and positive on E. This implies the first conclusion of the lemma. Inequality (137) follows (
4.11
) in Lemma 4.2 of Dalang et al. [8].
It follows from Lemma 4.3 of Dalang et al. [8] that
(140)
det
⁡
⁡
Cov
(
u
(
s
,
x
)
,
u
(
t
,
y
)
)
≥
c
4.5
(
|
s
-
t
|
1
/
2
+
|
x
-
y
|
)
.
By using (8), (140), and the first inequality in Lemma 18, we can deduce that (138) holds. This finishes the proof of Lemma 18.Therefore, Lemma18 shows that Theorem 16 includes the corresponding conclusion of solutions of nonlinear stochastic heat equations in [8]. The following theorems, which are two new results in [6, 8, 10], are the consequences of Theorems 10, 11, and 17 with K
=
(
K
1
,
K
2
)
∈
(
0,1
)
2. Moreover, we can obtain very different results when the parameters K take different values.Theorem 19.
Letu
=
{
u
(
s
,
x
)
,
s
∈
[
0
,
T
]
,
x
∈
[
0,1
]
} be the solution of (131) and let I be a rectangle on (
0
,
T
]
×
[
0,1
]. The following conclusions hold.(i)
Ifd
>
∑
ℓ
=
1
2
‍
(
4
/
(
ℓ
∧
(
4
K
ℓ
)
)
), then for all f
∈
ℒ
(
K
), we have f
∈
𝒫 and
(141)
dim
⁡
⁡
{
(
s
,
x
)
∈
I
:
u
(
s
,
x
)
=
f
(
s
,
x
)
}
=
0
a.s.
(ii)
Ifd
<
∑
ℓ
=
1
2
‍
(
4
/
(
ℓ
∧
(
4
K
ℓ
)
)
), then for all f
∈
𝒬
(
K
), we have f
∈
𝒞
∖
𝒫, and
(142)
Dim
⁡
{
(
s
,
x
)
∈
I
:
u
(
s
,
x
)
=
f
(
s
,
x
)
}
=
dim
⁡
⁡
{
(
s
,
x
)
∈
I
:
u
(
s
,
x
)
=
f
(
s
,
x
)
}
=
{
2
-
(
1
4
∧
K
1
)
d
,
i
f
≤
d
<
4
1
∧
(
4
K
1
)
,
1
+
2
∧
(
4
K
2
)
1
∧
(
4
K
1
)
-
(
1
2
∧
K
2
)
d
,
i
f
4
1
∧
(
4
K
1
)
≤
d
<
∑
ℓ
=
1
2
4
ℓ
∧
(
4
K
ℓ
)
,
on an event of positive probability.
(iii)
IfE
⊆
I is a Borel set and f
∈
𝒬
(
K
), then there exist positive constants c
4.6 and c
4.7, such that
(143)
c
4.6
𝒞
d
ρ
(
E
)
≤
ℙ
{
∃
(
s
,
x
)
∈
E
such
that
X
(
s
,
x
)
=
f
(
s
,
x
)
}
≤
c
4.7
ℋ
d
ρ
(
E
)
,
where ρ is the metric on [
0
,
T
]
×
[
0,1
] defined by
(144)
ρ
(
(
s
,
x
)
,
(
t
,
y
)
)
=
|
s
-
t
|
(
1
/
4
)
∧
K
1
+
|
x
-
y
|
(
1
/
2
)
∧
K
2
.Proof.
As shown by Proposition 4.1 in [8], it is sufficient to prove the results for the case of b
≡
0 and σ
=
I
d in (131). Note that K
=
(
K
1
,
K
2
)
∈
(
0,1
)
2 and H
=
(
1
/
4,1
/
2
). Therefore, the conclusions follow from Theorems 9, 10, 11, 16, 17 and Lemma 18.As we showed, in Theorem19, we can also apply these theorems in this paper to recover the same results such as the Brownian sheet [15], fractional Brownian motion [14], fractional Brownian sheets [4], linear hyperbolic SPDEs considered by Dalang and Nualart [10], linear SPDEs considered by Mueller and Tribe [6], and operator-scaling stable Gaussian random fields with stationary increments constructed in [22].



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*Source: 947171-2014-03-24.xml*