Abstract
Let X 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.
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 . 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 be a fixed vector and, for with (), let denote a compact interval (or a rectangle). For example, we may take , where is a fixed constant.
Let , , be a Gaussian random field on a probability space with mean zero and whose components , , are independent. Suppose that for each , satisfies the following general conditions.(C1)There exist positive and finite constants , , and such that for all and (C2)There exists a positive and finite constant such that, for all ,
Here denotes the conditional variance of given . We will call an -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 Section 3.
Let . As usual, a function is said to be a polar function for the random field if Let denote the collection of the continuous functions satisfying (3).
Let be a fixed vector, and let denote the collection of all Hölder continuous functions of any order less than along the th direction in time; that is, there exists a finite and positive constant , depending only on and , such that for all , , and ,
Moreover, let denote the collection of all functions satisfying the following condition: there exist finite and positive constants and , depending only on and , such that for all and ,
Note that if , then the functions in are called Hölder continuous of any order less than , and the functions in are called quasi-spiral with order ; see Kahane [11]. Hence and 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 function , when is it nonpolar for in the sense that ? When is it polar for in the sense that ?(ii)Given a nonrandom Borel set , what is the probability for the random set ? What is the Hausdorff and packing dimensions of the set ifis nonpolar or ?
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 for 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 . 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 induces far richer fractal structure into the properties of the nonpolar functions for .
The rest of the paper is organized as follows. In Section 2, we derive a few preliminary estimates and lemmas for 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 . We also give upper and lower bounds for the probabilities for a nonpolar function intersecting and determine the Hausdorff and packing dimensions of the times points for a nonpolar function intersecting . 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 use 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 are numbered as .
2. Some Preliminary Estimates
Because of the complex dependence structure for the anisotropic Gaussian random fields, the proofs of the main results in Sections 3 and 4 are quite involved. Therefore, we split the proofs into several lemmas to be used in Sections 3 and 4.
Let be a compact set in . denotes the covariance matrix of the random vector . Then, for all , where .
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. Let be an -Gaussian random field satisfying conditions (C1) and (C2) and let be a compact set on . Then there exist positive constants and , such that for all , ,
Proof. Since is a compact set in , then there exists a positive constant , such that . In order to prove (7), it suffices to show that (7) holds for all with . We claim that for all, with ,
If , then by using the expression for the characteristic functions and the density functions of Gaussian random fields, it turns out that
By applying the fact that the conditional distribution of given is still Gaussian with mean and variance , one can evaluate the integral in the right-hand side of (9) and thus deduce that (8) holds.
If , then we can deduce that the related coefficient of and is equal to , so there exists such that 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 is a mean zero Gaussian vector. Since is a positive continuous function on , then there exists a positive constant such that, for all ,
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. Let be an -Gaussian random field satisfying conditions (C1) and (C2) and let be a compact set on . Then there exist positive constants and , such that, for all , we have where , .
Proof. Since is a compact set on , then there exists a positive constant , such that . 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 By taking , then for all , It follows from (12) and (13) that Note that Then inequalities (14) and (15) imply Now we prove the upper bound in (11). By using (1) and (7) and repeating the procedure in (12), we can derive By taking , then for all , It follows from (17) and (18) that Combining (15), (18), and (19), we obtain By inequalities (16) and (20), we finish the proof of Lemma 2.
Let be a metric on defined by
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. Let be an -Gaussian random field satisfying conditions (C1) and (C2). Then there exist positive constants and , such that, for all , , and all , where denotes the ball of radius centered at in the metric defined by (21).
Proof. Using the Gaussian regressions, we have Note that, for all , the Gaussian random variables and are independent. By using the triangle inequality, we can deduce that, for all , where . Then By the Cauchy-Schwarz inequality, (1), and (23), we have Therefore, there exists a positive constant such that, for all and , we can deduce that . Recall that, for the unimodality of the centered Gaussian process , we have Note that and are independent. It follows from (27) that In order to estimate , we denote that the Gaussian process − and note that and the canonical metric Therefore, by the Hölder inequality and the Cauchy-Schwarz inequality, we have By using (1), (10), (30), and the fact that , we have Then where is the metric entropy number of and . It follows from Dudley’s theorem of Kahane [11] that Combining (25), (27), (28), and (33) and using the coordinate processes independence of , we have This finishes the proof of Lemma 3.
Lemma 4. Let be an -Gaussian random field satisfying conditions (C1) and (C2) and let be a compact set on . Then there exists a positive constant such that, for all , and ,
Proof. Note that
Denote by the identity matrix of order 2 and let . Then the inverse of is given by
where denotes the determinant of .
By (36), Lemma 2, Fubini’s theorem, and some elementary calculations, we derive
If , then
For all , we can deduce
If , by the inequality above and taking and , we have
It follows from Lemma 1 that, for all ,
By using (42) and the fact that , we have
Combining (36) through (43), we prove that Lemma 4 holds.
For proving the lower bound in Theorem 11, we will use two lemmas below, which are slightly more general results, by modifying the argument in [3, 17].
Lemma 5. Let , and be given constants. Then for all constants , and , there exists a positive and finite constant , depending on , , , , , , , and only, such that, for all ,
Proof. Let and . By the symmetry of the integrand, we get
Putting and using the fact that , we see that the above integral is bound by
where we have used the substitution .
Let . If , then for , it follows from (46) and Hölder's inequality that there exists a positive and finite constant , which depends on , , , , , , , and only, such that
where we have used the fact that .
If , then some elementary calculations imply that, for all ,
where depends on , , , , , , , and only. By (47) and (48), the proof of Lemma 5 is finished.
Lemma 6. Let , , , and be positive constants. For and , let Then there exist positive and finite constants and , depending on , , , and only, such that the following hold for all reals satisfying :(i)if , then (ii)if , then (iii)if and , then
Proof. If , by using Lemma 10 in [3], we can prove that inequalities (50), (51), and (52) hold. If , then we can split the integral in (49) so that Let . Since and , , and are positive constants, we get By using (53), (54), and Lemma 10 in [3] again, we can also prove (50), (51), and (52); in this case . Thus, the proof of Lemma 6 is finished.
Let and be given vectors. For convenience, we may further assume
Lemma 7. Let be an -Gaussian random field satisfying conditions (C1) and (C2). If , then there exists a positive and finite constant , depending on , , , , , and only, such that, for all ,
Proof. Note that . Then, by using Lemma 1, we have Let be the unique positive integer such that Then, we choose positive constants such that for each and Applying Lemma 5 to (57) with we obtain that By repeatedly using Lemma 5 to the integral in above inequality for steps, we have Since the satisfy (59), we have Thus, the integral in the right-hand side of (62) is finite. This completes the proof of Lemma 7.
Lemma 8. Let be an -Gaussian random field satisfying conditions (C1) and (C2). If , then there exist positive and finite constants and , such that for all , , where and depends on , , , , , , and only.
Proof. For our purpose, let us note that (55) implies
Then, there exist such that for all and , we have . By using Lemma 1 and , we have
By a change of variable, we have
In order to show the integral in (67) is finite, we will integrate iteratively. We only need to consider the case when
Here and in the sequel . Then, by using (55), we can deduce that
where is the unique integer satisfying (68).
If in (68), we integrate . Note that and = . Then we can use (52) of Lemma 6 with and to get
since .
If in (68), we integrate first. Since , we can use (50) of Lemma 6 with and to get
We can repeat this procedure for integrating .
Note that if , then . We need to use (51) of Lemma 6 with and to integrate and obtain
since .
On the other hand, if , then . By using (50) of Lemma 6 with and to integrate , we can deduce
Note that and = for a small enough . Applying (52) to integrate in (73), we see that
since . 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 function to be polar for . We also give the intersection probabilities for a nonpolar function and and determine the Hausdorff and packing dimensions of the set .
Let us note that If and only if there exists a rectangle , such that For our purpose, it suffices to consider the polar functions of in a rectangle with .
Theorem 9. Let be an -Gaussian random field satisfying conditions (C1) and (C2) on . If , then .
Proof. For any constants and any rectangle with , it follows from a similar argument as in the proof of Theorem 4.2 in [2] that there is a random variable of finite moments of all orders and an event of probability 1 such that, for all ,
For any , in order to prove , it suffices to prove that, for any and any rectangle ,
Fix , and choose such that . By , a.s., then there exist and such that and for any , . For any integer , divide the rectangle into subrectangles with sides parallel to the axes and side lengths . Let be the lower-left vertex of .
Let , be fixed. If there exists such that , then by (77) and ,
where .
We can choose a positive such that . It follows from (79) that
In the above, we can get the last inequality as is big enough. This proves Theorem 9.
Theorem 10. Let be an -Gaussian random field satisfying conditions (C1) and (C2) and . If , then with probability 1,
Proof. Let , . In order to prove inequality (81), it suffices to prove that, for any and any ,
We choose such that . Then for all , we have
For any integer , divide the interval into subrectangles of side lengths . Let be the lower-left vertex of . It follows from (83) that
Let
Then can be covered by . For every , can be covered by
cubes of side length . Then, we can cover the by a sequence of cubes of side length .
Repeating this procedure in (77) and (79) in Theorem 9, we can deduce that, for all , if there exists such that , then
where . Hence,
where . We can choose a positive such that . It follows from (84)(88) and lemma of Fatou thatTherefore, there exists such that , and for all we have . Then, . Since , we obtain (82).
Theorem 11. Let be an -Gaussian random field satisfying conditions (C1) and (C2) and . If , then with positive probability,
Proof. Let us assume that for some , and let be a positive constant such that
and hence
Note that, if we can prove that there is a constant , independent of and , such that
then the lower bound in (90) will follow by letting . The proof of (93) is based on the capacity argument due to Kahane [11].
Let be the space of all nonnegative measures on with finite -energy. It is known that is a complete metric space under the metric
We define a sequence of random positive measures on the Borel sets of by
It follows from Kahane [11] or Testard [16] that if there are positive constants and such that
where , then there is a subsequence of , say , such that in and is strictly positive with probability . In this case, it follows from (95) that the measure has its support in a.s. Hence, Frostman's theorem yields (93) with . It remains to verify (96).
By using Fubini's theorem and (10), for all , we have
Using Lemmas 4 and 7 and Fubini’s theorem, we can deduce that
Similar to (98), we have
where the last inequality follows from Lemma 8. This completes the proof of Theorem 11.
By using Theorems 10 and 11, we can derive the following corollaries.
Corollary 12. If and , then with positive probability, In particular, we have the following.
Corollary 13. If , then .
The following corollary presents the Hausdorff dimension about the fixed points of , .
Corollary 14. Let and . Then, with positive probability,
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. Let be Brownian motion. Then for any , there exists a function satisfying the Hölder condition with index such that .
The following theorem provides the intersection probability for a nonpolar function and in terms of Hausdorff measure and capacity, respectively.
Theorem 16. Let be an -Gaussian random field satisfying conditions (C1) and (C2). If is a compact set on and , then there exist positive constants and depending on , , and only, such that where the metric is defined in (21), is the capacity of on generated by the kernel function , and is defined as the -dimensional Hausdorff measure of in the metric space .
Proof. We first prove the lower bound in (102). When , the lower bound in (102) holds automatically. On the other hand, when , by the definition of , then there exists a finite positive measure supported on , such that
For all , we define a family of random measures on the Borel sets of by
We claim that there are positive constants and , such that
where .
For any , by Fubini’s theorem and (10), we have
Let . It follows from Lemmas 1 and 2 that
By Fubini’s theorem, (103), and (107), we have
By modifying an argument from Kahane [11] or Testard [16], we can verify that there is a subsequence of , say , such that and is strictly positive with probability that at least . It follows from (104) and the continuity of that has its support on a.s. Then, we apply the Paley-Zygmund inequality, (103), and (105), to deduce that
Next we prove that the upper of (102) holds. When , the result follows immediately; when , we can choose and fix an arbitrary constant . By using the definition of and modifying an argument from Theorem 32 in Rogers [18], there is a sequence of balls , in the metric space such that
By (110) and Lemma 3, we have
This implies that the upper of (102) holds in this case.
When , by using Theorem 32 in Rogers [18] again, we can deduce that there exist sequences of open balls , in the metric space such that
Let
and then by Lemma 3 and (112) we have
Therefore the Borel-Cantelli Lemma implies
On the other hand, by (112) we have
Then (115) and (116) imply the upper bound of (102) when . Thus, the proof of Theorem 16 is finished.
Finally, we discuss the packing dimension for the -anisotropic Gaussian random fields.
For any and any bounded set , we use to denote the smallest number of cubes of side lengths that are needed to cover . Then the upper box-counting dimension of is defined as The packing dimension of is defined as It is proved in Tricot Jr. [19] that, for any bounded set ,
Theorem 17. Let be an -Gaussian random field satisfying conditions (C1) and (C2). If , then for any , with positive probability
Proof. The lower bound of (120) follows from (90) and (119). In order to prove the upper bound in (120), let us assume that for some . Then, by (119) we only need to prove that
where .
For any integer , divide the into subrectangles with sides parallel to the axes and side lengths . Then can be covered by and each is equivalent to a ball of radius under the metric . It follows from Lemma 3 that
For every , let
Then can be covered by . For every , can be covered by
cubes of side length . Thus, we can cover the by a sequence of cubes of side length . Denote the number of such cubes by . Using (122) and (124), we have
Now let be fixed and let be the constant defined by
We consider the sequence of integers . By using (125) and Markov inequality, we have
Then it follows from the Borel-Cantelli lemma that a.s.
For any , we can choose some positive integer such that . Then, this, together with (128), implies that a.s.
Letting along rational numbers and optimizing over , 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 for nonlinear stochastic heat equations in [8].
Let be a space-time white noise in . That is, the components are independent space-time white noises, which are generalized Gaussian processes with covariance given by where is the Dirac delta function. For all , let be globally Lipschitz and bounded functions, and let be a deterministic invertible matrix.
Consider the system of SPDEs for , and , with the initial conditions for all , and the Neumann boundary conditions where . Equation (131) can be interpreted rigorously as in Dalang et al. [8].
A random field is a solution of (131) if is adapted to and if for every , and , where is the Green kernel for the heat equation with Neumann boundary conditions (see Walsh [20]).
For the linear form of (131) (i.e., and (the identity matrix)), Mueller and Tribe [6] found necessary and sufficient conditions (in terms of the dimension ) for its solution 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 and obtained the Hausdorff dimensions of the level sets and the set of double times of . Recently, Chen [5] studied the fractal properties of the algebraic sum of the image sets for 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 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., ): The solution of (134) is the mean zero Gaussian random field with values in defined by Moreover, since the matrix is invertible, a change of variables shows (see proof of Proposition 4.1 in [8]) that solves the following uncoupled system of SPDE: Note that if and only if ; that is, they belong or do not belong to the same functional class . Thus, both processes and have the same intersection probability, Hausdorff dimension and packing dimension properties. Therefore, without loss of generality, we will assume that 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 satisfies conditions (C1) and (C2) with and .
Lemma 18. Let be the solution of (134). Then for any compact set , there exist positive and finite constants such that the following hold. (i)For all , and for all , (ii)For all ,
Proof. Since is a compact set on , then there exists a positive constant , such that . By (135), we have
Note that is a continuous function in and positive on . This implies the first conclusion of the lemma. Inequality (137) follows in Lemma 4.2 of Dalang et al. [8].
It follows from Lemma 4.3 of Dalang et al. [8] that
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, Lemma 18 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 . Moreover, we can obtain very different results when the parameters take different values.
Theorem 19. Let be the solution of (131) and let be a rectangle on . The following conclusions hold.(i)If , then for all , we have and (ii)If , then for all , we have , and on an event of positive probability.(iii)If is a Borel set and , then there exist positive constants and , such that where is the metric on defined by
Proof. As shown by Proposition 4.1 in [8], it is sufficient to prove the results for the case of and in (131). Note that and . Therefore, the conclusions follow from Theorems 9, 10, 11, 16, 17 and Lemma 18.
As we showed, in Theorem 19, 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].
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author thanks Professor Yimin Xiao for stimulating discussions and for helpful comments on the paper. The research was supported by the National Natural Foundation of China (11371321) and the Key Research Base for Humanities and Social Sciences of Zhejiang Provincial High Education Talents (Statistics of Zhejiang Gongshang University).