Regularity of Gaussian Processes on Dirichlet Spaces

Abstract

We study the regularity of centered Gaussian processes $(Z_x( \omega ))_{x\in M}$, indexed by compact metric spaces $(M, \rho )$. It is shown that the almost everywhere Besov regularity of such a process is (almost) equivalent to the Besov regularity of the covariance $K(x,y) = {\mathbb E}(Z_x Z_y)$ under the assumption that (i) there is an underlying Dirichlet structure on M that determines the Besov regularity, and (ii) the operator K with kernel K(x, y) and the underlying operator A of the Dirichlet structure commute. As an application of this result, we establish the Besov regularity of Gaussian processes indexed by compact homogeneous spaces and, in particular, by the sphere.

Appendices

Appendices

1.1 Appendix I: Positive and Negative Definite Functions

We recall in this appendix some well-known (or not so well-known) facts about positive definite and negative definite functions. For details, we refer the reader to [5, 7, 9, 17, 43].

Recall first the definitions of positive and negative definite functions:

Definition 8.1

Given a set M, a real-valued function K(x, y) defined on $M\times M$ is said to be positive definite (P.D.) if $K(x,y)=K(y,x)$, $\forall x, y\in M$, and

$$\begin{aligned} \forall \alpha _1,\ldots ,\alpha _n \in {\mathbb R},\; \forall x_1,\ldots ,x_n\in M, \quad \sum _{i,j=1}^n\alpha _i\alpha _j K(x_i,x_j)\ge 0. \end{aligned}$$

Clearly, if K(x, y) is P.D., then $|K(x,y)| \le \sqrt{K(x,x)}\sqrt{K(y,y)}.$ It is well known that the following characterization is valid:

$$\begin{aligned} K(x,y) \; \hbox {is P.D.}\quad \Longleftrightarrow \quad K(x,y)= {\mathbb E}(Z_xZ_y), \end{aligned}$$

where $(Z_x)_{x\in M}$ is some (centered ) Gaussian process.

Definition 8.2

For any $u\in M$, we associate with K(x, y) the following P.D. function:

$$\begin{aligned} K_u(x,y) := K(x,y)+ K(u,u)-K(x,u)-K(y,u)= {\mathbb E}[(Z_x-Z_u)(Z_y-Z_u)], \end{aligned}$$

where $(Z_x-Z_u)$ is the process “killed” at the point $u\in M.$

Clearly,

$$\begin{aligned} K_u \equiv K\Longleftrightarrow K(u,u)=0. \end{aligned}$$

Definition 8.3

Given a set M, a real-valued function $\psi (x,y)$ defined on $M\times M$ is said to be negative definite (N.D.) if

$$\begin{aligned}&\psi (x,y)=\psi (y,x),\;\; \forall x, y\in M, \quad \psi (x,x)\equiv 0, \quad \hbox {and}\\&\forall \alpha _1,\ldots ,\alpha _n \in {\mathbb R}\;\;\hbox {s.t.}\;\; \sum _i \alpha _i=0, \;\; \forall x_1,\ldots , x_n\in M, \quad \sum _{i,j=1}^n \alpha _i\alpha _j \psi (x_i,x_j)\le 0. \end{aligned}$$

The following characterization is valid (see, e.g., [7, Proposition 3.2]):

$$\begin{aligned} \psi (x,y) \;\hbox { is N.D.}\quad \Longleftrightarrow \quad \psi (x,y)= {\mathbb E}(Z_x-Z_y)^2, \end{aligned}$$

where $(Z_x)_{x\in M}$ is some Gaussian process.

Consequently, if $\psi (x,y)$ is N.D., then $\psi (x,y) \ge 0$, $\forall x,y \in M$, and $\sqrt{\psi (x,y)}$ verifies the triangular inequality.

The following proposition is easy to verify.

Proposition 8.4

(a) Let K(x, y) be a P.D. kernel on a set M, and set

$$\begin{aligned} \psi _K(x,y):= K(x,x)+ K(y,y)- 2K(x,y). \end{aligned}$$

(8.1)

Then $\psi _K$ is negative definite. The function $\psi _K$ will be termed the N.D. function associated with K. In fact, if $K(x,y)= {\mathbb E}(Z_x Z_y)$, then $\psi _K(x,y)= {\mathbb E}(Z_x-Z_y)^2$. Furthermore, $\psi _K \equiv \psi _{K_u}$, $\forall u\in M$.

(b) Let $\psi$ be a N.D. function, and for any $u\in M$, define

$$\begin{aligned} N(u,\psi )(x,y):=\frac{1}{2} [\psi (x,u)+\psi (y,u)-\psi (x,y)]. \end{aligned}$$

Thus, if $\psi (x,y)= {\mathbb E}(Z_x-Z_y)^2$, then $N(u,\psi )(x,y) :={\mathbb E}\big [(Z_x-Z_u)(Z_y-Z_u)\big ]$. Then $N(u,\psi )$ is P.D. Moreover,

$$\begin{aligned} N(u,\psi _K)=K_u. \end{aligned}$$

(c) If K is P.D., then $K(x,y) \equiv {\text {constant}}\Longleftrightarrow \psi _K \equiv 0$.

Proposition 8.5

Let $\psi (x,y)$ be a real-valued continuous N.D. function on the compact space M, $\mu$ a positive Radon measure, with support M, and set

$$\begin{aligned} \tilde{K}(x,y):= \frac{1}{2|M|} \int _M [\psi (x,u) + \psi (y,u) -\psi (x,y)] d\mu (u). \end{aligned}$$

Then

(a) $\tilde{K}$ is positive definite, and $\psi _{\tilde{K}}=\psi$.

(b) ${\mathbb {1}}$ is an eigenfunction of the operator $\tilde{K}$ with kernel $\tilde{K}(x, y)$; that is,

$$\begin{aligned} \int _M \tilde{K}(x,y) d\mu (y) \equiv \tilde{\lambda } , \quad \tilde{\lambda }= \frac{1}{2|M|}\int _M\int _M \psi (u,y)d\mu (u) d\mu (y) (\ge 0). \end{aligned}$$

(c)

$$\begin{aligned} \exists z \in M \;\;\hbox {s.t.}\;\; \tilde{K}(z,z)=0 \;\Longleftrightarrow \; \tilde{K}(x,y) \equiv 0 \;\Longleftrightarrow \; \psi (x,y) \equiv 0. \end{aligned}$$

Proof

Parts (a) and (b) are straightforward. For the proof of (c), we first observe the obvious implications:

$$\begin{aligned} \psi (x,y) \equiv 0 \; \Longrightarrow \; \tilde{K}(x,y) \equiv 0 \; \Longrightarrow \; \tilde{K}(z,z)=0, \; \forall z\in M. \end{aligned}$$

Now, let $\tilde{K}(z,z)=0$ for some $z\in M$. Then

$$\begin{aligned} \frac{1}{2|M|} \int _M [\psi (z,u) + \psi (z,u) -\psi (z,z)] d\mu (u)=0. \end{aligned}$$

By definition, $\psi (z,z)=0$, and hence $\int _M \psi (z,u)d\mu (u)=0$. However, $\psi (z,u)$ is continuous, $\psi (z,u)\ge 0$, and ${\text {supp}}\,(\mu )=M$. Therefore, $\psi (z,u)=0$, $\forall u\in M$. Now, by the triangle inequality, we obtain for $x,y \in M$,

$$\begin{aligned} 0\le \sqrt{\psi (x,y)} \le \sqrt{\psi (x,z)} +\sqrt{\psi (z,y)} =0, \end{aligned}$$

and hence $\psi (x,y) \equiv 0$. This completes the proof. $\square$

Remark 8.6

One can verify easily that if K(x, y) is P.D. on M, then

$$\begin{aligned} K_u(x,y)&:= K(x,y) +K(u,u)-K(x,u)-K(y,u)\\&= \frac{1}{2} [\psi _K(x,u) + \psi _K(y,u) -\psi _K(x,y)]. \end{aligned}$$

The proof of the following proposition is straightforward.

Proposition 8.7

Let M be a compact space, equipped with a Radon measure $\mu$. Assume that K(x, y) is a continuous P.D. kernel, and as previously, let

$$\begin{aligned} \psi (x, y):=\psi _K(x, y)= K(x,x)+ K(y,y)- 2K(x,y) \; \hbox { be the associated N.D. kernel}, \end{aligned}$$

$$\begin{aligned} K_u(x,y)&:= K(x,y) +K(u,u)-K(x,u)-K(y,u) \\&= \frac{1}{2} [\psi (x,u) + \psi (y,u) -\psi (x,y)],\\ \tilde{K}(x,y)&:= \frac{1}{2|M|} \int _M [\psi (x,u) + \psi (y,u) -\psi (x,y)] d\mu (u)\\&= \frac{1}{|M|}\int _M K_u(x,y) d\mu (u). \end{aligned}$$

Denote by K and $\tilde{K}$ the operators with kernels K(x, y) and $\tilde{K}(x, y)$. Then

$$\begin{aligned} \tilde{K}(x,y)= K(x,y) + |M|^{-1}{\text {Tr}}(K) - |M|^{-1} K{\mathbb {1}}(x) - |M|^{-1}K{\mathbb {1}}(y). \end{aligned}$$

(8.2)

Moreover, $\psi _{\tilde{K}} =\psi$, $\tilde{K}_u=K_u$, and $\tilde{K}{\mathbb {1}}= \tilde{\lambda } {\mathbb {1}}$, where

$$\begin{aligned} \tilde{\lambda }&= Tr(K)-\frac{1}{|M|}\int _M\int _M K(x,y)d\mu (x)d\mu (y) \\&= \frac{1}{2|M|}\int _M\int _M \psi (u,y)d\mu (u) d\mu (y) \ge 0. \end{aligned}$$

In addition,

$$\begin{aligned} K= \tilde{K} + C \; \Longleftrightarrow \; K{\mathbb {1}}= \lambda {\mathbb {1}}, \end{aligned}$$

(8.3)

and, if so, $\tilde{\lambda } = ({\text {Tr}}(K)- \lambda )$, $C=\frac{1}{|M|}(Tr(K)-2\lambda )$.

Remark 8.8

The following useful assertions can be found in, e.g., [7, 9, 42, 43]. For N.D. functions, there exists a functional calculus that has no equivalent for P.D. functions:

• (1) Let F be a bounded completely continuous function, i.e.,

  $$\begin{aligned} \forall z>0, \;\forall n \in {\mathbb N}, \; D^nF(z) \ge 0 \end{aligned}$$

  or equivalently,

  $$\begin{aligned} F(z)= \int _0^\infty e^{-tz} d\mu (t), \;\; \mu \ge 0,\;\; \mu ([0, \infty ))<\infty . \end{aligned}$$

  Then

  $$\begin{aligned} \psi \; \hbox {is N.D.} \Longrightarrow F(\psi )\; \hbox {is P.D.} \end{aligned}$$

• (2) If G is a Bernstein function, i.e.,

  $$\begin{aligned} G(z)= az + \int _0^\infty (1-e^{-tz}) d\mu (t), \;\; a \ge 0, \;; \mu \ge 0, \;\; \int _0^\infty \frac{t}{1+t}d\mu (t) <\infty , \end{aligned}$$

  then

  $$\begin{aligned} \psi \hbox { N.D.} \Longrightarrow G(\psi ) \hbox { is N.D.} \end{aligned}$$

  For instance, we have:

  $$\begin{aligned} \psi \;\hbox {is N.D.} \;&\Longleftrightarrow \forall t>0, \; e^{-t \psi } \;\hbox {is P.D.},\\ \psi \; \hbox {is N.D.} \;&\Longrightarrow \forall \; 0<\alpha \le 1, \; \psi ^{\alpha } \;\hbox {is N.D.},\\ \psi \; \hbox {is N.D.} \;&\Longrightarrow \; \log (1+\psi ) \;\hbox {is N.D.} \end{aligned}$$

1.2 Appendix II: Gaussian Probability on Separable Banach Spaces

For a detailed account of the material in this section, we refer the reader to [10].

Let E be a Banach space, and let $\mathcal {B}(E)$ be the sigma-algebra of Borel sets on E. Let $E^*$ be its topological dual, and assume $\mathcal {F}$ is a vector space of real-valued functions defined on E, and $\gamma (\mathcal {F},E)$ is the sigma-algebra generated by $\mathcal {F}$.

If $\mathcal {F}= \mathcal {C}_{b}(E,{\mathbb R})$ is the vector space of continuous bounded functions on E, then $\gamma (\mathcal {C}_{b}(E,{\mathbb R}),E)= \mathcal {B}(E)$ is the Borel sigma-algebra.

If E is separable, it is well known that the sigma-algebra $\gamma (E^*,E)$ generated by $E^*$ is $\mathcal {B}(E)$ .

Proposition 8.9

Let E be a separable Banach space. Let H be a subspace of $E^*$, endowed with the $\sigma (E^*,E)$ topology. Then

$$\begin{aligned} H \hbox { is closed }\; \Longleftrightarrow \; H\hbox { is stable by simple limit}. \end{aligned}$$

Proof

The implication $\Rightarrow$ is obvious. We now prove $\Leftarrow$. By the Banach–Krein–Smulian theorem, H is $\sigma (E^*,E)$-closed if and only if $\forall R>0$, $B(0,R) \cap H$ is $\sigma (E^*,E)$-closed. As E is a separable Banach space, we have: For all $R>0$,

$$\begin{aligned} B(0,R)=\{f\in E^*: \Vert f\Vert _{E^*} \le R\} \; \hbox {is metrizable (and compact) for }\; \sigma (E^*,E). \end{aligned}$$

Hence we only have to verify that for every sequence $(f_n) \subset B(0,R) \cap H$ such that $\lim _{n \mapsto \infty } f_n= f$ in the $\sigma (E^*,E)-$topology, we have $f \in B(0,R) \cap H$. But clearly, this implies $\forall x\in E, \; \lim _{n\mapsto \infty } f_n(x)=f(x)$, so we have $f \in B(0,R) \cap H$. $\square$

Corollary 8.10

Let E be a separable Banach space and H a subspace of $E^*$. Then:

• (1) $\overline{H}^{\sigma (E^*,E)}$ coincides with the smallest vector space of functions on E, stable by simple limits containing H.

• (2)
  $$\begin{aligned} \gamma (H,E)= \gamma (\overline{H}^{\sigma (E^*,E)},E). \end{aligned}$$

• (3) If H is a subspace of $E^*$ separating E, then

  $$\begin{aligned} \gamma (H,E)= \gamma (E^*,E)=\mathcal {B}(E). \end{aligned}$$

Proof

(1) Clearly, as $E^*$ is stable by simple limits (by the Banach–Steinhaus theorem), the smallest vector space of functions on E, stable by simple limits containing H, is contained in $E^*$; hence, by the preceding proposition, it is $\overline{H}^{\sigma (E^*,E)}$.

(2) Let $\gamma (H,E)$ is the sigma-algebra generated by H. The vector subspace $V= \{u \in E^*: u, \; \gamma (H,E)- \hbox {measurable} \}$ is stable by simple limits. Hence, $\overline{H}^{\sigma (E^*,E)} \subset V$.

(3) By the Hahn–Banach theorem, if H is separating, $\overline{H}^{\sigma (E^*,E)}= E^*$, and hence

$$\begin{aligned} \gamma (H,E)= \gamma (E^*,E)=\mathcal {B}(E). \end{aligned}$$

$\square$

Lemma 8.11

Let E be a separable Banach space, and H be a subspace of $E^*$ separating E. There is at most one probability measure P on the Borel sets of E such that, under P, $\gamma \in H$ is a centered Gaussian variable with a given covariance $K(\gamma ,\gamma '):$

$$\begin{aligned} K(\gamma ,\gamma ')=\int _E \gamma (\omega ) \gamma '(\omega ) DP(\omega ) \end{aligned}$$

on H. Moreover, if such a probability exists, then:

• (1) $E^*$ is a Gaussian space, and $\overline{E^*}^{L^2(E,P)}$ is the Gaussian space generated by H.

• (2) There exists $\alpha >0$ such that

  $$\begin{aligned} \int _{E} e^{\alpha \Vert x \Vert _E^2} dP(x) <\infty . \end{aligned}$$

  (8.4)

Proof

If $K(\gamma ,\gamma ')$ is a positive definite function on H, it determines an additive function on the algebra of cylindrical sets related to H:

$$\begin{aligned} \big \{x\in E: (\gamma _1(x),\ldots ,\gamma _n(x) ) \in C\big \}, \; \gamma _i \in H, \; C \; \hbox { Borel set of}\; {\mathbb R}^n. \end{aligned}$$

Now, the sigma-algebra generated by this algebra is the Borel sigma-algebra of E.

Assume that such a probability P exists. Let $\mathcal {H}= E^* \cap \overline{H}^{L^2(E,P)}$. Clearly, $\overline{H}^{L^2(E,P)}$ is the Gaussian space generated by H, and if $(\gamma _n)_{n\ge 1} \in \mathcal {H}$ is such that $\forall x \in E$, $\lim _{n\mapsto \infty }\gamma _n(x)= \gamma (x)$ exists, then clearly $\gamma \in E^*$ by the Banach–Stheinhauss theorem, and $\gamma \in \overline{H}^{L^2(E,P)}$ since a simple limit of random variables in a closed Gaussian space belongs to this Gaussian space. Therefore, $\gamma \in \mathcal {H}$, which by Proposition 8.9 implies that $\mathcal {H}$ is closed. But $H \subset \mathcal {H}$ and $\overline{H}^{\sigma (E^*,E)}= E^*$ leads to $\mathcal {H}= E^*$.

Finally, (8.4) is just the Fernique theorem. $\square$

1.2.1 Cameron–Martin Space

Let us recall that, due to the Fernique theorem and Bochner integration, we have the following map from $E^*$ to E :

$$\begin{aligned} I: \gamma \in E^* \mapsto \int ^E_E \omega \gamma (\omega )dP(\omega ) \in E \end{aligned}$$

as

$$\begin{aligned} \Big \Vert \int ^E_E \omega \gamma (\omega )dP(\omega )\Big \Vert _E \le \int ^E \Vert \omega \Vert |\gamma (\omega )|dP(\omega ) \le \Big (\int ^E \Vert \omega \Vert ^2 |dP(\omega )\Big )^{\frac{1}{2}} \Vert \gamma \Vert _{L^2(P,E)} \end{aligned}$$

and

$$\begin{aligned} \gamma '(I(\gamma )) = \int _E \gamma '(\omega ) \gamma (\omega )dP(\omega ), \quad \forall \gamma , \gamma ' \in E^*. \end{aligned}$$

Therefore, I can be extended to $\bar{I}: \overline{E^*}^{L^2(E,P)} \mapsto E$. The subspace

$$\begin{aligned}{\mathbb H}\subset E, \quad {\mathbb H}= \bar{I}(\overline{E^*}^{L^2(E,P)}) \end{aligned}$$

with the induced Hilbert structure is the Cameron–Martin space associated with the Gaussian probability space $(E, \mathcal {B}(E),P)$ (see [10]).

1.3 Important Special Case

Let M be a set, and let E be a separable Banach space of real-valued functions on M. Let

$$\begin{aligned} \forall x \in M, \; f \in E \xrightarrow {\delta _x} f(x) \in {\mathbb R}. \end{aligned}$$

Suppose $\delta _x \in E^*$. So, $\mathcal {H}=\{ \sum _\mathrm{finite} \alpha _i \delta _{x_i}\}$ is dense in $E^*$ in the $\sigma (E^*,E)-$ topology.

Let K(x, y) be a positive definite function on $M\times M$. There is at most one probability measure P on the Borel sets of E such that, under P, $E^*$ is a Gaussian space and $(\delta _x)_{x\in M}$ is a centered Gaussian process with covariance

$$\begin{aligned} K(x,y)= \int _E \delta _x(\omega ) \delta _y(\omega )dP(\omega ), \;\;\hbox {i.e.}, \;\; \int _E e^{-i t \delta _x(\omega )} dP(\omega ) = e^{-\frac{1}{2}t^2 K(x,x)},\;\; \forall t \in {\mathbb R}. \end{aligned}$$

The Cameron–Martin space is identified with the Reproducing Kernel Hilbert Space ${\mathbb H}_K$ associated with K, i.e., the closure of

$$\begin{aligned} \Big \{ y \in M \mapsto f(y)=\sum _i \lambda _i K(x_i,y) \}; \quad \Vert f \Vert ^2_{{\mathbb H}_K}= \sum _{i,j} \lambda _i \lambda _j K(x_i,x_j) \Big \}. \end{aligned}$$

${\mathbb H}_K$ is characterized as a Hilbert space of functions on M such that

$$\begin{aligned} \forall x\in M, f \in {\mathbb H}_K \mapsto f(x)= \langle K(x,.), f \rangle _{{\mathbb H}_K} \; (\hbox {is continuous}). \end{aligned}$$

Therefore, if such a P exists on E, then ${\mathbb H}_K \subseteq M$.