Brownian Motion

4 Kolmogorov-Chentsov Theorem

We follow the proof of the Kolmogorov-Chentsov theorem from [ KU23 ] . That proof notably uses the chaining technique developed by Talagrand [ Tal22 ] .

That theorem is about stochastic processes \(X : T \to \Omega \to E\), where \(\Omega \) is a measurable space with a probability measure \(\mathbb {P}\), the index set \(T\) is a metric space with distance \(d_T\), and \(E\) is also a metric space with distance \(d_E\), on which we put the Borel \(\sigma \)-algebra.

The main result is Theorem 4.48. Under an assumption on the covering number of \(T\), for a process \(X\) that satisfies the Kolmogorov condition \(\mathbb {E}[d_E(X_s, X_t)^p] \le M d_T(s, t)^q\) (see Definition 4.29),the theorem gives a finite bound on the expectation of the supremum of the ratio

\begin{align*} \mathbb {E}\left[ \sup _{s, t \in T'} \frac{d_E(X_s, X_t)^p}{d_T(s, t)^{\beta p}} \right] \: , \end{align*}

for \(T'\) a countable subset of \(T\). As a corollary, we obtain that there exists a version of \(X\) with Hölder continuous paths.

In Lean, we will use the typeclass PseudoEMetricSpace for both \(T\) and \(E\) as long as possible, and then specialize to EMetricSpace (or perhaps even MetricSpace) when we need the stronger properties of a metric space. For example, to prove the existence of a version of a stochastic process, we will eventually use the fact that \(d_E(x, y) = 0\) implies \(x = y\), which does not hold in a pseudo-metric space. All distances will be expressed with edist, which takes values in ENNReal, and the integrals refer to Lebesgue integrals.

4.1 Covers and covering numbers

Let \((E, d_E)\) be a pseudo-metric space.

Definition 4.1 \(\varepsilon \)-cover
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A set \(C \subseteq E\) is an \(\varepsilon \)-cover of a set \(A \subseteq E\) if for every \(x \in A\), there exists \(y \in C\) such that \(d_E(x, y) \le \varepsilon \).

Definition 4.2 External covering number
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The external covering number of a set \(A \subseteq E\) for \(\varepsilon \ge 0\) is the smallest cardinality of an \(\varepsilon \)-cover of \(A\). Denote it by \(N^{ext}_\varepsilon (A)\).

Definition 4.3 Internal covering number
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The internal covering number of a set \(A \subseteq E\) for \(\varepsilon \ge 0\) is the smallest cardinality of an \(\varepsilon \)-cover of \(A\) which is a subset of \(A\). Denote it by \(N^{int}_\varepsilon (A)\).

\(N^{ext}_\varepsilon (A) \le N^{int}_\varepsilon (A)\).

Proof
Definition 4.5 Bounded internal covering number
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Let \(\mathrm{diam}(A)\) be the diameter of \(A \subseteq E\), i.e. \(\mathrm{diam}(A) = \sup _{x,y \in A} d_E(x, y)\). A set \(A \subseteq E\) has bounded internal covering number with constant \(c{\gt}0\) and exponent \(t{\gt}0\) if for all \(\varepsilon \in (0, \mathrm{diam}(A))\), \(N^{int}_\varepsilon (A) \le c \varepsilon ^{-t}\).

Lemma 4.6

The unit interval \(I = [0, 1] \subseteq \mathbb {R}\) has bounded internal covering number with constant \(1\) and exponent \(1\): for \(\varepsilon \le 1\), \(N^{int}_\varepsilon (I) \le 1/\varepsilon \).

Proof

4.2 Chaining

4.2.1 Chaining sequence

Definition 4.7
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Let \(S\) be a finite set of \(E\) and \(x \in E\). We denote by \(\pi (x, S)\) the point in \(S\) which is closest to \(x\), i.e. a point such that \(d_E(x, S) = \min _{y \in S} d_E(x, y)\) (chosen arbitrarily among the minima if there are several).

Lemma 4.8
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Let \(S\) be a finite set of \(E\) and \(x \in E\). Then for all \(y \in S\), \(d_E(x, \pi (x, S)) \le d_E(x, y)\).

Proof

By definition.

Lemma 4.9

Let \(C_\varepsilon \) be a finite \(\varepsilon \)-cover of \(A \subseteq E\) (assuming such a finite cover exists). Then for all \(x \in A\), \(d_E(x, \pi (x, C_\varepsilon )) \le \varepsilon \).

Proof
Definition 4.10 Chaining sequence
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Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(A \subseteq E\) with \(C_n \subseteq A\) and \(x \in C_k\) for some \(k \in \mathbb {N}\). We define the chaining sequence of \(x\), denoted \((\bar{x}_i)_{i \le k}\), recursively as follows: \(\bar{x}_k = x\) and for \(i {\lt} k\), \(\bar{x}_i = \pi (\bar{x}_{i+1}, C_i)\).

Lemma 4.11
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Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(A \subseteq E\) with \(C_n \subseteq A\) and \(x \in C_k\) for some \(k \in \mathbb {N}\). Then for all \(i \le k\), \(\bar{x}_i\in C_i\).

Proof

By definition.

Lemma 4.12

Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(A \subseteq E\) with \(C_n \subseteq A\) and \(x \in C_k\) for some \(k \in \mathbb {N}\). Then for all \(i {\lt} k\), \(d_E(\bar{x}_i, \bar{x}_{i+1}) \le \varepsilon _i\).

Proof

Apply Lemma 4.9 with \(S = C_i\) and \(x = \bar{x}_{i+1}\).

Lemma 4.13

Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(A \subseteq E\) with \(C_n \subseteq A\) and \(x \in C_k\) for some \(k \in \mathbb {N}\). Then for \(m \le k\), \(d_E(\bar{x}_m, x) \le \sum _{i=m}^{k-1} \varepsilon _i\).

Proof

By the triangle inequality and Lemma 4.12,

\begin{align*} d_E(\bar{x}_m, x) \le \sum _{i=m}^{k-1} d_E(\bar{x}_i, \bar{x}_{i+1}) \le \sum _{i=m}^{k-1} \varepsilon _i \: . \end{align*}
Lemma 4.14
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Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(A \subseteq E\) with \(C_n \subseteq A\). Let \(m, k, \ell \in \mathbb {N}\) with \(m \le k\) and \(m \le \ell \) and let \(x \in C_k\) and \(y \in C_\ell \). Then

\begin{align*} d_E(\bar{x}_m, \bar{y}_m) & \le d_E(x, y) + \sum _{i=m}^{k-1} \varepsilon _i + \sum _{j=m}^{\ell -j} \varepsilon _j \end{align*}
Proof

Triangle inequality and Lemma 4.13.

Corollary 4.15

For \(\varepsilon _n = \varepsilon _0 2^{-n}\), with the hypothesis of Lemma 4.14, we have

\begin{align*} d_E(\bar{x}_m, \bar{y}_m) & \le d_E(x, y) + \varepsilon _0 2^{-m+2} \: . \end{align*}
Proof

4.2.2 A subset of pairs

We will be interested in bounding expressions of the form \(\sup _{s,t\in J, d_T(s,t) \le c} d_E(f(s), f(t))\) for a finite set \(J\) and some function \(f : T \to E\). This is a supremum over pairs in \(J\) and there could be \(\vert J \vert ^2\) such pairs. We will build a subset \(K\) of \(J^2\) which is much smaller, of size linear in \(\vert J \vert \), such that its points are not too far apart and

\begin{align*} \sup _{s,t\in J, d_T(s,t) \le c} d_E(f(s), f(t)) & \le 2 \sup _{(s,t) \in K} d_E(f(s), f(t)) \: . \end{align*}

The pairs \((s, t) \in K\) will still be close together, in the sense that \(d_T(s, t) \le c n\) for some \(n\) that is logarithmic in the size of \(J\).

For \(t \in V \subseteq T\) and \(u\ge 0\), we denote by \(B_V(t, u)\) the closed ball with center \(t\) and radius \(u\) in \(V\). That is, \(B_V(t, u) = \{ s \in V \mid d_T(s, t) \le u\} \).

We want to cover \(J\) with balls that have radius logarithmic in the number of points of the ball.

Definition 4.16
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Let \(V\) be a finite subset of a metric space and let \(t \in V\) and \(a {\gt} 1\), \(c {\gt} 0\). Let the log-size radius of \(t\) in \(V\), denoted by \(r_{V,t}\), be the smallest positive integer \(r\) such that \(\vert B_V(t, r c) \vert \le a^{r}\).

Lemma 4.17
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\(a^{r_{V,t}-1} \le \vert B_V(t, (r_{V,t}-1)c) \vert \) .

Proof
Lemma 4.18
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\(\vert B_V(t, r_{V,t}c) \vert \le a^{r_{V,t}}\) .

Proof
Definition 4.19 Log-size ball sequence
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Let \((T,d_T)\) be a metric space and let \(J \subseteq T\) be finite, \(a,c \in \mathbb R_+\) with \(a \ge 1\) and \(n \in \{ 1, 2, ...\} \) such that \(|J| \le a^n\). An log-size ball sequence for \((J, a, c, n)\) is a sequence of \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) such that

  • \(V_0 = J\), \(t_0\) is an arbitrary point in \(J\),

  • for all \(i\), \(r_i\) is the log-size radius of \(t_i\) in \(V_i\),

  • \(V_{i+1} = V_i \setminus B_{V_i}(t_i, (r_i - 1)c)\), \(t_{i+1}\) is arbitrarily chosen in \(V_{i+1}\).

A log-size ball sequence gives a partition of \(J\) into sets which are contained in balls of radius \((r_i - 1)c\) around \(t_i\), and satisfy cardinality constraints.

Lemma 4.20
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The radius of a log-size ball sequence \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) for \((J, a, c, n)\) satisfies \(r_i \le n\) for all \(i \in \mathbb {N}\).

Proof

Since \(|J| \le a^n\), we have \(\vert B_{V_i}(t_i, n c) \vert \le \vert J \vert \le a^{n}\).

Lemma 4.21

The sets \(V_i\) of a log-size ball sequence \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) are a decreasing sequence of sets. That is, \(V_{i+1} \subseteq V_i\) for all \(i \in \mathbb {N}\).

Proof

\(V_{i+1} = V_i \setminus B_{V_i}(t_i, (r_i - 1)c)\) hence \(V_{i+1} \subseteq V_i\).

Lemma 4.22

For any log-size ball sequence \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) for \((J, a, c, n)\), for all \(k \ge \vert J \vert \), \(V_k = \emptyset \).

Proof

\(V_{i+1} = V_i \setminus B_{V_i}(t_i, (r_i - 1)c)\) and since \(t_i \in B_{V_i}(t_i, (r_i - 1)c)\), we have \(\vert V_{i+1} \vert {\lt} \vert V_i \vert \) and the cardinal eventually reaches \(0\), in at most \(\vert J \vert \) steps.

Lemma 4.23

For \(i \ne j\), the balls \(B_{V_i}(t, (r_i-1)c)\) and \(B_{V_j}(t_j, (r_j-1)c)\) of a log-size ball sequence \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) are disjoint.

Proof

Assume w.l.o.g. that \(i {\lt} j\). Then \(B_{V_j}(t_j, (r_j-1)c) \subseteq V_j \subseteq V_{i+1}\). It suffices to show that \(B_{V_i}(t_i, (r_i-1)c)\) and \(V_{i+1}\) are disjoint. This follows from the definition of \(V_{i+1} = V_i \setminus B_{V_i}(t_i, (r_i-1)c)\).

Definition 4.24
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Let \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) be a log-size ball sequence for \((J, a, c, n)\). For \(i \in \mathbb {N}\), let \(K_i = \{ t_i\} \times B_{V_i}(t_i, r_i c)\) be the set of pairs \((t_i, s)\) for \(s\) in the ball \(B_{V_i}(t_i, r_i c)\). We define \(K = \bigcup _{i=0}^{\vert J \vert -1} K_i\), set of all pairs from the log-size ball sequence.

Lemma 4.25

The cardinal of the pair set \(K\) of a log-size ball sequence for \((J, a, c, n)\) satisfies \(|K| \le a |J|\).

Proof

Using Lemma 4.18, the cardinal of \(K\) is bounded by

\begin{align*} \vert K \vert & \le \sum _{i=0}^{m-1} \vert K_i \vert \le \sum _{i=0}^{m-1} a^{r_i} \: . \end{align*}

Since the sets \(B_{V_i}(t_i, (r_i-1)c)\) are disjoint by Lemma 4.23, we can use Lemma 4.17 to get

\begin{align*} \sum _{i=0}^{m-1} a^{r_i - 1} \le \sum _{i=0}^{m-1} \vert B_{V_i}(t_i, (r_i-1)c) \vert = \left\vert \bigcup _{i=0}^{m-1} B_{V_i}(t_i, (r_i-1)c) \right\vert \le \vert J \vert \: . \end{align*}

We obtained the inequality \(\vert K \vert \le a \vert J \vert \)

Lemma 4.26
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Let \((s, t)\) be a pair in the pair set \(K\) of a log-size ball sequence for \((J, a, c, n)\). Then \(d_T(s, t) \le c n\).

Proof

A pair \((t, s) \in K\) is of the form \((t_i, s)\) for \(s \in B_V(t_i, r_i c)\) and satisfies

\begin{align*} d_T(t_i, s) \le c r_i \le c n \: . \end{align*}

The last inequality is from Lemma 4.20.

Lemma 4.27
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Let \(K\) be the pair set of a log-size ball sequence \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) for \((J, a, c, n)\). Then for any function \(f : T \to E\) with \((E,d_E)\) a metric space,

\begin{align*} \sup _{s,t\in J, d_T(s,t) \le c} d_E(f(s), f(t)) & \le 2 \sup _{(s,t) \in K} d_E(f(s), f(t)) \: . \end{align*}
Proof

Let \((s, t) \in J^2\) such that \(d_T(s, t) \le c\). Then there exists a largest \(\ell \in \mathbb {N}\) such that \(s, t \in V_\ell \). Assume w.l.o.g. that \(s \notin V_{\ell + 1}\). Then \(s \in B_{V_\ell }(t_\ell , (r_\ell -1)c)\) (since \(V_{\ell + 1} = V_\ell \setminus B_{V_\ell }(t_\ell , (r_\ell -1)c)\)), which implies \(d_T(s, t_\ell ) \le (r_\ell - 1)c\).

Since \(d_T(s, t) \le c\), \(d_T(t, t_\ell ) \le d_T(t, s) + d_T(s, t_\ell ) \le r_\ell c\), hence \(t \in B_{V_\ell }(t_\ell , r_\ell c)\) and we have that both \(s\) and \(t\) are in \(B_{V_\ell }(t_\ell , r_\ell c)\). Thus both \((t_\ell , s)\) and \((t_\ell , t)\) are in \(K_\ell \subseteq K\). Finally

\begin{align*} d_E(f(s), f(t)) & \le d_E(f(s), f(t_\ell )) + d_E(f(t_\ell ), f(t)) \\ & \le 2\sup _{(s',t') \in K} d_E(f(s’), f(t’)) \: . \end{align*}
Lemma 4.28

Let \((T,d_T)\) be a metric space. Let \(J \subseteq T\) be finite, \(a {\gt} 1\), \(c{\gt}0\) and \(n \in \{ 1, 2, ...\} \) such that \(|J| \le a^n\). Then, there is \(K \subseteq J^2\) such that for any function \(f : T \to E\) with \((E,d_E)\) a metric space,

\begin{align} |K| & \le a |J| \: , \label{eq:chain1} \\ \forall (s,t) \in K, & \: d_T(s,t) \le c n \: , \label{eq:chain2} \\ \sup _{s,t\in J, d_T(s,t) \le c} d_E(f(s), f(t)) & \le 2 \sup _{(s,t) \in K} d_E(f(s), f(t)) \: . \label{eq:chain3} \end{align}
Proof

Let \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) be a log-size ball sequence for \((J, a, c, n)\). We show that its pair set satisfies the conditions of the lemma.

Equation 1 is given by Lemma 4.25. The second property 2 is Lemma 4.26. Equation 3 was proved in Lemma 4.27.

4.3 Chaining for stochastic processes under Kolmogorov conditions

4.3.1 Kolmogorov condition

Definition 4.29 Kolmogorov condition
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Let \(X : T \to \Omega \to E\) be a stochastic process, where \((T, d_T)\) and \((E, d_E)\) are pseudo-metric spaces and \((\Omega , \mathbb {P})\) is a measure space. Let \(p, q {\gt} 0\). We say that \(X\) satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\) if for all \(s, t \in T\), \((X_s, X_t)\) is \(\mathbb {P}\)-a.e. measurable for the Borel \(\sigma \)-algebra on \(E^2\) and

\begin{align*} \mathbb {E}[d_E(X_s, X_t)^p] \le M d_T(s, t)^q \: . \end{align*}

Remark: the measurability condition on the pair could be replaced by the simpler condition of measurability of \(X_t\) for all \(t \in T\) if we assumed that \(E\) is separable (SecondCountableTopology in Lean), which implies that the Borel \(\sigma \)-algebra on the product is equal to the product of the Borel \(\sigma \)-algebras. We follow [ KU23 ] and do not require separability.

Measurability

Lemma 4.30

If \(X : T \to \Omega \to E\) is a process that satisfies the Kolmogorov condition, then for all \(t \in T\), \(X_t\) is \(\mathbb {P}\)-a.e. measurable.

Proof
Lemma 4.31

If \(E\) is separable and \(X : T \to \Omega \to E\) is a process such that \(X_t\) is \(\mathbb {P}\)-a.e. measurable for all \(t \in T\), then for all \(s, t \in T\), the pair \((X_s, X_t)\) is \(\mathbb {P}\)-a.e. measurable for the Borel \(\sigma \)-algebra on \(E^2\).

Proof
Lemma 4.32

If \(X : T \to \Omega \to E\) is a process that satisfies the Kolmogorov condition, then for all \(s,t \in T\) the function \(\omega \mapsto d_E(X_s(\omega ), X_t(\omega ))\) is \(\mathbb {P}\)-a.e. measurable.

Proof

Distance bounds

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \(\varepsilon {\gt} 0\) and \(C \subseteq T^2\) be a finite set such that for all \((s, t) \in C\), \(d_T(s, t) \le \varepsilon \). Then

\begin{align*} \mathbb {E}\left[\sup _{(s,t) \in C} d_E(X_s, X_t)^p \right] & \le \vert C \vert M \varepsilon ^q \: . \end{align*}
Proof
\begin{align*} \mathbb {E}\left[\sup _{(s,t) \in C} d_E(X_s, X_t)^p \right] & \le \mathbb {E}\left[\sum _{(s,t) \in C} d_E(X_s, X_t)^p \right] \\ & \le M \sum _{(s,t) \in C} d_T(s, t)^q \\ & \le \vert C \vert M \varepsilon ^q \: . \end{align*}

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \(J \subseteq T\) be finite, \(a, c \in \mathbb R_+\) with \(a \ge 1\) and \(n \in \{ 1, 2, ...\} \) such that \(|J| \le a^n\). Then

\begin{align*} \mathbb {E} \left[ \sup _{s, t \in J; d_T(s, t) \le c} d_E(X_s, X_t)^p \right] & \le 2^p a |J| M (cn)^q \: . \end{align*}
Proof

By Lemma 4.28, there exists \(K \subseteq J^2\) such that

\begin{align*} |K| & \le a |J| \: , \\ \forall (s,t) \in K, & \ d_T(s,t) \le c n \: , \\ \sup _{s,t\in J, d_T(s,t) \le c} d_E(X_s, X_t) & \le 2 \sup _{(s,t) \in K} d_E(X_s, X_t) \: . \end{align*}

Hence for such a set \(K\),

\begin{align*} \mathbb {E} \left[ \sup _{s, t \in J; d_T(s, t) \le c} d_E(X_s, X_t)^p \right] & \le 2^p \mathbb {E} \left[ \sup _{(s, t) \in K} d_E(X_s, X_t)^p \right] \: . \end{align*}

Then by Lemma 4.33,

\begin{align*} \mathbb {E} \left[ \sup _{(s, t) \in K} d_E(X_s, X_t)^p \right] & \le |K| M (cn)^q \le a |J| M (cn)^q \: . \end{align*}

4.3.2 Bound for a set of points that are close together

For a finite index set \(T\), we want to obtain a bound on

\begin{align*} \mathbb {E}\left[ \sup _{s, t \in T; d_T(s, t) \le \delta } d_E(X_s, X_t)^p \right] \: . \end{align*}

Note the condition that the supremum is taken over pairs \((s, t)\) such that \(d_T(s, t) \le \delta \).

We consider covers of \(T\) at different scales. \(C_n\) is a finite \(\varepsilon _n\)-cover of \(T\) with \(\varepsilon _n = \varepsilon _0 2^{-n}\). \(T\) is equal to \(C_k\) for some \(k\) large enough, so the supremum over \(T\) is a supremum at that scale \(k\). We will change scale to some \(m \le k\) that depends on the distance bound \(\delta \) (\(m\) is of order \(\log _2 \delta \)) and consider the supremum over \(C_m\) (plus a term due to the scale change). Then for the supremum over a set in \(C_m\), we use the reduction in the number of pairs of Lemma 4.28.

Lemma 4.35
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Let \(X : T \to E\). Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(J \subseteq T\) with \(C_n \subseteq J\). For \(m \le k\),

\begin{align*} \sup _{s, t \in C_k; d_T(s, t) \le \delta } d_E(X_s, X_t) & \le \sup _{s, t \in C_k; d_T(s, t) \le \delta } d_E(X_{\bar{s}_m}, X_{\bar{t}_m}) + 2 \sup _{s \in C_k} d_E(X_s, X_{\bar{s}_m}) \: . \end{align*}
Proof

By the triangle inequality,

\begin{align*} d_E(X_s, X_t) & \le d_E(s, X_{\bar{s}_m}) + d(X_{\bar{s}_m}, X_{\bar{t}_m}) + d_E(X_{\bar{t}_m}, t) \: . \end{align*}
Corollary 4.36
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Let \(X : T \to E\). Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(J \subseteq T\) with \(C_n \subseteq J\). For \(m \le k\),

\begin{align*} \sup _{s, t \in C_k; d_T(s, t) \le \delta } d_E(X_s, X_t)^p & \le 2^p \sup _{s, t \in C_k; d_T(s, t) \le \delta } d_E(X_{\bar{s}_m}, X_{\bar{t}_m})^p + 4^p \sup _{s \in C_k} d_E(X_s, X_{\bar{s}_m})^p \: . \end{align*}
Proof

This is Lemma 4.35, together with the fact that for \(a, b \ge 0\),

\begin{align*} (a + b)^p \le (2\max (a,b))^p = 2^p \max (a^p,b^p) \le 2^p (a^p + b^p) \: . \end{align*}

First term

Lemma 4.37

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \(C\) be a finite \(\varepsilon \)-cover of \(J \subseteq T\) with \(C \subseteq J\), with minimal cardinal. Then for \(c \ge 0\),

\begin{align*} \mathbb {E} \left[ \sup _{s, t \in C; d_T(s, t) \le c} d_E(X_s, X_t)^p \right] & \le 2^{p+2} M \left(2 c \log _2 N^{int}_{\varepsilon }(J) \right)^q N^{int}_{\varepsilon }(J) \: . \end{align*}

Note the logarithm has base \(2\).

Proof

Let \(\bar{r} = \min \{ r \in \mathbb {N} \mid N^{int}_{\varepsilon }(J) \le 2^r\} \).

\begin{align*} \vert C \vert = N^{int}_{\varepsilon }(J) \le 2^{\bar{r}} \: . \end{align*}

By Lemma 4.34 with \(J = C\), \(a = 2\), \(b = 1\), \(c = c\), \(n = \bar{r}\),

\begin{align*} \mathbb {E} \left[ \sup _{s, t \in C; d_T(s, t) \le c} d_E(X_s, X_t)^p \right] & \le 2^{p+1} |C_m| M (c \bar{r})^q \le 2^{p+1} M (c \bar{r})^q 2^{\bar{r}} \: . \end{align*}

By definition, \(2^{\bar{r}} \le 2 N^{int}_{\varepsilon }(J)\) and \(\bar{r} \le 1 + \log _2 N^{int}_{\varepsilon }(J)\). Suppose \(N^{int}_{\varepsilon }(J) \ge 2\) (if it equals one the result is trivial). Then \(\bar{r} \le 2 \log _2 N^{int}_{\varepsilon }(J)\).

\begin{align*} \mathbb {E} \left[ \sup _{s, t \in C; d_T(s, t) \le c} d_E(X_s, X_t)^p \right] & \le 2^{p+2} M \left(2 c \log _2 N^{int}_{\varepsilon }(J) \right)^q N^{int}_{\varepsilon }(J) \: . \end{align*}

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). For all \(n \in \mathbb {N}\), let \(C_n\) a finite \(\varepsilon _n\)-cover of \(J \subseteq T\) with \(C_n \subseteq J\) for \(\varepsilon _n = \varepsilon _0 2^{-n}\), with minimal cardinal. Let \(\delta {\gt} 0\) and let \(n_2\) be such that \(\varepsilon _0 2^{-n_2} {\lt} \delta /2 \le \varepsilon _0 2^{-n_2+1}\). Let \(k = \min \{ j \in \mathbb {Z} \mid \varepsilon _0 2^{-j} {\lt} \inf _{s, t \in J; s \ne t}d_T(s, t)\} \). For \(m = \min (n_2, k)\),

\begin{align*} \mathbb {E} \left[ \sup _{s, t \in C_k; d_T(s, t) \le \delta } d_E(X_{\bar{s}_m}, X_{\bar{t}_m})^p \right] & \le 2^{p+2} M \left(16 \delta \log _2 N^{int}_{\delta /4}(J) \right)^q N^{int}_{\delta /4}(J) \: . \end{align*}

TODO: \(C_k = J\)

Proof

If \(\delta \le \varepsilon _0 2^{-k}\), then \(\{ (s, t) \in C_k; d_T(s, t) \le \delta \} = \{ (s, s) \mid s \in C_k\} \) and the inequality holds trivially. We can thus assume \(\delta {\gt} \varepsilon _0 2^{-k}\).

For \(s, t \in C_k\) with \(d_T(s, t) \le \delta \), \(d_T(\bar{s}_m, \bar{t}_m) \le \delta + \varepsilon _0 2^{-m+2} \le \varepsilon _0 2^{-m+3}\) (Corollary 4.15). It thus suffices to get a bound on \(\mathbb {E} \left[ \sup _{s, t \in C_m; d_T(s, t) \le \varepsilon _0 2^{-m+3}} d_E(X_s, X_t)^p \right]\).

We can apply Lemma 4.37 with \(\varepsilon = \varepsilon _m\), \(c = \varepsilon _0 2^{-m+3}\). We obtain

\begin{align*} \mathbb {E} \left[ \sup _{s, t \in C_m; d_T(s, t) \le \varepsilon _0 2^{-m+3}} d_E(X_s, X_t)^p \right] & \le 2^{p+2} M \left(16 \varepsilon _0 2^{-m} \log _2 N^{int}_{\varepsilon _m}(J) \right)^q N^{int}_{\varepsilon _m}(J) \: . \end{align*}

By definition of \(m\) and \(n_2\), \(\varepsilon _m = \varepsilon _0 2^{-m} \ge \varepsilon _0 2^{-n_2} \ge \delta /4\), hence \(N^{int}_{\varepsilon _m}(J) \le N^{int}_{\delta / 4}(J)\).

If \(m = n_2\) then \(\varepsilon _0 2^{-m} = \varepsilon _0 2^{-n_2} {\lt} \delta /2\). Otherwise, \(m = k\) and \(\varepsilon _0 2^{-m} = \varepsilon _0 2^{-k} {\lt} \delta \) as argued at the start of the proof. We thus get \(\varepsilon _0 2^{-m} \le \delta \).

Second term

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers and \(C_n\) a finite \(\varepsilon _n\)-cover of \(T\) with \(C_n \subseteq T\). Then for \(j {\lt} k\),

\begin{align*} \mathbb {E}\left[\sup _{t \in C_k} d_E(X_{\bar{t}_j}, X_{\bar{t}_{j+1}})^p \right] & \le \vert C_{j+1} \vert M \varepsilon _j^q \: . \end{align*}
Proof
\begin{align*} \mathbb {E}\left[\sup _{t \in C_k} d_E(X_{\bar{t}_j}, X_{\bar{t}_{j+1}})^p \right] & \le \mathbb {E}\left[\sum _{u \in C_{j+1}} d_E(X_{\bar{u}_j}, X_{u})^p \right] \: . \end{align*}

We then apply Lemma 4.33 to the set \(C = \{ (\bar{u}_j, u) \mid u \in C_{j+1}\} \), which satisfies the condition \(d_T(\bar{u}_j, u) \le \varepsilon _j\) and has cardinal \(\vert C_{j+1} \vert \).

Lemma 4.40

Let \(X : T \to \Omega \to E\) be a stochastic process. Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers and \(C_n\) a finite \(\varepsilon _n\)-cover of \(T\) with \(C_n \subseteq T\). For \(m \le k\),

\begin{align*} \mathbb {E}\left[\sup _{t \in C_k} d_E(X_t, X_{\bar{t}_m})^p \right] & \le \left(\sum _{i=m}^{k-1} \left( \mathbb {E}\left[\sup _{t \in C_k} d_E(X_{\bar{t}_i}, X_{\bar{t}_{i+1}})^p\right] \right)^{1/p}\right)^p \: . \end{align*}
Proof
\begin{align*} \sup _{t \in C_k} d_E(X_t, X_{\bar{t}_m})^p & \le \sup _{t \in C_k} \left( \sum _{i=m}^{k-1} d_E(X_{\bar{t}_i}, X_{\bar{t}_{i+1}}) \right)^p \\ & \le \left( \sum _{i=m}^{k-1} \sup _{t \in C_k} d_E(X_{\bar{t}_i}, X_{\bar{t}_{i+1}}) \right)^p \: . \end{align*}

And then, by Minkowski’s inequality,

\begin{align*} \left(\mathbb {E} \left[\sup _{t \in C_k} d_E(X_t, X_{\bar{t}_m})^p \right]\right)^{1/p} & \le \sum _{i=m}^{k-1} \mathbb {E} \left[\sup _{t \in C_k} d_E(X_{\bar{t}_i}, X_{\bar{t}_{i+1}}) \right] \\ & \le \sum _{i=m}^{k-1} \left( \mathbb {E}\left[\sup _{t \in C_k} d_E(X_{\bar{t}_i}, X_{\bar{t}_{i+1}})^p \right] \right)^{1/p} \: . \end{align*}

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers and \(C_n\) a finite \(\varepsilon _n\)-cover of \(T\) with \(C_n \subseteq T\). Then for \(m \le k\),

\begin{align*} \mathbb {E} \left[\sup _{t \in C_k} d_E(X_t, X_{\bar{t}_m})^p \right] & \le M \left( \sum _{j=m}^{k-1} \vert C_{j+1} \vert ^{1/p} \varepsilon _j^{q/p} \right)^p \: . \end{align*}
Proof

Put together Lemma 4.39 and Lemma 4.40.

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers in \((0, \mathrm{diam}(T))\) and \(C_n\) a finite \(\varepsilon _n\)-cover of \(T\) with \(C_n \subseteq T\), and with minimal cardinality. Suppose that \(T\) has bounded internal covering number with constant \(c_1{\gt}0\) and exponent \(d {\gt} 0\). Then for \(m \le k\),

\begin{align*} \mathbb {E} \left[\sup _{t \in C_k} d_E(X_t, X_{\bar{t}_m})^p \right] & \le M c_1 \left( \sum _{j=m}^{k-1} \varepsilon _{j+1}^{-d/p} \varepsilon _j^{q/p} \right)^p \: . \end{align*}
Proof

By Lemma 4.41, we have

\begin{align*} \mathbb {E} \left[\sup _{t \in C_k} d_E(X_t, X_{\bar{t}_m})^p \right] & \le M \left( \sum _{j=m}^{k-1} \vert C_{j+1} \vert ^{1/p} \varepsilon _j^{q/p} \right)^p \: . \end{align*}

Then by the minimality of the cardinality of \(C_n\) and the bounded internal covering number hypothesis, we have

\begin{align*} \vert C_{j+1} \vert & \le N^{int}_{\varepsilon _{j+1}}(T) \le c_1 \varepsilon _{j+1}^{-d} \: . \end{align*}

Under the assumptions of Lemma 4.42, for \(\varepsilon _n = \varepsilon _0 2^{-n}\), then for \(m \le k\),

\begin{align*} \mathbb {E} \left[\sup _{t \in C_k} d_E(X_t, X_{\bar{t}_m})^p \right] & \le M c_1 \varepsilon _0^{q - d} 2^{d - m(q-d)}\left( \frac{1 - 2^{- (k - m) (q -d)/p}}{1 - 2^{- (q -d)/p}}\right)^p \\ & \le 2^d M c_1 (\varepsilon _0 2^{-m})^{q - d} \frac{1}{\left( 1 - 2^{- (q -d)/p}\right)^p} \: . \end{align*}
Proof

Putting it all together

Suppose that \(T\) is a finite set with bounded internal covering number with constant \(c_1{\gt}0\) and exponent \(d {\gt} 0\). Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\), with \(q {\gt} d\) and \(p {\gt} 0\). For all \(\delta {\gt} 0\),

\begin{align*} & \mathbb {E}\left[ \sup _{s, t \in T; d_T(s, t) \le \delta } d_E(X_s, X_t)^p \right] \\ & \le 4^{p+2q+1} M \delta ^{q-d} \left(\delta ^d \left(\log _2 N^{int}_{\delta /4}(T) \right)^q N^{int}_{\delta /4}(T) + \frac{c_1}{\left( 2^{(q -d)/p} - 1\right)^p}\right) \: . \end{align*}
Proof

Let \(\varepsilon _0 \in (0, \mathrm{diam}(T))\). For all \(n \in \mathbb {N}\), let \(C_n\) a finite \(\varepsilon _n\)-cover of \(T\) with \(C_n \subseteq T\) for \(\varepsilon _n = \varepsilon _0 2^{-n}\), with minimal cardinal. Let \(k = \min \{ j \in \mathbb {Z} \mid \varepsilon _0 2^{-j} {\lt} \inf _{s, t \in J; s \ne t}d_T(s, t)\} \). Note that \(C_k = T\). By Corollary 4.36,

\begin{align*} & \mathbb {E}\left[ \sup _{s, t \in C_k; d_T(s, t) \le \delta } d_E(X_s, X_t)^p \right] \\ & \le 2^p \mathbb {E}\left[ \sup _{s, t \in C_k; d_T(s, t) \le \delta } d_E(X_{\bar{s}_m}, X_{\bar{t}_m})^p \right] + 4^p \mathbb {E}\left[ \sup _{s \in C_k} d_E(X_s, X_{\bar{s}_m})^p \right] \: . \end{align*}

By Lemma 4.38 and Corollary 4.43,

\begin{align*} \mathbb {E} \left[ \sup _{s, t \in C_k; d_T(s, t) \le \delta } d_E(X_{\bar{s}_m}, X_{\bar{t}_m})^p \right] & \le 2^{p+2} M \left(16 \delta \log _2 N^{int}_{\delta /4}(T) \right)^q N^{int}_{\delta /4}(J) \: . \\ \mathbb {E} \left[\sup _{t \in C_k} d_E(X_t, X_{\bar{t}_m})^p \right] & \le 2^d M c_1 (\varepsilon _0 2^{-m})^{q - d} \frac{1}{\left( 1 - 2^{- (q -d)/p}\right)^p} \: . \end{align*}

Putting these two inequalities together, we obtain

\begin{align*} & \mathbb {E}\left[ \sup _{s, t \in C_k; d_T(s, t) \le \delta } d_E(X_s, X_t)^p \right] \\ & \le 4^p M \left(4\left(16 \delta \log _2 N^{int}_{\delta /4}(T) \right)^q N^{int}_{\delta /4}(T) + 2^d c_1 (\varepsilon _0 2^{-m})^{q - d} \frac{1}{\left( 1 - 2^{- (q -d)/p}\right)^p}\right) \: . \end{align*}

TODO: it remains to argue that w.l.o.g. \(\varepsilon _0 2^{-m} \le \delta \) as in another lemma above (todo: refactor that). We obtain

\begin{align*} & \mathbb {E}\left[ \sup _{s, t \in T; d_T(s, t) \le \delta } d_E(X_s, X_t)^p \right] \\ & \le 4^p M \left(4^{2q+1}\delta ^q \left(\log _2 N^{int}_{\delta /4}(T) \right)^q N^{int}_{\delta /4}(T) + c_1 2^d \delta ^{q - d} \frac{1}{\left( 1 - 2^{- (q -d)/p}\right)^p}\right) \\ & = 4^p M \left(4^{2q+1}\delta ^q \left(\log _2 N^{int}_{\delta /4}(T) \right)^q N^{int}_{\delta /4}(T) + c_1 2^q \delta ^{q - d} \frac{1}{\left( 2^{(q -d)/p} - 1\right)^p}\right) \\ & \le 4^{p+2q+1} M \delta ^{q-d} \left(\delta ^d \left(\log _2 N^{int}_{\delta /4}(T) \right)^q N^{int}_{\delta /4}(T) + \frac{c_1}{\left( 2^{(q -d)/p} - 1\right)^p}\right) \: . \end{align*}

With the same assumptions and notations as in Theorem 4.44, for all \(\delta \in (0, 4\mathrm{diam}(T))\),

\begin{align*} \mathbb {E}\left[ \sup _{s, t \in T; d_T(s, t) \le \delta } d_E(X_s, X_t)^p \right] & \le 4^{p+2q+1} M c_1 \delta ^{q-d} \left(4^d \left(\log _2 \left(c_1 \delta ^{-d} 4^d \right) \right)^q + \frac{1}{\left( 2^{(q -d)/p} - 1\right)^p}\right) \: . \end{align*}
Proof

We apply Theorem 4.44 and then remark that for \(\delta \le 4\mathrm{diam}(T)\), we can use the bounded internal covering number hypothesis to bound \(N^{int}_{\delta /4}(T)\) :

\begin{align*} N^{int}_{\delta /4}(T) \le c_1 \left(\frac{\delta }{4}\right)^{-d} \: . \end{align*}

4.4 Kolmogorov-Chentsov Theorem

4.4.1 Sets with bounded internal covering number

Lemma 4.46

Let \(J \subseteq T\) be a finite set and suppose that \(T\) has finite diameter. For \(k \in \mathbb {N}\), let \(\eta _k = 2^{-k}(\mathrm{diam}(T) + 1)\). For \(X : T \to \Omega \to E\) a stochastic process and \(\beta \in (0, (q - d)/p)\),

\begin{align*} \mathbb {E}\left[ \sup _{s, t \in J;\: s \ne t} \frac{d_E(X_s, X_t)^p}{d_T(s, t)^{\beta p}} \right] & \le \sum _{k=0}^\infty 2^{k \beta p} \mathbb {E}\left[ \sup _{s, t \in J;\: s \ne t, \: d_T(s, t) \le 2 \eta _k} d_E(X_s, X_t)^p \right] \: . \end{align*}
Proof

We introduce for each \(k \in \mathbb {N}\) the set of pairs \((s, t)\) such that \(\eta _k {\lt} d_T(s, t) \le 2 \eta _k\). Note that \(\eta _k \ge 2^{-k}\).

\begin{align*} \mathbb {E}\left[ \sup _{s, t \in J;\: s \ne t} \frac{d_E(X_s, X_t)^p}{d_T(s, t)^{\beta p}} \right] & \le \sum _{k=0}^\infty \mathbb {E}\left[ \sup _{s, t \in J;\: s \ne t, \: \eta _k {\lt} d_T(s, t) \le 2 \eta _k} \frac{d_E(X_s, X_t)^p}{d_T(s, t)^{\beta p}} \right] \\ & \le \sum _{k=0}^\infty \eta _k^{-\beta p} \mathbb {E}\left[ \sup _{s, t \in J;\: s \ne t, \: d_T(s, t) \le 2 \eta _k} d_E(X_s, X_t)^p \right] \\ & \le \sum _{k=0}^\infty 2^{k \beta p} \mathbb {E}\left[ \sup _{s, t \in J;\: s \ne t, \: d_T(s, t) \le 2 \eta _k} d_E(X_s, X_t)^p \right] \: . \end{align*}

Suppose that \(J \subseteq T\) is a finite set and that \(T\) has bounded internal covering number with constant \(c_1{\gt}0\) and exponent \(d {\gt} 0\). Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\), with \(q {\gt} d\) and \(p {\gt} 0\). Let \(\beta \in (0, (q - d)/p)\). Then there exists a constant \(L(T, c_1, d, p, q, \beta ) {\lt} \infty \) (that depends on \(T\) bot not \(J\)) such that

\begin{align*} \mathbb {E}\left[ \sup _{s, t \in J;\: s \ne t} \frac{d_E(X_s, X_t)^p}{d_T(s, t)^{\beta p}} \right] \le M L(T, c_1, d, p, q, \beta ) \: . \end{align*}
Proof

Let \(\eta _k = 2^{-k}(\mathrm{diam}(T) + 1)\) for \(k \in \mathbb {N}\). By Lemma 4.46, we have

\begin{align*} \mathbb {E}\left[ \sup _{s, t \in J;\: s \ne t} \frac{d_E(X_s, X_t)^p}{d_T(s, t)^{\beta p}} \right] & \le \sum _{k=0}^\infty 2^{k \beta p} \mathbb {E}\left[ \sup _{s, t \in J;\: s \ne t, \: d_T(s, t) \le 2 \eta _k} d_E(X_s, X_t)^p \right] \: . \end{align*}

We want to show that the sum is finite.

Let \(k_0 = \min \{ k \in \mathbb {N} \mid \eta _{k+1} \le \mathrm{diam}(T)\} \). We deal separately with the parts of the sum for \(k {\lt} k_0\) and \(k \ge k_0\).

Large \(k\).

For \(k \ge k_0\), \(2\eta _k / 4 = \eta _{k+1} \le \mathrm{diam}(T)\), which means that we can apply Corollary 4.45 to bound each expectation in the sum.

\begin{align*} & \mathbb {E}\left[ \sup _{s, t \in J;\: s \ne t, \: d_T(s, t) \le 2 \eta _k} d_E(X_s, X_t)^p \right] \\ & \le 4^{p+2q+1} M c_1 (2 \eta _k)^{q-d} \left(4^d \left(\log _2 \left(c_1 (2 \eta _k)^{-d} 4^d \right) \right)^q + \frac{1}{\left( 2^{(q -d)/p} - 1\right)^p}\right) \\ & \le 4^{p+2q+1} M c_1 (\mathrm{diam}(T)+1)^{q-d} 2^{(-k + 1)(q-d)} \left(4^d \left(\log _2 \left(c_1 2^{(k + 1)d} \right) \right)^q + \frac{1}{\left( 2^{(q -d)/p} - 1\right)^p}\right) \\ & = 4^{p+2q+1} M c_1 (\mathrm{diam}(T)+1)^{q-d} 2^{(-k + 1)(q-d)} \left(4^d \left(\log _2(c_1) + (k + 1)d \right)^q + \frac{1}{\left( 2^{(q -d)/p} - 1\right)^p}\right) \: . \end{align*}

Let \(a_k = 2^{k \beta p} 4^{p+2q+1} M c_1 (\mathrm{diam}(T)+1)^{q-d} 2^{(-k + 1)(q-d)} \left(4^d \left(\log _2(c_1) + (k + 1)d \right)^q + \frac{1}{\left( 2^{(q -d)/p} - 1\right)^p}\right)\). To show that the sum \(\sum _{k=k_0}^\infty a_k\) is finite, we can use the ratio test.

\begin{align*} \frac{\vert a_{k+1} \vert }{\vert a_k \vert } & = 2^{\beta p - (q - d)} \frac{\left(4^d \left(\log _2(c_1) + (k + 2)d \right)^q + \frac{1}{\left( 2^{(q -d)/p} - 1\right)^p}\right)}{\left(4^d \left(\log _2(c_1) + (k + 1)d \right)^q + \frac{1}{\left( 2^{(q -d)/p} - 1\right)^p}\right)} \end{align*}

The limit at infinity of that ratio is \(2^{\beta p - (q - d)} {\lt} 1\), hence the series \(\sum _{k=k_0}^\infty a_k\) converges.

Small \(k\).

For \(k {\lt} k_0\), \(N^{int}_{\eta _{k+1}}(T) = 1\). Applying Theorem 4.44 with \(\delta = 2 \eta _k\), we obtain

\begin{align*} & \mathbb {E}\left[ \sup _{s, t \in T; d_T(s, t) \le 2 \eta _k} d_E(X_s, X_t)^p \right] \\ & \le 4^{p+2q+1} M (2 \eta _k)^{q-d} \frac{c_1}{\left( 2^{(q -d)/p} - 1\right)^p} \\ & \le 4^{p+2q+1} M (\mathrm{diam}(T) + 1)^{q-d} 2^{(-k+1)q-d} \frac{c_1}{\left( 2^{(q -d)/p} - 1\right)^p} \: . \end{align*}

This is an expression of the form \(K 2^{-k(q-d)}\) for some constant \(K {\lt} \infty \). Coming back to the sum, we have

\begin{align*} \sum _{k=1}^{k_0 - 1} 2^{k \beta p} \mathbb {E}\left[ \sup _{s, t \in J;\: s \ne t, \: d_T(s, t) \le 2 \eta _k} d_E(X_s, X_t)^p \right] & \le K \sum _{k=1}^{k_0 - 1} 2^{k (\beta p - (q-d))} \\ & \le K \frac{1}{1 - 2^{(\beta p - (q-d))}} \: . \end{align*}

The sum is finite since \(\beta p {\lt} (q - d)\) by assumption.

Suppose that \(T\) has bounded internal covering number with constant \(c_1{\gt}0\) and exponent \(d {\gt} 0\). Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\), with \(q {\gt} d\) and \(p {\gt} 0\). Let \(\beta \in (0, (q - d)/p)\). Then there exists a constant \(L(T, c_1, d, p, q, \beta ) {\lt} \infty \) such that for every countable subset \(T' \subseteq T\) with positive diameter,

\begin{align*} \mathbb {E}\left[ \sup _{s, t \in T';\: s \ne t} \frac{d_E(X_s, X_t)^p}{d_T(s, t)^{\beta p}} \right] \le M L(T, c_1, d, p, q, \beta ) \: . \end{align*}
Proof

Build a monotone sequence of finite sets \(T_n \subseteq T'\), use Lemma 4.47 to obtain a bound for each \(T_n\) that does not depend on \(T_n\), and then use monotone convergence.

Corollary 4.49

Under the same assumptions as in Theorem 4.48, for every countable subset \(T' \subseteq T\) with positive diameter, for \(L(T, c_1, d, p, q, \beta ) {\lt} \infty \) the same constant,

\begin{align*} \mathbb {E}\left[ \sup _{s, t \in T';\: d_T(s, t) \le \delta } d_E(X_s, X_t)^p \right] \le M L(T, c_1, d, p, q, \beta ) \delta ^{\beta p} \: . \end{align*}
Proof

Immediately follows from Theorem 4.48.

Under the assumptions of Theorem 4.48, for \(E\) a complete space and \(\beta \in (0, (q - d)/p)\), there exists a version \(Y\) of \(X\) (i.e., a process \(Y\) with \(\mathbb {P}(Y_t \ne X_t) = 0\) for all \(t\)) such that the paths of \(Y\) are Hölder continuous of order \(\beta \).

Proof

Let \(T'\) be a countable dense subset of \(T\). Let \(A\) be the event

\begin{align*} \left\{ \sup _{s, t \in T';\: s \ne t} \frac{d_E(X_s, X_t)^p}{d_T(s, t)^{\beta p}} {\lt} \infty \right\} \: . \end{align*}

As a consequence of Theorem 4.48, we have \(\mathbb {P}(A) = 1\).

On the event \(A\), \((X_t)_{t \in T'}\) has Hölder continuous paths of order \(\beta \). Let \(x_0 \in E\) be arbitrary and let \(Y: T \to \Omega \to E\) be the process defined by

\begin{align*} Y_t(\omega ) & = \begin{cases} \lim _{s \to t, s \in T'} X_s(\omega ) & \text{if } \omega \in A \\ x_0 & \text{otherwise} \end{cases} \: . \end{align*}

Then \(Y\) has Hölder continuous paths of order \(\beta \) almost surely.

We can show that \((Y_s, Y_t)\) is \(\mathbb {P}\)-a.e. measurable for all \(s, t \in T\).

It remains to show that \(Y\) is a version of \(X\). Let then \(t \in T\) and let \((t_n)_{n \in \mathbb {N}}\) be a sequence in \(T'\) that converges to \(t\). We want to show that \(\mathbb {P}(Y_t \ne X_t) = 0\). It suffices to show that \(\mathbb {P}(d_E(Y_t, X_t) {\gt} 0) = 0\), which itself would follow from \(\mathbb {P}(d_E(Y_t, X_t) {\gt} \varepsilon ) = 0\) for all \(\varepsilon {\gt} 0\).

\begin{align*} \mathbb {P}(d_E(Y_t, X_t) {\gt} \varepsilon ) & \le \mathbb {P}(d_E(Y_t, X_{t_n}) + d_E(X_{t_n}, X_t) {\gt} \varepsilon ) \\ & \le \mathbb {P}(d_E(Y_t, X_{t_n}) {\gt} \varepsilon /2) + \mathbb {P}(d_E(X_{t_n}, X_t) {\gt} \varepsilon /2) \: . \end{align*}

TODO

Under the assumptions of Theorem 4.48, for \(E\) a complete space, there exists a version \(Y\) of \(X\) (i.e., a process \(Y\) with \(\mathbb {P}(Y_t \ne X_t) = 0\) for all \(t\)) such that the paths of \(Y\) are Hölder continuous of all orders \(\gamma \in (0, (q - d)/p)\).

Proof

Let \((\beta _n)\) be an increasing sequence of numbers in \((0, (q - d)/p)\) such that \(\beta _n \to (q - d)/p\). For each \(n\), let \(Y^n\) be the version of \(X\) given by Lemma 4.50 for \(\beta = \beta _n\). Then by Lemma 2.33, the processes \(Y^0\) and \(Y^n\) are indistinguishable for all \(n\). That is, there exists an event \(A_n\) such that \(\mathbb {P}(A_n) = 1\) and such that for all \(\omega \in A_n\), \(Y^0_t(\omega ) = Y^n_t(\omega )\) for all \(t \in T\).

Let \(A = \bigcap _{n \in \mathbb {N}} A_n\) and let \(x_0 \in E\) be arbitrary. Then \(\mathbb {P}(A) = 1\) and the process \(Y(\omega ) = Y^0(\omega )\) for \(\omega \in A\) and \(Y(\omega ) = x_0\) for \(\omega \notin A\) has paths that are Hölder continuous of all orders \(\gamma \in (0, (q - d)/p)\).

4.4.2 Localized version of the Kolmogorov-Chentsov theorem

Definition 4.52 Cover with bounded covering numbers

A set \(T\) is said to have a cover with bounded covering numbers if there exists a monotone sequence of totally bounded open subsets \((T_n)_{n \in \mathbb {N}}\) of \(T\) such that for all \(n\), \(T_n\) has bounded internal covering number with constant \(c_n{\gt}0\) and exponent \(d_n {\gt} 0\), and such that \(\bigcup _{n \in \mathbb {N}} T_n = T\).

Lemma 4.53

\(\mathbb {R}_+\) has a cover with bounded covering numbers for the sets \(T_n = [0,n)\), constants \(c_n = n\) and exponents \(d_n = 1\).

Proof
Definition 4.54 Localized Kolmogorov conditions

Let \(T\) be a metric space with a cover \((T_n)\) with bounded covering numbers with constants \(c_n\) and exponents \(d_n\). A stochastic process \(X : T \to \Omega \to E\) satisfies the localized Kolmogorov conditions if \((X_s, X_t)\) is \(\mathbb {P}\)-a.e. measurable for all \(s, t \in T\) and if for all \(n \in \mathbb {N}\), there are exponents \((p_n, q_n)\) with \(p_n {\gt} 0\) and \(q_n {\gt} d_n\) and a constant \(M_n {\gt} 0\) as well as \(\rho _n {\gt} 0\) such that for all \(s, t \in T_n\) with \(d_T(s, t) \le \rho _n\),

\begin{align*} \mathbb {E}\left[ d_E(X_s, X_t)^{p_n} \right] & \le M_n d_T(s, t)^{q_n} \: . \end{align*}

Let \(T\) be a metric space with a cover \((T_n)\) with bounded covering numbers with constants \(c_n\) and exponents \(d_n\). Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\), with \(q {\gt} \sup _n d_n\). Then \(X\) satisfies the localized Kolmogorov conditions with exponents \((p_n, q_n) = (p, q)\), constants \(M_n = M\) and any \(\rho _n {\gt} 0\).

Proof

Let \(T\) be a metric space with a cover \((T_n)\) with bounded covering numbers with constants \(c_n\) and exponents \(d_n\). Let \(X : T \to \Omega \to E\) be a process that satisfies the localized Kolmogorov conditions with exponents \((p_n, q_n)\) with \(q_n {\gt} d_n\). Then \(X\) has a version \(Y\) such that almost surely the paths of \(Y\) are Hölder continuous of all orders \(\gamma \in \bigcap _n [0, (q_n - d_n)/p_n)\), in which “Hölder continuous of order 0” means “uniformly continuous”. Moreover, for \(n \in \mathbb {N}\), \(t \in T_n\), there is an open neighborhood \(V_t\) of \(t\) in \(T\) such that for all \(\beta \in [0, (q_n - d_n)/p_n)\),

\begin{align*} \mathbb {E}\left[ \sup _{s, t \in V_t} \frac{d_E(Y_s, Y_t)^{p_n}}{d_T(s, t)^{\beta p_n}} \right] & {\lt} \infty \: . \end{align*}
Proof