Brownian Motion

5 Kolmogorov-Chentsov Theorem

5.1 Covers

Let \((E, d_E)\) be a pseudometric space.

Definition 5.1 \(\varepsilon \)-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 \(d_E(x, y) {\lt} \varepsilon \).

Definition 5.2 External covering number
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Lean declarations

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

Lemma 5.4
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\(N^{ext}_\varepsilon (A) \le N^{int}_\varepsilon (A)\).

Proof ▶
Lemma 5.5
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For \(I = [0, 1] \subseteq \mathbb {R}\), \(N^{int}_\varepsilon (I) \le 1/\varepsilon \).

Proof ▶

5.2 Chaining

Lemma 5.6
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Let \((I,d_I)\) and \((E,d_E)\) be metric spaces, and \(f : I \to E\). Moreover, let \(J \subseteq I\) be finite, \(a,b,c \in \mathbb R_+\) with \(a \ge 1\) and \(n \in \{ 1, 2, ...\} \) such that \(|J| \le b a^n\). Then, there is \(K \subseteq J^2\) such that

\begin{align} |K| & \le a |J|, \label{eq:chain1} \\ (s,t) \in K & \implies d_I(s,t) \le cn, \label{eq:chain2} \\ \sup _{s,t\in J, d_I(s,t) \le c} |f(t) - f(s)| & \le 2 \sup _{(s,t) \in K} |f(s) - f(t)|. \label{eq:chain3} \end{align}
Proof ▶

5.3 Kolmogorov-Chentsov Theorem

Theorem 5.7 Continuous version; Kolmogorov, Chentsov
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Let \((I, d_I)\) be a compact metric space. Suppose that there is \(c_1{\gt}0\) and \(d \in \mathbb {N}\) such that for all \(\varepsilon {\gt} 0\) small enough, \(N^{int}_\varepsilon (I) \le c_1 \varepsilon ^{-d}\). Assume that \(X = (X_t)_{t\in I}\) is an \(E\)-valued stochastic process and there are \(\alpha , \beta , c_2{\gt}0\) with

\begin{align*} \mathbb {E}[d_E(X_s, X_t)^\alpha ] \le c_2 d_I(s,t)^{d+\beta }, \qquad s,t\in I \: . \end{align*}

Then, there exists a version \(Y = (Y_t)_{t\in I}\) of \(X\) such that, for some random variables \(H{\gt}0\) and \(K{\lt}\infty \),

\begin{align*} \mathbb {P}\Big(\sup _{s\neq t, d_I(s,t) \leq H} d_E(Y_s, Y_t)/d_I(s,t)^\gamma \le K\Big) = 1 \: , \end{align*}

for every \(\gamma \in (0,\beta /\alpha )\). In particular, \(Y\) almost surely is locally Hölder of all orders \(\gamma \in (0,\beta /\alpha )\), and has continuous paths.

Proof ▶

5.4 Brownian motion