Let \(s \in J\). Since \(C\) is an \(\varepsilon \)-cover of \(J\), there exists \(t \in C\) such that \(d_T(s, t) \le \varepsilon \). Since \(\varepsilon {\lt} \inf _{s, t \in J; s \ne t}d_T(s, t)\), we have \(s = t\). Thus, \(s \in C\). Since \(s\) was arbitrary, we have \(J \subseteq C\). And by assumption \(C \subseteq J\).

Since \(C\) is a cover of \(A\), \(A\) is a subset of the union of the closed balls \(B_\varepsilon (c)\) for \(c \in C\). Then

Use Lemma 5.13 with a cover of minimal cardinal.

Since \(S\) is \(\varepsilon \)-separated, the closed balls \(B_{\varepsilon /2}(s)\) for \(s \in S\) are pairwise disjoint. Furthermore, these balls are contained in \(A + B_{\varepsilon /2}\). Thus, we have

Use Lemma 5.15 with \(S\) an \(\varepsilon \)-separated set of maximal cardinality.

We have \(\frac{V(A)}{V(B_\varepsilon )} \le N^{ext}_\varepsilon (A)\) by Lemma 5.14 and \(N^{ext}_\varepsilon (A) \le N^{int}_\varepsilon (A)\) by Lemma 5.6.

We have \(N^{int}_\varepsilon (A) \le P_\varepsilon (A)\) by Lemma 5.7 and \(P_\varepsilon (A) \le \frac{V(A + B_{\varepsilon /2})}{V(B_{\varepsilon /2})}\) by Lemma 5.16.

By Lemma 5.17,

By Lemma 5.18,

By definition.

By definition.

Apply Lemma 5.26 with \(S = C_i\) and \(x = \bar{x}_{i+1}\).

By the triangle inequality and Lemma 5.29,

Triangle inequality and Lemma 5.30.

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

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

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

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

Using Lemma 5.35, the cardinal of \(K\) is bounded by

Since the sets \(B_{V_i}(t_i, (r_i-1)c)\) are disjoint by Lemma 5.40, we can use Lemma 5.34 to get

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

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

The last inequality is from Lemma 5.37.

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

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 5.42. The second property 2 is Lemma 5.43. Equation 3 was proved in Lemma 5.44.

It suffices to show that \(d_E(X_s, X_t)^p = 0\) almost everywhere, which is in turn implied by \(\mathbb {E}[d_E(X_s, X_t)^p] \le M d_t(s, t)^q = 0\).

Since \(T'\) is countable, we get from Lemma 5.47 that almost surely, for all \(s, t \in T'\), \(d_E(X_s, X_t)^p = 0\). In particular the expectation of the supremum is \(0\).

By Lemma 5.45, there exists \(K \subseteq J^2\) such that

Hence for such a set \(K\),

Then by Lemma 5.52,

By the triangle inequality,

This is Lemma 5.54, together with the fact that for \(a, b \ge 0\),

Let \(\bar{r} = 1 + \log _2 N^{int}_{\varepsilon }(J)\). Then

By Lemma 5.53 with \(J = C\), \(a = 2\), \(c = c\), \(n = \bar{r}\),

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

By definition of \(m\), \(\delta \le \varepsilon _0 2^{-m+2}\). 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 5.32). 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 5.56 with \(\varepsilon = \varepsilon _m\), \(c = \varepsilon _0 2^{-m+3}\). We obtain

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

Finally, by definition of \(m\) we have \(\varepsilon _0 2^{-m} \le \delta \).

We then apply Lemma 5.52 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 \(\vert C_{j+1} \vert \).

By the triangle inequality,

We thus have

And then, by Minkowski’s inequality, since \(p \ge 1\),

Finally, we raise to the \(p\)-th power to obtain the result.

Put together Lemma 5.58 and Lemma 5.59.

By Lemma 5.60, we have

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

Applying first Lemma 5.61, we get

For \(0 {\lt} p \le 1\), the power function is sub-additive, i.e. for \(a, b \ge 0\),

We can thus apply the triangle inequality to obtain

Put together Lemma 5.58 and Lemma 5.63.

By Lemma 5.64, we have

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

Applying first Lemma 5.65, we get

This is the max of the two bounds obtained \(p \ge 1\) and \(p \le 1\).

First, remark that \(C\) is actually a \(0\)-cover of \(J\). For \(s, t \in J\), let \(s', t' \in C\) be such that \(d_T(s, s') = 0\) and \(d_T(t, t') = 0\). Then by the triangle inequality,

and by Lemma 5.47, we have \(d_E(X_s, X_{s'}) = 0\) and \(d_E(X_t, X_{t'}) = 0\) almost surely, hence \(d_E(X_s, X_t) \le d_E(X_{s'}, X_{t'})\). Since \(J\) is finite, almost surely we have that inequality for all pairs \((s, t) \in J\) and their corresponding \((s', t') \in C\). Note that \(d_T(s', t') = d_T(s, t)\), hence \(d_T(s, t) \le \delta \) is equivalent to \(d_T(s', t') \le \delta \). We obtain

The reverse inequality holds because \(C\) is a subset of \(J\).

Let \(\varepsilon _0 = \mathrm{diam}(T)\). 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 \(k\) be a natural number such that \(\varepsilon _0 2^{-k} {\lt} \inf _{s, t \in T; d_T(s,t){\gt}0} d_T(s, t)\), which exists since \(T\) is finite. By Lemma 5.69, the supremum over \(T\) can be replaced by a supremum over \(C_k\).

By Corollary 5.55,

Since \(\varepsilon _0 = \mathrm{diam}(T)\), \(C_0\) is a singleton and \(d_E(X_{\bar{s}_0}, X_{\bar{t}_0}) = 0\) for all \(s, t\). We thus have

By Lemma 5.68,

Let \(\varepsilon _0 = \mathrm{diam}(T)\). 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 \(k\) be a natural number such that \(\varepsilon _0 2^{-k} {\lt} \inf _{s, t \in T; d_T(s,t){\gt}0} d_T(s, t)\), which exists since \(T\) is finite. If \(\delta \le \varepsilon _0 2^{-k}\), then \(\{ (s, t) \in C_k; d_T(s, t) \le \delta \} = \{ (s, t) \mid s,t \in C_k, d_T(s,t) = 0\} \) and the inequality holds trivially (by Lemma 5.48). We can thus assume \(\delta {\gt} \varepsilon _0 2^{-k}\).

By Lemma 5.69, the supremum over \(T\) can be replaced by a supremum over \(C_k\).

By Corollary 5.55, for any \(m \le k\),

First term

We have \(\delta \le 4\varepsilon _0\) by assumption. Let \(n_2 = \lfloor \log _2(4\varepsilon _0/\delta ) \rfloor \) and \(m = \min \{ n_2, k\} \). 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 \(\varepsilon _0 2^{-m} \le \delta \). We can also verify that \(\delta \le \varepsilon _0 2^{-n_2+2} \le \varepsilon _0 2^{-m+2}\). By Lemma 5.57,

Second term

By Lemma 5.68 and then the inequality \(\varepsilon _0 2^{-m} \le \delta \),

Putting the two terms together, we obtain

We apply Theorem 5.71 and then remark that for \(\delta \le 4\mathrm{diam}(T)\), we can use the bounded internal covering number hypothesis to bound \(N^{int}_{\delta /4}(T)\) :

We combine Corollary 5.72 and Theorem 5.70.

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 \(\eta _k \ge 2^{-k}\).

Let \(a_k = 2^{2p+5q+1} M c_1 (\mathrm{diam}(T)+1)^{q-d} 2^{k (\beta p - (q-d))} \left(4^d \left(\max \left\{ 0, \log _2(c_1) + (k + 2)d \right\} \right)^q + C_p\right)\). Then \(L(T, c_1, d, p, q, \beta ) = \sum _{k=0}^\infty a_k\). To show that the sum is finite, we can use the ratio test.

The limit at infinity of that ratio is \(2^{\beta p - (q - d)} {\lt} 1\), hence the series \(\sum _{k=0}^\infty a_k\) converges.

Since \(J \subseteq T\), \(J\) has bounded internal covering number with constant \(2^d c_1\) and exponent \(d\) (Lemma 5.23).

Let \(\eta _k = 2^{-k}(\mathrm{diam}(T) + 1)\) for \(k \in \mathbb {N}\). By Lemma 5.74, we have

We apply Corollary 5.73 to bound each expectation in the sum.

The sum is then less than \(M\) times \(L(T, c_1, d, p, q, \beta )\).

Build a monotone sequence of finite sets \(T_n \subseteq T'\), use Lemma 5.77 to obtain a bound for each \(T_n\) that does not depend on \(T_n\), and then use monotone convergence.

Immediately follows from Theorem 5.78.

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

As a consequence of Theorem 5.78, 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

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 modification 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 \(\varepsilon {\gt} 0\).

TODO

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 modification of \(X\) given by Lemma 5.80 for \(\beta = \beta _n\). Then by Lemma 2.7, 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 \(\gamma \in (0, (q - d)/p)\).

For each \(n\), by Theorem 5.81 there is a modification \(Y_n\) of \(X\) seen as a process on \(T_n\) such that the paths of \(Y_n\) are Hölder continuous of all orders \(\gamma \in (0, (q - d)/p)\). By Lemma 2.7, \(Y_n\) and \(Y_{n+1}\) are indistinguishable on \(T_n\). That is, almost surely \(Y_n = Y_{n+1}\) on \(T_n\). Since there are countably many such almost sure equalities, we get that almost surely there is equality for all \(n\). Let \(A\) be the event that this happens, let \(x_0 \in E\) be arbitrary and define a process \(Y : T \to \Omega \to E\) by

Then \(Y\) is a modification of \(X\) and has paths that are Hölder continuous of all orders \(\gamma \in (0, (q - d)/p)\) almost surely.

For each \(n\), by Theorem 5.84 there is a modification \(Y_n\) of \(X\) such that the paths of \(Y_n\) are Hölder continuous of all orders \(\gamma \in (0, (q_n - d)/p_n)\). By Lemma 2.7, any two processes \(Y_n, Y_m\) are indistinguishable. That is, almost surely \(Y_n = Y_m\). Since there are countably many such almost sure equalities, we get that almost surely there is equality for all \(n, m\). Let then \(Y\) be the process equal to \(Y_0\) on the event that the equalities hold, and equal to an arbitrary point \(x_0 \in E\) otherwise. Then first, \(Y\) is a modification of \(X\). Then for any \(\gamma {\lt} \sup _n (q_n - d)/p_n\) there is \(n\) such that \(\gamma {\lt} (q_n - d)/p_n\) and thus since \(Y = Y_n\) the paths of \(Y\) are Hölder continuous of order \(\gamma \) almost surely.