Dependencies

Under the same assumptions as in Theorem 5.78, for every countable subset \(T' \subseteq T\) with positive diameter, for \(L(T, c_1, d, p, q, \beta ) {\lt} \infty \) the same constant,

For \(\varepsilon _n = \varepsilon _0 2^{-n}\), with the hypothesis of Lemma 5.31, we have

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 \(\delta {\gt} 0\),

With the same assumptions and notations as in Theorem 5.71, for all \(\delta \in (0, 4\mathrm{diam}(T)]\),

Under the assumptions of Lemma 5.61, for \(\varepsilon _n = \varepsilon _0 2^{-n}\), then for \(m \le k\),

Under the assumptions of Lemma 5.65, for \(\varepsilon _n = \varepsilon _0 2^{-n}\), then for \(m \le k\),

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 \(m \le k\),

By Theorem 5.85, there exists a modification \(B\) of the pre-Brownian process such that all the paths of \(B\) are Hölder continuous of all orders \(\gamma \in (0, 1/2)\). We call \(B\) the Brownian motion on \(\mathbb {R}_+\).

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 \(i {\lt} k\), \(\bar{x}_i = \pi (\bar{x}_{i+1}, C_i)\).

The characteristic function of a measure \(\mu \) on an inner product space \(E\) is the function \(E \to \mathbb {C}\) defined by

This is equal to the normed space version of the characteristic function applied to the linear map \(x \mapsto \langle t, x \rangle \).

The characteristic function of a measure \(\mu \) on a normed space \(E\) is the function \(E^* \to \mathbb {C}\) defined by

The covariance bilinear form of a measure \(\mu \) on \(F\) with finite second moment is the continuous bilinear form \(C_\mu : F^* \times F^* \to \mathbb {R}\) with

The covariance bilinear form of a finite measure \(\mu \) with finite second moment on a Hilbert space \(E\) is the continuous bilinear form \(C_\mu : E \times E \to \mathbb {R}\) with

This is \(C_\mu \) applied to the linear maps \(L_x, L_y \in E^*\) defined by \(L_x(z) = \langle x, z \rangle \) and \(L_y(z) = \langle y, z \rangle \).

The covariance matrix of a finite measure \(\mu \) with finite second moment on a finite dimensional inner product space \(E\) is the positive semidefinite matrix \(\Sigma _\mu \) such that for \(u, v \in E\),

This is the covariance bilinear form \(C'_\mu (u, v)\), as a matrix.

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

We denote by \(P_B\) the projective limit of the projective family of the Brownian motion given by Theorem 4.3. This is a probability measure on \(\mathbb {R}^{\mathbb {R}_+}\).

For \(I = \{ t_1, \ldots , t_n\} \) a finite subset of \(\mathbb {R}_+\), let \(P^B_I\) be the multivariate Gaussian measure on \(\mathbb {R}^n\) with mean \(0\) and covariance matrix \(C_{ij} = \min (t_i, t_j)\) for \(1 \leq i,j \leq n\). We call the family of measures \(P^B_I\) the projective family of the Brownian motion.

The real Gaussian measure with mean \(\mu \in \mathbb {R}\) and variance \(\sigma ^2 {\gt} 0\) is the measure on \(\mathbb {R}\) with density \(\frac{1}{\sqrt{2 \pi \sigma ^2}} \exp \left(-\frac{(x - \mu )^2}{2 \sigma ^2}\right)\) with respect to the Lebesgue measure. The real Gaussian measure with mean \(\mu \in \mathbb {R}\) and variance \(0\) is the Dirac measure \(\delta _\mu \). We denote this measure by \(\mathcal{N}(\mu , \sigma ^2)\).

Let \(v_1, \ldots , v_n\) be vectors in an inner product space \(E\). The Gram matrix of \(v_1, \ldots , v_n\) is the matrix in \(\mathbb {R}^{n \times n}\) with entries \(G_{ij} = \langle v_i, v_j \rangle \) for \(1 \leq i,j \leq n\).

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

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 \(\varepsilon \in (0, \mathrm{diam}(A)]\), \(N^{int}_\varepsilon (A) \le c \varepsilon ^{-t}\).

We say that a stochastic processes \(Y\) is a indistinguishable from \(X\) if \(\mathbb {P}\)-a.e., for all \(t \in T\), \(X_t = Y_t\).

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

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 \(d_E(x, y) \le \varepsilon \).

A measure \(\mu \) on \(F\) is Gaussian if for every continuous linear form \(L \in F^*\), the pushforward measure \(L_* \mu \) is a Gaussian measure on \(\mathbb {R}\).

A process \(X : T \to \Omega \to E\) is Gaussian if for every finite subset \(t_1, \ldots , t_n \in T\), the random vector \((X_{t_1}, \ldots , X_{t_n})\) has a Gaussian distribution.

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

A measure \(\mu \) on \(E^T\) is the projective limit of a projective family of measures \(P\) indexed by finite sets of \(T\) if, for every finite set \(I \subseteq T\), the projection from \(E^T\) to \(E^I\) maps \(\mu \) to \(P_I\).

A family of measures \(P\) indexed by finite sets of \(T\) is projective if, for finite sets \(J \subseteq I\), the projection from \(E^I\) to \(E^J\) maps \(P_I\) to \(P_J\).

A set \(c \subseteq E\) is \(\varepsilon \)-separated if for all \(x, y \in c\), \(d_E(x, y) {\gt} \varepsilon \).

We introduce the constant

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 \(V_{i+1}\).

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 \(\vert B_V(t, r c) \vert \le a^{r}\).

The identity is a measurable equivalence between the continuous functions of \(\mathbb {R}^{\mathbb {R}_+}\) with the subset sigma-algebra obtained from the product sigma-algebra, and \(C(\mathbb {R}_+, \mathbb {R})\) with the Borel sigma-algebra coming from the compact-open topology.

Mathematically this says nothing more than the equality of sigma-algebras of Theorem 6.12 but in Lean we have two different types so we need an equivalence.

We say that a stochastic process \(Y\) is a modification of another stochastic process \(X\) if for all \(t \in T\), \(Y_t =_{\mathbb {P}\text{-a.e.}} X_t\).

The multivariate Gaussian measure on \(\mathbb {R}^d\) with mean \(m \in \mathbb {R}^d\) and covariance matrix \(\Sigma \in \mathbb {R}^{d \times d}\), with \(\Sigma \) positive semidefinite, is the pushforward measure of the standard Gaussian measure on \(\mathbb {R}^d\) by the map \(x \mapsto m + \Sigma ^{1/2} x\). We denote this measure by \(\mathcal{N}(m, \Sigma )\).

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).

The packing number of a set \(A \subseteq E\) for \(\varepsilon {\gt} 0\) is the largest cardinality of an \(\varepsilon \)-separated subset of \(A\). Denote it by \(P_\varepsilon (A)\).

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.

Let \(\Omega = \mathbb {R}^{\mathbb {R}_+}\) and consider the probability space \((\Omega , P_B)\) (where \(P_B\) is the measure defined in Definition 4.10). The identity on that space is a function \(\Omega \to \mathbb {R}_+ \to \mathbb {R}\). We reorder the arguments to define a stochastic process \(X : \mathbb {R}_+ \to \Omega \to \mathbb {R}\), which we call the pre-Brownian process.

The law of a stochastic process \(X\) is the measure on the measurable space \(E^T\) obtained by pushing forward the measure \(\mathbb {P}\) by the map \(\omega \mapsto X(\cdot , \omega )\).

Let \((e_1, \ldots , e_d)\) be an orthonormal basis of \(E\) and let \(\mu \) be the standard Gaussian measure on \(\mathbb {R}\). The standard Gaussian measure on \(E\) is the pushforward measure of the product measure \(\mu \times \ldots \times \mu \) by the map \(x \mapsto \sum _{i=1}^d x_i \cdot e_i\).

The Wiener measure on \(C(\mathbb {R}_+, \mathbb {R})\) with the Borel sigma-algebra is the map of the auxiliary Wiener measure by the measurable equivalence of definition 6.13.

The pushforward of the measure \(P_B\) of Definition 4.10 by the Brownian motion \(B\) is a measure on the continuous functions on \(\mathbb {R}^{\mathbb {R}_+}\), with the sigma-algebra induced by the product sigma-algebra on \(\mathbb {R}^{\mathbb {R}_+}\).

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 \(E^2\).

Let \(I = \{ t_1, \ldots , t_n\} \) be a finite subset of \(\mathbb {R}_+\). For \(i \le n\), let \(v_i = \mathbb {I}_{[0, t_i]}\) be the indicator function of the interval \([0, t_i]\), as an element of \(L^2(\mathbb {R})\). Then the Gram matrix of \(v_1, \ldots , v_n\) is equal to the matrix \(C_{ij} = \min (t_i, t_j)\) for \(1 \leq i,j \leq n\).

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

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

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

The central moment of order \(2n\) of a real Gaussian measure \(\mathcal{N}(\mu , \sigma ^2)\) is given by

in which \((2n - 1)!! = (2n - 1)(2n - 3) \cdots 3 \cdot 1\) is the double factorial of \(2n - 1\).

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 \(i \le k\), \(\bar{x}_i\in C_i\).

The characteristic function of a real Gaussian measure with mean \(\mu \) and variance \(\sigma ^2\) is given by \(x \mapsto \exp \left(i \mu x - \frac{\sigma ^2 x^2}{2}\right)\).

Let \(\mu \) be a measure on \(F\) and let \(L \in F^*\). Then

The characteristic function of the standard Gaussian measure on \(E\) is given by

Let \(\mu \) be a measure on a normed space \(E\) and let \(L\) be a continuous linear map from \(E\) to \(F\). Then for all \(L' \in F^*\),

The paths of the Brownian motion are continuous.

For \(\mu \) a measure on \(F\) with finite second moment and \(L \in F^*\), \(C_\mu (L, L) = \mathbb {V}_\mu [L]\).

Let \(C\) be a \(\varepsilon \)-cover of a \(J\) with \(C \subseteq J\). If \(\varepsilon {\lt} \inf _{s, t \in J; s \ne t} d_T(s, t)\) then \(C = J\).

Let \(E\) and \(F\) be two Hilbert spaces with \(F\) finite dimensional, \(\mu \) a finite measure on \(E\) with finite second moment, and \(L : E \to F\) a continuous linear map. Then the covariance bilinear form of the measure \(L_*\mu \) is given by

in which \(L^\dagger : F \to E\) is the adjoint of \(L\).

Let \(E\) and \(F\) be two finite dimensional inner product spaces, \(\mu \) a measure on \(E\) with finite second moment, and \(L : E \to F\) a continuous linear map. Then the covariance matrix of the measure \(L_*\mu \) has entries

in which \(L^\dagger : F \to E\) is the adjoint of \(L\).

The covariance matrix of the multivariate Gaussian measure \(\mathcal{N}(m, \Sigma )\) is \(\Sigma \).

The covariance matrix of the standard Gaussian measure is the identity matrix.

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 \(i {\lt} k\), \(d_E(\bar{x}_i, \bar{x}_{i+1}) \le \varepsilon _i\).

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

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 \(m \le k\), \(d_E(\bar{x}_m, x) \le \sum _{i=m}^{k-1} \varepsilon _i\).

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

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

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

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

For \(B \subseteq A\), \(N^{ext}_\varepsilon (B) \le N^{ext}_{\varepsilon }(A)\).

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

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

If \(A\) has bounded internal covering number with constant \(c{\gt}0\) and exponent \(d{\gt}0\), then for all \(B \subseteq A\), \(B\) has bounded internal covering number with constant \(2^d c\) and exponent \(d\).

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

Under the assumptions of Theorem 5.78, for \(E\) a complete space and \(\beta \in (0, (q - d)/p)\), there exists a modification \(Y\) 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 \(\beta \).

If \(Y\) is indistinguishable from \(X\), then \(Y\) is a modification of \(X\).

Let \(T\) and \(E\) be topological spaces and suppose that \(T\) is separable Hausdorff. Let \(X, Y : T \to \Omega \to E\) be two stochastic processes that are modifications of each other and are almost surely continuous. Then \(X\) and \(Y\) are indistinguishable.

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 \(\beta \in (0, (q - d)/p)\),

For \(\mu _1, \ldots , \mu _d\) probability measures on \(\mathbb {R}\) and \(f : \mathbb {R} \to \mathbb {R}\) integrable with respect to \(\mu _i\), we have

The mean of the multivariate Gaussian measure \(\mathcal{N}(m, \Sigma )\) is \(m\).

The mean of the standard Gaussian measure is \(0\).

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 \(p \ge 1\) and \(m \le k\),

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 \(0 {\lt} p \le 1\) and \(m \le k\),

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 \(c \ge 0\),

Note the logarithm has base \(2\).

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. Suppose \(\varepsilon _0 {\lt} \infty \), let \(\delta \in (0, 4 \varepsilon _0]\) and let \(m\) be a natular number such that \(\varepsilon _0 2^{-m} \le \delta \) and \(\delta \le \varepsilon _0 2^{-m+2}\). Then for \(k \ge m\),

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

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 \(p \ge 1\) and \(m \le k\),

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 \(p \le 1\) and \(m \le k\),

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 \(p \ge 1\) and \(m \le k\),

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 \(C_n \subseteq T\). For \(0 {\lt} p \le 1\) and \(m \le k\),

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

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 \(j {\lt} k\),

\(N_\varepsilon ^{int}(B_1) \ge \frac{1}{\varepsilon ^d}\).

\(N_\varepsilon ^{int}(B_1) \le \left(\frac{2}{\varepsilon } + 1\right)^d\).

If \(\mathrm{diam}(A) \le \varepsilon \) and \(A\) is nonempty, then \(N^{int}_\varepsilon (A) = 1\).

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

\(N^{int}_\varepsilon (A) \le \frac{V(A + B_{\varepsilon /2})}{V(B_{\varepsilon /2})}\).

For \(B \subseteq A\), \(N^{int}_\varepsilon (B) \le N^{int}_{\varepsilon /2}(A)\).

The standard Gaussian measure on \(E\) is centered, i.e., \(\mu [L] = 0\) for every \(L \in E^*\).

The characteristic function of a Gaussian measure \(\mu \) on \(E\) is given by

Two Gaussian measures \(\mu \) and \(\nu \) on a separable Hilbert space are equal if and only if they have same mean and same covariance.

A Gaussian measure is a probability measure.

For a Gaussian measure \(\mu \), \(\mu \times \mu \) is invariant by rotation.

A Gaussian measure \(\mu \) has finite moments of all orders. In particular, there is a well defined mean \(m_\mu := \mu [\mathrm{id}]\), and for all \(L \in F^*\), \(\mu [L] = L(m_\mu )\).

Let \(\mu \) be a Gaussian measure on \(F\) and let \(c \in F\). Then the measure \(\mu \) translated by \(c\) (the map of \(\mu \) by \(x \mapsto x + c\)) is a Gaussian measure on \(F\).

The convolution of two Gaussian measures is a Gaussian measure.

A finite measure \(\mu \) on a Hilbert space \(E\) is Gaussian if and only if for every \(t \in E\), the characteristic function of \(\mu \) at \(t\) is

A finite measure \(\mu \) on \(E\) is Gaussian if and only if there exists \(m \in E\) and \(C\) positive semidefinite such that for all \(t \in E\), the characteristic function of \(\mu \) at \(t\) is

If that’s the case, then \(m = m_\mu \) and \(C = C'_\mu \).

Let \(F, G\) be two Banach spaces, let \(\mu \) be a Gaussian measure on \(F\) and let \(T : F \to G\) be a continuous linear map. Then \(T_*\mu \) is a Gaussian measure on \(G\).

A multivariate Gaussian measure is a Gaussian measure.

The standard Gaussian measure on \(E\) is a Gaussian measure.

The Brownian motion is a Gaussian process.

Let \(X, Y : T \to \Omega \to E\) be two stochastic processes that are modifications of each other (that is, for all \(t \in T\), \(X_t =_{a.e.} Y_t\)). If \(X\) is a Gaussian process, then \(Y\) is a Gaussian process as well.

The pre-Brownian process \(X\) of Definition 6.1 is a Gaussian process.

The paths of the Brownian motion are Hölder continuous of all orders \(\gamma \in (0, 1/2)\).

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

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.

If \(X : T \to \Omega \to E\) is a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\) and \(s, t \in T\) are such that \(d_T(s, t) = 0\), then \(\mathbb {P}\)-a.e. \(d_E(X_s, X_t) = 0\).

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \(T'\) be a countable subset of \(T\) such that for all \(s, t \in T'\), \(d_T(s, t) = 0\). Then

The pre-Brownian process \(X\) of Definition 6.1 satisfies the Kolmogorov condition for exponents \((2n,n)\) with constant \((2n - 1)!!\) for all \(n \in \mathbb {N}\). That is, for all \(s, t \in \mathbb {R}_+\), we have

The standard Gaussian measure is a probability measure.

The projective family of the Brownian motion is a projective family of measures.

For \(\mathrm{diam}(T) {\lt} \infty \), \(p{\gt} 0\), \(q {\gt} d {\gt} 0\) and \(\beta \in (0, (q-d)/p)\), the constant \(L(T, c_1, d, p, q, \beta )\) is finite.

For \(t \in \mathbb {R}_+\), the law of \(B_t\) (the Brownian motion at time \(t\)) is the real Gaussian measure \(\mathcal{N}(0,t)\).

For \(s, t \in \mathbb {R}_+\), the law of \(B_t - B_s\) is the real Gaussian measure \(\mathcal{N}(0,\vert t - s \vert )\).

Let \(A \subseteq E\) and let \(S \subseteq A\) be an \(\varepsilon \)-separated set. Then \(\vert S \vert V(B_{\varepsilon /2}) \le V(A + B_{\varepsilon /2})\).

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\) and let \(J\) be a finite subset of \(T\). Let \(C\) be an \(\varepsilon \)-cover of \(J\) with \(C \subseteq J\). If \(\varepsilon {\lt} \inf _{s, t \in J; d_T(s, t){\gt}0} d_T(s, t)\) then

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.

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

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 \(i \in \mathbb {N}\).

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 \(i \in \mathbb {N}\).

Let \(X, Y : T \to \Omega \to E\) be two stochastic processes. Then \(X\) and \(Y\) have same finite-dimensional distributions if and only if they have the same law.

Let \(X, Y : T \to \Omega \to E\) be two stochastic processes that are modifications of each other. Then for all \(t_1, \ldots , t_n \in T\), the random vector \((X_{t_1}, \ldots , X_{t_n})\) has the same distribution as the random vector \((Y_{t_1}, \ldots , Y_{t_n})\). That is, \(X\) and \(Y\) have same finite-dimensional distributions.

Let \(X\) be the pre-Brownian process of Definition 6.1. Then, for all \(s, t \in \mathbb {R}_+\), the random variable \(X_t - X_s\) is a Gaussian random variable with mean \(0\) and variance \(|t - s|\).

\(P_\varepsilon (A) V(B_{\varepsilon /2}) \le V(A + B_{\varepsilon /2})\).

\(P_{2\varepsilon }(A) \le N^{ext}_\varepsilon (A)\).

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,

For \(I = \{ t_1, \ldots , t_n\} \) a finite subset of \(\mathbb {R}_+\), let \(C \in \mathbb {R}^{n \times n}\) be the matrix \(C_{ij} = \min (t_i, t_j)\) for \(1 \leq i,j \leq n\). Then \(C\) is positive semidefinite.

A gram matrix is positive semidefinite.

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 \(m \le k\),

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \(C_n\) a finite \((\varepsilon _0 2^{-n})\)-cover of \(T\) for \(\varepsilon _0 \le \mathrm{diam}(T)\) 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 \(m \le k\),

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,

\(\frac{V(A)}{V(B_\varepsilon )} \le N^{int}_\varepsilon (A)\).

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

Let \(A \subseteq E\) and \(C \subseteq E\) be a finite \(\varepsilon \)-cover of \(A\). Denote by \(V(A)\) the volume of \(A\). Then \(V(A) \le \vert C \vert V(B_\varepsilon )\), in which \(B_\varepsilon \) is the closed ball of radius \(\varepsilon \) in \(E\).

The characteristic function of a multivariate Gaussian measure \(\mathcal{N}(m, \Sigma )\) is given by

The borel sigma-algebra on \(C(\mathbb {R}_+, \mathbb {R})\) coming from the compact-open topology is equal to the smallest sigma-algebra for which the evaluation maps \(f \mapsto f(t)\) are measurable for every \(t \in \mathbb {R}_+\).

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 for every countable subset \(T' \subseteq T\) with positive diameter,

Let \(\mu \) be a finite measure on \(F\) such that \(\mu \times \mu \) is invariant under the rotation of angle \(-\frac{\pi }{4}\). Then there exists \(C {\gt} 0\) such that the function \(x \mapsto \exp (C \Vert x \Vert ^2)\) is integrable with respect to \(\mu \).

In a separable Hilbert space, if two finite measures have same characteristic function, they are equal.

In a separable Banach space, if two finite measures have same characteristic function, they are equal.

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 \(\delta \ge 4\mathrm{diam}(T)\),

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 \(\delta \in (0, 4\mathrm{diam}(T)]\),

Under the assumptions of Theorem 5.78, for \(E\) a complete space, there exists a modification \(Y\) 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 \(\gamma \in (0, (q - d)/p)\).

For a Gaussian measure, there exists \(C {\gt} 0\) such that the function \(x \mapsto \exp (C \Vert x \Vert ^2)\) is integrable.

A finite measure \(\mu \) on \(F\) is Gaussian if and only if for every continuous linear form \(L \in F^*\), the characteristic function of \(\mu \) at \(L\) is

in which \(\mathbb {V}_\mu [L]\) is the variance of \(L\) with respect to \(\mu \).

Let \(\mathcal{X}\) be a Polish space, equipped with the Borel \(\sigma \)-algebra, and let \(T\) be an index set. Let \(P\) be a projective family of finite measures on \(\mathcal{X}\). Then the projective limit \(\mu \) of \(P\) exists, is unique, and is a finite measure on \(\mathcal{X}^T\). Moreover, if \(P_I\) is a probability measure for every finite set \(I \subseteq T\), then \(\mu \) is a probability measure.

Let \(T\) be a metric space with a cover \((T_n)\) with bounded covering numbers with constants \(c_n\) and the same exponent \(d\). Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition with exponents \((p, q)\) with \(q {\gt} d\). Then \(X\) has a modification \(Y\) such that almost surely the paths of \(Y\) are Hölder continuous of all orders \(\gamma \in (0, (q - d)/p)\).

Let \(T\) be a metric space with a cover \((T_n)\) with bounded covering numbers with constants \(c_n\) and the same exponent \(d\). Let \((p_n, q_n)_{n \in \mathbb {N}}\) be a sequence of pairs of positive numbers such that \(q_n {\gt} d\) for all \(n \in \mathbb {N}\). Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition with exponents \((p_n, q_n)\) for all \(n \in \mathbb {N}\). Then \(X\) has a modification \(Y\) such that almost surely the paths of \(Y\) are Hölder continuous of all orders \(\gamma \in (0, \sup _n (q_n - d)/p_n)\).