Dependencies

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,

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

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

Under the assumptions of Lemma 4.42, 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 4.56, there exists a version \(Y\) of the pre-Brownian process such that all the paths of \(Y\) are Hölder continuous of all orders \(\gamma \in (0, 1/2)\). We call \(Y\) 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 3.3. 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 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 \(\bigcup _{n \in \mathbb {N}} T_n = T\).

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

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 measure \(\mu \) on \(F\) is centered if for every continuous linear form \(L \in F^*\), \(\mu [L] = 0\).

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

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 \(d_T(s, t) \le \rho _n\),

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 \(P_J\).

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 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).

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 3.10). 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.

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 \(x \mapsto \sum _{i=1}^d x_i \cdot e_i\).

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 \(L' \in F^*\),

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

Let \(E\) and \(F\) be two finite dimensional Hilbert spaces, \(\mu \) 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 \(x \in A\), \(d_E(x, \pi (x, C_\varepsilon )) \le \varepsilon \).

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

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

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

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 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 \(\beta \).

Let \(X, Y : T \to \Omega \to E\) be two stochastic processes that are versions of each other (that is, for all \(t \in T\), \(X_t =_{a.e.} Y_t\)) and are almost surely continuous. Then \(X\) and \(Y\) are indistinguishable. That is, almost surely, \(X_t = Y_t\) for all \(t \in T\) simultaneously.

TODO: hypotheses on \(T, E\)?

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

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 \(m \le k\),

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. 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 \(m = \min (n_2, k)\),

TODO: \(C_k = J\)

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 \(m \le k\),

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 \(m \le k\),

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 \(j {\lt} k\),

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

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 \(E\) is Gaussian if and only if there exists \(m \in E\) and \(\Sigma \) 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 \(\Sigma = \Sigma _\mu \).

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

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 versions 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 5.1 is a Gaussian process.

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.

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.

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 \(\rho _n {\gt} 0\).

The pre-Brownian process \(X\) of Definition 5.1 satisfies the Kolmogorov condition for exponents \((2n,n)\) with some constant \(M_n\) 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 \(i \ne j\), 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 that are versions of each other (that is, for all \(t \in T\), \(X_t =_{a.e.} Y_t\)). 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 \((Y_{t_1}, \ldots , Y_{t_n})\).

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 \(K\) 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,

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

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,

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

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 \(\gamma \in (0, (q - d)/p)\).

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 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 \(\beta \in [0, (q_n - d_n)/p_n)\),