Dependencies

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

The covariance matrix of a Gaussian measure \(\mu \) on \(E\) is the positive semidefinite matrix \(\Sigma _\mu \) with

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

Let \(0 \le t_1 \le \ldots \le t_n\) be non-negative reals. Let \(\mu _0\) be the real Gaussian distribution \(\mathcal{N}(0, t_1)\). For \(i \in \{ 1, \ldots , n-1\} \), let \(\kappa _i\) be the Markov kernel from \(\mathbb {R}\) to \(\mathbb {R}\) defined by \(\kappa _i(x) = \mathcal{N}(x, t_{i+1} - t_i)\) (the Gaussian increment \(\kappa ^G_{t_{i+1} - t_i}\)). Let \(P_{t_1, \ldots , t_n}\) be the measure on \(\mathbb {R}^n\) defined by \(\mu _0 \otimes \kappa _1 \otimes \ldots \otimes \kappa _{n-1}\).

(Gaussian increment) For \(v \ge 0\), the map from \(\mathbb {R}\) to the probability measures on \(\mathbb {R}\) defined by \(x \mapsto \mathcal{N}(x, v)\) is a Markov kernel. We call that kernel the Gaussian increment with variance \(v\) and denote it by \(\kappa ^G_v\).

We denote by TODO 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 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 \(d_E(x, y) {\lt} \varepsilon \).

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.

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 \(P_J\).

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 \((e_1, \ldots , e_d)\) be an orthonormal basis of \(E\) and let \(\mu _1, \ldots , \mu _d\) be independent standard Gaussian measures on \(\mathbb {R}\). The standard Gaussian measure on \(E\) is the pushforward measure of the product measure \(\mu _1 \times \ldots \times \mu _d\) by the map \(x \mapsto \sum _{i=1}^d x_i \cdot e_i\).

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

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

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

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

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

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.

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 \(L \in F^*\), \(\mu [L] = L(m_\mu )\).

The convolution of two Gaussian measures is a Gaussian measure.

\(P_{t_1, \ldots , t_n}\) is a Gaussian measure on \(\mathbb {R}^n\) with mean \(0\) and covariance matrix \(C_{ij} = \min (t_i, t_j)\) for \(1 \leq i,j \leq n\).

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 standard Gaussian measure is a probability measure.

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

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.

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

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.

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 \((I, d_I)\) 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

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

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.

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.