In this blog post, we will focus on the some analysis on the space of probability measures as it is the main technical component arising in Mc-Kean Vlasov SDEs.

This blog post is mainly inspired from the excellent book of Carmona-Delarue : Probabilistic Theory of Mean Field Games with Applications I available here

Table of contents

1. Metric Spaces of Probability Measures
   1. Distance Between Measures
   2. Some topological properties of the Wasserstein space $\mathcal{P}_2(\mathbb{R}^d)$
2. Differentiability of Functions of Probability Measures
   1. Some notions of derivatives on Probability Measures
   2. Itô Formula Along Flow of Measures

Some Metrics on Probability Measures

We will denote by $(E,d)$ a complete separable metric space and the corresponding $\sigma$-algebra $\mathcal{E}$ making $(E,\mathcal{E})$ a mesurable space. In the following, it will be assumed to be the Borel $\sigma$-field $\mathcal{B}(E)$ determined by the topology of $(E,d)$. We will also denote by $\mathcal{P}(E)$ the space of probability measures on $(E,\mathcal{E})$.

The notion of distance between measures

The Lévy-Prokhorov distance

The Lévy-Prokhorov distance between 2 measures $\nu$ and $\mu$ on $(E,\mathcal{E})$ is defined as the following :

\(\begin{align} d_{LP}(\mu,\nu) = \text{ inf } &\lbrace \epsilon > 0 : \forall A \in \mathcal{E}, \mu(A) \leq \nu(A^{\epsilon}) + \epsilon \\ &\text{ and } \nu(A) \leq \mu(A^{\epsilon}) + \epsilon \rbrace \notag \end{align}\) where $A^{\epsilon} = \lbrace x \in E : \exists y \in A : d(x,y) < \epsilon \rbrace$. Note that $A \subset A^{\epsilon}$ but we want the distance $d$ to control in the same way the distance between $\mu$ and $\nu$. There exists a simplest caracterization of ths result defined as the following :

\(\begin{align} d_{LP}(\mu,\nu) = \text{ inf }\lbrace \epsilon > 0 : \underset{\pi \in \Pi(\mu,\nu)}{\text{ inf}} \int_{E \times E} \mathbb{1}_{d(x,y) > \epsilon} \pi(dx,dy) < \epsilon \rbrace \end{align}\) where $\Pi(\mu,\nu)$ denotes the set of probability measures on $E \times E$ with $\mu$ and $\nu$ beeing his first and second marginals.

This representation is more practicable as it leads to easiers computations. For example, when $\mu = \delta_{x}$ and $\nu = \delta_{y}$ for $x,y \in E$. We can see that $d_{LP}(\mu,\nu) = 1 \wedge d(x,y)$.

$\textit{Proof : }$ Consider the case when $d(x,y) \geq 1$. Therefore, in this case, we have for $\epsilon < d(x,y)$ \(\int_{E \times E} \mathbb{1}_{d(x,y) > \epsilon} \pi(dx,dy) = 1\) Therefore, it is clear that $d_{LP}(\mu,\nu) = 1$.

Now if we consider the cas when $d(x,y) < 1$. Therefore, in this case, we still have, for $\epsilon < d(x,y) < 1$ but for $\epsilon \geq d(x,y)$ the integral becomes null meaning that $d_{LP}(\mu,\nu) = d(x,y)$. Finally, we have $d_{LP}(\mu,\nu) = 1 \wedge d(x,y)$.

The total variation distance

The total variation distance between 2 measures $\mu$ and $\nu$ on $(E,\mathcal{E})$ is defined as the following :

When choosing \(\epsilon = \underset{A \in B(E)}{\text{ sup }} | \mu(A) - \nu(A) |\) for $\mu,\nu \in \mathcal{P}(E)$ in the definition of $d_{LP}(\mu,\nu)$ , we have :

\(\begin{align} d_{LP}(\mu,\nu) \leq \frac{1}{2} d_{TV}(\mu,\nu) \notag \end{align}\) Moreover, we have another representation of $d_{TV}(\mu,\nu)$ as follows :

Representation in terms of random variables.

If we assume that the probability space ($\Omega,\mathcal{F},\mathbb{P})$ is atomless in the sense that $\forall A \in \mathcal{F}$, if $\mathbb{P}(A) > 0$ there exists $B \subset A$ and $B \in \mathcal{F}$ such that $\mathbb{P}(B) > 0$. Then, for any distribution $\mu \in \mathcal{P}(E)$, we can construct a random variable $X : \Omega \to E$ such that $\mathcal{L}(X) = \mu$. (This result is admitted). If we apply this result to the space $E \times E$ equipped with a product distance, we see that whenever $\mu$ and $\nu$ are probability distrubtions and $\pi \in \Pi(\mu,\nu)$, we can find a pair of random variables $(X,Y) : \Omega \to E \times E$ such that $\pi=\mathcal{L}(X,Y)$. Therefore, we have 2 probabilistic representations for $d_{LP}(\mu,\nu)$ and $d_{TV}(\mu,\nu)$ given by :

The Wasserstein distance

Now, we introduce the most famous notion of distance in the space of measures / probabilities. First, we now introduce the notation $\mathcal{P}{p}(E)$ for the subspace of probability measures with a moment of order p meaning that $\mu \in \mathcal{P}{p}(E)$ if we have $\int_{E} d(x,0)^{p} \mu(dx) < + \infty$ or for any other choic as it does not play any role in the definition.

Now, for any $p \geq 1$ and $\mu,\nu \in \mathcal{P}_p(E)$, the $p-th$ Wasserstein distance is defined by :

\(\begin{align} W_{p}(\mu,\nu) = \underset{\pi \in \Pi(\mu,\nu)}{\text{ inf }}\left[\int_{E \times E} d(x,y)^{p} \pi(dx,dy)\right]^{\frac{1}{p}} \end{align}\) Assuming that there exists atleast one $\pi \in \Pi(\mu,\nu)$ realizing the infimum in (6). If we denote the notation $\Pi_p^{opt}(\mu,\nu)$ for all the couplings realizing ths infimum, we can write

\(\begin{align} \Pi^{opt}_{\mu,\nu} = \lbrace \pi \in \Pi(\mu,\nu) : W_{p}(\mu,\nu)^p = \int_{E \times E} d(x,y)^{p} \pi(dx,dy) \rbrace \end{align}\) This set is therefore nonempty.

The following proposition is the Duality Kantorovich duality theorem as is particularly important in optimal transport theory.

$\textit{Proposition : }$

Suppose $(E,d)$ is a metric complete space and $\mu,\nu \in \mathcal{P}_{p}(E)$, therefore we have :

\(\begin{align} W_{p}(\mu,\nu)^{p} = \underset{(\phi,\psi), \phi(x) + \psi(y) \leq d(x,y)^{p}}{\text{ sup }} \left[\int_{E} \phi(x) \mu(dx) + \int_{E} \psi(y)\nu(dy) \right]. \end{align}\) where the supremum is taken over all the bounded continuous functions $(\phi,\psi)$ defined on $E$. Moreover, if $\pi \in \Pi^{opt}(\mu,\nu)$ there exists $\phi \in L^1(E,\mu)$ and $\psi \in L^1(E,\nu)$ such that we have for $\pi$ almost $(x,y) \in E \times E$, \(\begin{align} \phi(x) + \psi(y) = d(x,y)^p \end{align}\)

Representation in terms of random variables

Consider now an atomless probability space $(\Omega,\mathcal{F},\mathbb{P})$. A random variable $X : \Omega \to E$ is said to be of order p for $p \geq 1$ if $\mathbb{E}[d(x_0,X)^{p}] < + \infty$ for any $x_0 \in E$. If $X$ and $Y$ are 2 random variables of order p, then we have :

\(\begin{align} W_{p}(\mathcal{L}(X),\mathcal{L}(Y))^{p} \leq \mathbb{E}[d(X,Y)^{p}] \end{align}\) Moreover, for $\mu$ and $\nu$ of order $p$, we have :

Some topological properties of the Wasserstein space $\mathcal{P}_2(\mathbb{R}^d)$

To be done asap

Differentiability of derivatives on Probability Measures

Some notions of derivatives on Probability Measure

To be done asap.

An Itô Formula Along the flow of measures

To be done asap.