Probability paths and velocity fields

Fix an integer $d\ge 1$, a sigma-finite measure $\lambda$ on the Borel subsets of ${\mathbb{R}}^d$, and a Borel probability measure $\pi$ on ${\mathbb{R}}^d$. Here sigma-finite means that there are Borel sets $E_n$ with ${\mathbb{R}}^d={\bigcup }_{n\ge 1}{E}_n$ and $\lambda (E_n)<\mathrm{\infty }$ for each $n$. A conditional probability-density path with endpoint law $\pi$ is a function measurable for $\mathcal{B}([0,1])\otimes \mathcal{B}({\mathbb{R}}^d)\otimes \mathcal{B}({\mathbb{R}}^d)$ on time, state, and endpoint variables $\rho :[0,1]\times {\mathbb{R}}^d\times {\mathbb{R}}^d\to [0,\mathrm{\infty }]$ such that, for every $t\in [0,1]$, ${\int }_{{\mathbb{R}}^d}{\rho }_t(x\mid x_1)\lambda (dx)=1$ for $\pi$-almost every endpoint $x_1$. Its mixture function is $p_t(x):={\int }_{{\mathbb{R}}^d}{\rho }_t(x\mid x_1)\pi (d{x}_1)\mathrm{.}$ For a vector field $u_t(x\mid x_1)=(u_t^1,\dots ,u_t^d)$ measurable for the same product sigma-algebra, call it a conditional velocity field for $\rho$ when, for $\pi$-almost every $x_1$, ${\rho }_t(\cdot \mid x_1)$ and every component of ${\rho }_t(\cdot \mid x_1)u_t(\cdot \mid x_1)$ are continuously differentiable in $(t,x)$ on $(0,1)\times {\mathbb{R}}^d$, the functions ${\rho }_t$ and ${\rho }_t{u}_t^j$ are locally integrable there with respect to $dt\lambda (dx)$, and ${\int }_0^1\text{\hspace{0.17em}⁣}{\int }_{{\mathbb{R}}^d}{\rho }_t(x\mid x_1)({\mathrm{\partial }}_t\phi (t,x)+{\sum }_{j=1}^d{u}_t^j(x\mid x_1){\mathrm{\partial }}_{x_j}\phi (t,x))\lambda (dx)dt=0$ for every continuously differentiable real function $\phi$ with compact support in $(0,1)\times {\mathbb{R}}^d$. Here $dt$ denotes the supplied Lebesgue measure on time, and local integrability means absolute integrability on every compact subset. This is the conditional continuity equation for the measures ${\rho }_t(\cdot \mid x_1)\lambda$. In the Lebesgue-reference case $\lambda (dx)=dx$, its classical form under the stated differentiability is ${\mathrm{\partial }}_t{\rho }_t(x\mid x_1)+{\sum }_{j=1}^d{\mathrm{\partial }}_{x_j}\text{\hspace{0.17em}⁣}({\rho }_t(x\mid x_1)u_t^j(x\mid x_1))=0$ on $(0,1)\times {\mathbb{R}}^d$. This unweighted divergence formula is not the convention for a general reference measure. Only the interior-time equation is imposed; no endpoint continuity or existence or uniqueness of an ODE flow is asserted. At any $(t,x)$ where $0<p_t(x)<\mathrm{\infty }$ and the numerator is a finite componentwise Lebesgue integral, the marginal velocity field is $u_t(x):=\frac{{\int }_{{\mathbb{R}}^d}{\rho }_t(x\mid x_1)u_t(x\mid x_1)\pi (d{x}_1)}{p_t(x)}\mathrm{.}$ This definition assigns no value to the quotient where $p_t(x)=0$ or $p_t(x)=\mathrm{\infty }$, or where the numerator is not finite.

Probability measures and probability spaces

Measures on sigma-algebras

The Borel sigma-algebra

Lipman et al., §§2–3

Holderrieth and Erives, §§3.1–3.2

Santambrogio, §4.1.2, Definition 4.1: the distributional continuity equation for measures