Marginal Velocity Field

• Marginal velocity field is the conditional mean velocity at each spatial location, ensuring compatibility with discrete position marginals.
• It employs multi-marginal Schrödinger bridge frameworks and variational principles to interpolate particle paths with minimal stochastic energy.
• Computational approaches like Bregman iterations and Bayesian inference enable smooth, high-dimensional reconstructions under observational constraints.

The marginal velocity field is a central construct in the mathematical and computational theory of population dynamics, stochastic systems, and transport processes constrained by spatial marginals at discrete time points. It captures the conditional or mean velocity at every spatial location and time, consistent with an ensemble of particles or trajectories whose positions are known only at selected time slices. Contemporary developments employ frameworks such as multi-marginal Schrödinger bridge problems and Bayesian field inference to rigorously define, compute, and interpret the marginal velocity field in high-dimensional and observationally underdetermined settings (Chen et al., 2023, Chen et al., 2019, Lavaux, 2015).

1. Definition and Mathematical Formalism

The marginal velocity field $v(t, x)$ is defined as the conditional mean of the velocity at position $x$ and time $t$ under a probability law on phase space paths that matches empirical or imposed position-marginals at a collection of times $\{t_0, \dots, t_N\}$. Formally, for a phase space density $\mu(t, x, v)$,

$$m(t, x) = \int v\,\mu(t, x, v)\,dv, \qquad \rho(t, x) = \int \mu(t, x, v)\,dv,$$

$v(t, x) = \frac{m(t, x)}{\rho(t, x)},$

where $m(t, x)$ is the first velocity moment and $\rho(t, x)$ is the spatial marginal at time $t$ (Chen et al., 2019, Chen et al., 2023). This mean velocity field is optimally consistent with path measures that are closest to a physical or stochastic prior process and matches the sequence of observed spatial marginals.

2. Multi-Marginal Schrödinger Bridge Problem

The multi-marginal Schrödinger bridge framework seeks a path measure $\pi$ on phase space $z_t = (x_t, v_t)$ that minimizes the sum of Kullback-Leibler divergences (relative entropies) to a reference process, typically a Langevin or inertial SDE, under the constraint that the projected spatial marginals at times $\{t_i\}$ recover the empirical distributions $\{\rho_{t_i}\}$: $$\min_\pi\, \sum_{i=0}^{N-1} \mathrm{KL}\bigl( \pi_{t_i:t_{i+1}}\,\|\,\xi_{t_i:t_{i+1}} \bigr) \quad \text{s.t.} \quad \int \pi(x_{t_i}, v_{t_i})\,dv_{t_i} = \rho_{t_i}(x_{t_i})\ \forall\, i.$$ The optimally interpolating path measure $\pi^*$ factorizes through forward and backward Schrödinger potentials $(\Psi_t, \widehat{\Psi}_t)$, with the forward and backward SDEs: $$dx_t = v_t\,dt, \quad dv_t = \pm\,g^2(t)\,\nabla_v \log \Psi_t(x_t, v_t)\,dt + g(t)\,dW_t.$$ The conditional mean velocity is then obtained as

$$v^*(t, x) = \mathbb{E}_{\pi^*}[v_t \mid x_t = x] = \frac{\int v\,\Psi_t(x, v)\,\widehat{\Psi}_t(x, v)\,dv}{\int \Psi_t(x, v)\,\widehat{\Psi}_t(x, v)\,dv}.$$

These constructions yield globally smooth, time-symmetric trajectories that interpolate the marginals with minimal stochastic “energy” in Wasserstein space (Chen et al., 2023, Chen et al., 2019).

3. Variational and PDE Characterization

The derivation of the marginal velocity field proceeds via a variational principle that regularizes the classical optimal transport action by entropy (Fisher information) terms. The kinetic (Benamou–Brenier) action is: $$\min_{\mu,\,\hat a} \int_0^1 \int \Bigl\{ \|\hat a\|^2\,\mu + \tfrac14 \|\nabla_v \log \mu\|^2\,\mu \Bigr\} dx\,dv\,dt + [\mu_1 \log \mu_1 - \mu_0 \log \mu_0 ].$$ The coupled Hamiltonian PDEs for the optimal phase-space density and Hamiltonian potential $(\mu, \phi)$ are: $$\begin{aligned} &\partial_t \mu + v \cdot \nabla_x \mu + \nabla_v \cdot \bigl(\tfrac12 \nabla_v \phi\,\mu\bigr) = 0, \ &\partial_t \phi + v \cdot \nabla_x \phi + \tfrac14 \|\nabla_v \phi\|^2 - \tfrac14 \|\nabla_v \log \mu\|^2 - \tfrac12 \Delta_v \phi = 0. \end{aligned}$$ Boundary and convex constraints enforce the empirical position marginals at discrete times. In the zero-diffusion limit, this system recovers the classical measure-valued spline in Wasserstein space, with the marginal velocity interpolating optimally constrained particle flows (Chen et al., 2019).

4. Bregman Iteration and Sinkhorn-like Algorithms

Numerical computation of the marginal velocity field is achieved via Bregman iterative projections (generalizing iterative proportional fitting) in the space of path measures. This methodology cyclically enforces the KL projections onto each marginal-constraint set, efficiently solving high-dimensional, multi-marginal entropy-regularized transport by alternating between forward and backward “half-bridge” updates for network parameters representing the drifts in velocity (Chen et al., 2023, Chen et al., 2019).

In discretized settings, the coupling $\pi_{i,i+1}(x_i, v_i, x_{i+1}, v_{i+1})$ is updated by renormalizations that enforce marginals: $\pi_{01} \mapsfrom \frac{\rho_0(x_0)}{\int \bar\pi_{01}(x_0, v_0; x_1, v_1) dv_0 dx_1 dv_1}\bar\pi_{01}(x_0, v_0; x_1, v_1),$

$\pi_{N-1, N} \mapsfrom \frac{\rho_N(x_N)}{\int \bar\pi_{N-1, N}(x_{N-1}, v_{N-1}; x_N, v_N) dv_N dx_{N-1} dv_{N-1}}\bar\pi_{N-1, N}.$

Interleaved central steps “join and split” the marginals at interior time slices, enabling consistent optimization across all marginals (Chen et al., 2019).

5. Bayesian Marginal Velocity Field Reconstruction

In cosmological and observational contexts, the marginal velocity field can be inferred as a posterior distribution under a hierarchical Bayesian framework, enforcing joint constraints from noisy redshift and distance modulus measurements, instrument calibration, selection effects, and non-linear dispersion. The Bayesian posterior for the velocity at a spatial location is

$$p[v(x_i)\mid\text{data}] = \int p\bigl(v(x_i), \Theta, \{d_i\}, H, \ldots \mid \text{data}\bigr)d\cdots,$$

where $\Theta$ denotes velocity divergence modes and the integral spans all cosmological, noise, and selection parameters. Posterior samples generated by block Gibbs sampling yield unbiased estimates for credible intervals of the marginal velocity field, mapping its uncertainty structure and propagation of selection biases (Lavaux, 2015).

6. Measure-Valued Spline Connection and Time Symmetry

In the low-noise (zero diffusion) limit, multi-marginal Schrödinger bridges converge to classical second-order measure-valued splines: $$\inf \int_0^1 \int \|a\|^2 \mu\,dx\,dv\,dt,\quad \partial_t \mu + v \cdot \nabla_x \mu + \nabla_v \cdot (a \mu)=0,\quad \int \mu(t_k)\,dv=\rho_k.$$ Here, the velocity field $v(t,x)$ provides the minimal-energy, time-symmetric interpolation through all prescribed marginals in the 2-Wasserstein sense (Chen et al., 2019, Chen et al., 2023). Entropic regularization in the Schrödinger bridge context induces smoothness and stochasticity, establishing the marginal velocity field as the mean flow of a globally consistent, regularized measure-valued spline.

7. Computational and Practical Considerations

Evaluation of the marginal velocity field in high-dimensional scenarios requires efficient parameterizations (e.g., neural nets for drift terms), scalable stochastic simulation, and concessions between full density matching and mean-matching for computational tractability. The Deep Momentum Multi-Marginal Schrödinger Bridge (DMSB) framework applies these principles, reconstructing both stochastic trajectories and conditional velocity fields from position snapshots alone, with practical applications demonstrated for both synthetic and real biological data (Chen et al., 2023).

The table below summarizes main frameworks for marginal velocity field recovery:

Framework/Approach	Core Principle	Key Reference
Multi-Marginal Schrödinger Bridge	Entropy-minimizing interpolation of marginals in phase space	(Chen et al., 2019, Chen et al., 2023)
Bayesian Velocity Field Inference	Posterior sampling for $v(x)$ given observational data	(Lavaux, 2015)
Measure-Valued Spline (Wasserstein)	Minimal-energy (acceleration-penalized) interpolation	(Chen et al., 2019)

Each approach characterizes the marginal velocity field as an optimal or most probable mean flow consistent with known marginals, with differing regularizations, computational strategies, and application domains.