Figalli-Trevisan Superposition Principle

• The Figalli-Trevisan Superposition Principle is a representation theorem that decomposes measure-valued solutions into convex combinations of absolutely continuous ODE trajectories.
• It reinterprets nonlocal continuity equations by disintegrating the velocity field into a barycentric form, thereby connecting macroscopic measures with microscopic dynamics.
• The principle underlies various approximation schemes, such as Lattice Approximate and Mean Velocity methods, which help elucidate uniqueness and convergence issues in weak solutions.

The Figalli–Trevisan Superposition Principle is a rigorous representation theorem that establishes a bridge between weak, measure-theoretic evolutions and the existence of probabilistic pathwise dynamics for a broad class of measure differential equations (MDEs). Originating as a generalization of the Ambrosio–Gigli–Savaré superposition principle in the context of Wasserstein gradient flows, it provides a method to “disintegrate” measure-valued solutions to (possibly nonlocal and nonlinear) continuity equations into convex combinations (superpositions) of absolutely continuous trajectories, each of which solves an underlying ordinary differential equation (ODE). This formulation is central in linking the macroscopic statistical description of evolving measures to microscopic Lagrangian dynamics and plays a decisive role in the analysis of uniqueness and approximation schemes for MDEs.

1. MDEs and Reformulation as Nonlocal Continuity Equations

Measure Differential Equations (MDEs) generalize ODEs by encoding the evolution of probability measures rather than pointwise trajectories. An MDE is given in weak form by

$$\frac{d}{dt} \langle \mu_t, f \rangle = \int_{T^d} \nabla f(x) \cdot v\ dV[\mu_t](x,v),$$

where $\mu_t \in \mathscr{P}(\mathbb{R}^d)$ is a curve in the space of probability measures, $V$ is a probability vector field assigning to each $\mu$ a probability measure $V[\mu]$ on the tangent bundle $T^d = \mathbb{R}^d \times \mathbb{R}^d$, and $f$ ranges over smooth test functions.

Disintegrating $V[\mu_t]$ with respect to the spatial coordinate yields $V[\mu_t] = \mu_t \otimes \nu_x[\mu_t]$, with $\nu_x[\mu_t]$ describing the conditional velocity distribution at position $x$. The barycentric velocity field is defined as

$w_t(x) = \int_{\mathbb{R}^d} v\ d\nu_x[\mu_t](v).$

This allows rewriting the MDE as a nonlocal continuity equation

$$\partial_t \mu_t + \operatorname{div} (w_t \mu_t) = 0,$$

making explicit the relation between the original measure-driven evolution and a transport driven by the averaged velocity.

2. Mathematical Formulation of the Superposition Principle for MDEs

The superposition principle in the MDE framework states that, under suitable integrability (e.g., $$\int_0^T \int_{T^d} |v|^p\, dV[\mu_t](x,v) dt < \infty$$, $p>1$), every weak solution $\{\mu_t\}_{t \in [0,T]}$ to the nonlocal continuity equation can be represented as

$\mu_t = e_t{}_\# \boldsymbol{\eta},$

where $\boldsymbol{\eta}$ is a probability measure on the space $\Gamma_{[0,T]}$ of continuous curves, $e_t$ is the evaluation map $e_t(\gamma) = \gamma(t)$, and $\boldsymbol{\eta}$-almost every trajectory $\gamma$ is absolutely continuous and satisfies

$\dot{\gamma}(t) = w_t(\gamma(t))$

for almost every $t \in (0,T)$.

This yields a precise decomposition of the measure-valued evolution as a convex combination of ODE flows, providing a robust link between the “macroscopic” Eulerian and “microscopic” Lagrangian descriptions.

3. Numerical Approximation Schemes and Their Role

The weak measure-theoretic setting of MDEs generally does not guarantee uniqueness. The paper analyzes the convergence and selection mechanisms of various approximation schemes, each with a distinct impact on the solution structure:

A. Lattice Approximate Solution (LAS) Scheme: Time, space, and velocity are discretized on a grid; the measure is updated by

$$\mu_{k+1}^N = \sum_{i,j} m_{ij}^v(V[\mu_{k\Delta t}^N]) \, \delta_{x_i + \Delta t v_j},$$

where $m_{ij}^v$ are the grid weights. As the grid refines ($N \rightarrow \infty$), the associated measures on trajectories converge (along subsequences) to a measure supported on curves realizing the superposition principle.

B. Semi-Discrete Lagrangian Scheme: Discrete in time but continuous in space. Starting from $\mu_0$, evolve via

$$\bar{\mu}_{t_{k+1}}^N = \int_{T^d} \delta_{x + v \Delta t} dV[\bar{\mu}_{t_k}^N](x,v),$$

with linear interpolation for intermediate times. Under suitable Lipschitz properties of the PVF, this scheme converges (up to subsequences) to a solution, producing a Lipschitz semigroup and thus a form of uniqueness among limits of these approximations.

C. Mean Velocity (Barycentric) Scheme: Transport the measure deterministically using the barycentric velocity computed as

$\bar{v}(x) = \int v\, d\nu_x[\mu](v),$

leading to the update

$$\hat{\mu}_{t_{k+1}}^N = \hat{\mu}_{t_k} \oplus \Delta t\,\delta_{\bar{v}(x)}.$$

This can yield different weak solutions than LAS or semi-discrete schemes, particularly when the map $x \mapsto \nu_x[\mu]$ is only Borel measurable rather than continuous.

A tabular summary:

Scheme	Discretization Type	Selection Mechanism for Velocities
Lattice Approximate	Time, space, velocity (grid)	Empirical update via grid weights
Semi-Discrete Lagrangian	Time-only	Transport by integrating over $V[\mu]$
Mean Velocity	Time-only	Deterministic shift by barycentric velocity

This suggests the choice of approximation scheme is an implicit selection mechanism when uniqueness is not available.

4. Uniqueness and Non-Uniqueness Phenomena

Uniqueness in the weak measure framework is closely tied to the regularity of the barycentric velocity field $w_t(x)$. If $x \mapsto w_t(x)$ is Lipschitz, standard continuity equation theory implies uniqueness. However, only Borel measurability of $x \mapsto \nu_x[\mu]$ is typically available, resulting in irregularity of $w_t(x)$ and possible non-uniqueness. Consequently, different schemes (or even different discretizations of the same type) can converge to distinct weak solutions.

Illustrative phenomena including "mass splitting" versus stationary transport arise, demonstrating that numerical approximation inherently selects specific solutions under non-uniqueness. This underlines the subtlety required in the interpretation of weak measure-theoretic evolutions in the absence of structural regularity.

5. Analytical and Representational Consequences

The superposition principle enables:

• Construction of probabilistic representations for solutions to MDEs, making them accessible to stochastic and Lagrangian methods;
• Fine analysis of the correspondence among solutions produced by distinct numerical schemes;
• In the case of approximation limits (e.g., LAS), enabling the existence of a semigroup of solutions exhibiting a unique, regular evolution.

However, the principle also reveals inherent limitations: when the measure–velocity field relationship lacks sufficient regularity, the class of weak solutions becomes large, and the principle alone does not select a unique evolution. Therefore, the combined use of the superposition representation and carefully designed numerical or analytic approximation methods is essential.

6. Synthesis and Broader Significance

The adaptation of the superposition principle to MDEs as conducted in "Superposition principle and schemes for Measure Differential Equations" (Camilli et al., 2019) synthesizes techniques from optimal transport, kinetic theory, and probability to provide a robust and flexible framework for representing and analyzing measure-driven evolutions. The interplay between macroscopic measure flows and their microscopic realizations via ODE trajectories constitutes a central methodological advance for both theoretical investigations and computational schemes. The principle demonstrates particular strength in assessing the convergence of particle methods, evaluating uniqueness, and understanding the dynamical structures underpinning mean-field and kinetic limits.

By explicitly connecting the weak-form macroscopic evolution to path-space representations, the Figalli–Trevisan superposition principle enables new approaches to measure-valued PDEs where classical techniques are ineffective, and it provides a toolset for addressing foundational questions of uniqueness, stability, and selection in generalized dynamical systems.