Measure Differential Equations (MDE)

• MDE is a mathematical framework where measures serve either as driving signals or the evolving state, unifying distinct approaches.
• It extends classical ODEs to include impulsive, distributional, and nonlocal dynamics through techniques like LAS and integral representations.
• The theory underpins applications in fractional calculus, mean-field models, and epidemiology, highlighting its broad methodological impact.

Measure Differential Equation (MDE) is a literature-dependent term covering several distinct but related mathematical frameworks. In one established usage, it denotes differential equations driven by a Stieltjes measure or, more generally, by a distributional derivative $Du$, so that equations such as $Dx=f(t,x)+g(t,x)\,Du$ or $-D^2x=f(t,x)+g(t,x)\,Du$ include ordinary differential equations, impulsive equations, and broader distributional models as special cases (Liu et al., 2017). In another now-standard usage, it denotes an evolution equation for a time-dependent probability measure $\mu_t$ on $\mathbb{R}^n$, formally $\dot\mu_t = V[\mu_t]$, where $V$ is a probability vector field on the tangent bundle (Piccoli, 2017). A separate fractional-calculus literature also uses the acronym “MDE” for equations involving the M-fractional derivative; that terminology is distinct from both measure-driven and measure-valued theories (Padmapriya et al., 2019).

1. Terminology and principal meanings

Two principal meanings dominate the research literature. In the classical measure-driven sense, the unknown is an ordinary function $x$, while the “measure” enters as a driver in the independent variable through a Stieltjes or distributional term. In the probability-measure sense, the unknown itself is a measure $\mu_t$, and the equation evolves in a space of measures rather than in a finite-dimensional state space (Liu et al., 2017).

In the probability-vector-field framework, a Measure Differential Equation is formally

$\dot\mu_t = V[\mu_t],$

with $Dx=f(t,x)+g(t,x)\,Du$0 and $Dx=f(t,x)+g(t,x)\,Du$1. Here the state is a Borel probability measure, and the tangent-bundle measure $Dx=f(t,x)+g(t,x)\,Du$2 assigns a distribution of velocities to the mass of $Dx=f(t,x)+g(t,x)\,Du$3 (Piccoli, 2017).

A separate usage appears in local fractional calculus, where “MDE” abbreviates equations based on the M-fractional derivative $Dx=f(t,x)+g(t,x)\,Du$4. That usage concerns “M-fractional differential equations” rather than measure differential equations in either of the senses above (Padmapriya et al., 2019).

Usage	Prototype	Unknown
Classical measure-driven equation	$Dx=f(t,x)+g(t,x)\,Du$5	Function $Dx=f(t,x)+g(t,x)\,Du$6
Probability-vector-field MDE	$Dx=f(t,x)+g(t,x)\,Du$7	Measure $Dx=f(t,x)+g(t,x)\,Du$8
M-fractional equation	$Dx=f(t,x)+g(t,x)\,Du$9	Function $-D^2x=f(t,x)+g(t,x)\,Du$0

2. Classical measure-driven and distributional formulations

In the classical line associated with Das–Sharma, Leela, Satco, Slavík, Federson, Mesquita, and related authors, a first-order distributional equation

$-D^2x=f(t,x)+g(t,x)\,Du$1

is called a measure differential equation when $-D^2x=f(t,x)+g(t,x)\,Du$2 is of bounded variation, so that $-D^2x=f(t,x)+g(t,x)\,Du$3 is a Stieltjes measure. When $-D^2x=f(t,x)+g(t,x)\,Du$4 is absolutely continuous, the same form reduces to an ordinary differential equation. In this setting the solution is typically of bounded variation, and differentiation is interpreted in the measure/Stieltjes sense (Liu et al., 2017).

A broader distributional framework studies the nonlinear second-order problem

$-D^2x=f(t,x)+g(t,x)\,Du$5

with three-point boundary conditions and unknown $-D^2x=f(t,x)+g(t,x)\,Du$6, the space of regulated functions. Distributional derivatives are understood in the sense of Schwartz, and the terms involving $-D^2x=f(t,x)+g(t,x)\,Du$7 are interpreted through the Kurzweil–Henstock–Stieltjes integral. If $-D^2x=f(t,x)+g(t,x)\,Du$8 has bounded variation and $-D^2x=f(t,x)+g(t,x)\,Du$9 has bounded variation in $\mu_t$0, then $\mu_t$1 is a finite signed measure and the equation reduces to a second-order boundary-value measure differential equation (Liu et al., 2017).

The integral representation is central. Writing

$\mu_t$2

the distributional boundary-value problem is rewritten as an integral equation for regulated functions. Existence is proved by combining the Leray–Schauder nonlinear alternative with compactness in $\mu_t$3 and KH–Stieltjes convergence theory. Under hypotheses $\mu_t$4, a sufficient condition is

$\mu_t$5

where $\mu_t$6 arises from the growth bound on $\mu_t$7 (Liu et al., 2017).

This framework contains impulsive MDEs as concrete examples. Taking $\mu_t$8 to be a Heaviside function yields $\mu_t$9 equal to a Dirac measure, hence an impulsive measure term. Taking $\mathbb{R}^n$0 to be the Weierstrass function produces a driving signal of unbounded variation, which lies beyond classical measure theory but remains admissible in the KH–Stieltjes setting. This suggests that the distributional approach strictly extends the traditional measure-driven theory (Liu et al., 2017).

3. Probability vector fields and measure-valued dynamics

In the modern measure-valued theory, the basic object is a Probability Vector Field (PVF),

$\mathbb{R}^n$1

Disintegrating $\mathbb{R}^n$2 as

$\mathbb{R}^n$3

the conditional measure $\mathbb{R}^n$4 describes the distribution of velocities at position $\mathbb{R}^n$5. A weak solution of

$\mathbb{R}^n$6

is a curve such that, for every $\mathbb{R}^n$7,

$\mathbb{R}^n$8

This formulation makes MDEs a measure-theoretic generalization of ODEs and relates them directly to transport and kinetic equations (Piccoli, 2017).

Classical ODEs are recovered by choosing

$\mathbb{R}^n$9

for a vector field $\dot\mu_t = V[\mu_t]$0. Then the MDE is exactly the measure formulation of the transport equation

$\dot\mu_t = V[\mu_t]$1

The same framework also supports more general PVFs that encode random velocities, finite-speed diffusion, and concentration phenomena. At the algebraic level, vector-field addition corresponds to fiber convolution of PVFs, and the map $\dot\mu_t = V[\mu_t]$2 is a monoid isomorphism from vector fields under $\dot\mu_t = V[\mu_t]$3 to PVFs under fiber convolution (Piccoli, 2017).

A key structural result is that an MDE is equivalent to a nonlocal continuity equation driven by the barycentric velocity

$\dot\mu_t = V[\mu_t]$4

so that

$\dot\mu_t = V[\mu_t]$5

This equivalence connects the PVF formalism with Ambrosio–Gigli–Savaré superposition arguments and with the broader theory of nonlocal transport equations (Camilli et al., 2019).

4. Well-posedness, semigroups, and approximation schemes

The foundational existence theory for PVF-driven MDEs uses support sublinearity and Wasserstein continuity. Under assumption $\dot\mu_t = V[\mu_t]$6,

$\dot\mu_t = V[\mu_t]$7

and continuity assumption $\dot\mu_t = V[\mu_t]$8, solutions are constructed as uniform-in-time limits of Lattice Approximate Solutions (LAS). The LAS scheme discretizes time, space, and velocity with

$\dot\mu_t = V[\mu_t]$9

pushes forward spatially discretized mass along discretized velocities, and yields existence of a solution for every compactly supported initial measure. Under the stronger fiber-based Lipschitz condition $V$0, the theory produces a Lipschitz semigroup with continuous dependence on initial data (Piccoli, 2017).

Uniqueness is subtler than existence. Weak solutions need not be unique, and the theory therefore distinguishes uniqueness of a weak solution from uniqueness of a Lipschitz semigroup. The notion of Dirac germ reduces semigroup uniqueness to the behavior of LAS limits for finite sums of Dirac masses (Piccoli, 2017).

The superposition principle and alternative schemes sharpen this picture. A solution can be represented by a probability measure on path space whose marginals solve the MDE, in the spirit of the Ambrosio–Gigli–Savaré superposition principle. The same work analyzes a semi-discrete Lagrangian scheme and a mean-velocity scheme, and shows that different convergent schemes may select different weak solutions. The “splitting particle” example is the standard illustration: LAS and semi-discrete Lagrangian approximations yield a symmetric splitting solution, whereas the mean-velocity scheme may remain stationary (Camilli et al., 2019).

The theory also extends to nonconservative dynamics. An MDE with source and nonlinear growth/decay is written

$V$1

with $V$2 a measure vector field, $V$3 a source term, and $V$4 a bounded Lipschitz growth/decay rate that may change sign. Existence is proved in the flat metric $V$5 by a modified approximation scheme that incorporates the exact exponential solution of $V$6 and therefore preserves nonnegativity. The same paper introduces the simpler condition $V$7 for continuity with respect to initial data, replacing the earlier, more involved optimal-transport condition (Düll et al., 2021).

5. Relations to PDEs, stochastic systems, and applications

The epidemiological ODE–MDE model for viral mutations couples ordinary differential equations for susceptible and removed populations with an MDE for the infected population, now represented as a measure $V$8 on a mutation parameter $V$9. The infected measure evolves through a PVF $x$0 and a source term involving variant-dependent infection and recovery rates. Well-posedness is established with the generalized Wasserstein distance $x$1, which is designed for variable total mass, and the framework recovers classical SIR dynamics when the rates are variant-independent or effectively time-dependent (Gong et al., 2022).

A different line of work interprets the law of a mean field SDE as an MDE. For

$x$2

Itô’s formula yields

$x$3

and, when $x$4 has density $x$5, this becomes a nonlinear Fokker–Planck equation. A recent numerical method truncates the Fokker–Planck problem to a bounded domain, applies an explicit–implicit finite difference scheme, and then uses the numerical density to replace the true measure in the mean field SDE. The resulting error analysis controls both the density approximation and the law approximation (Zhou et al., 23 Mar 2025).

With common noise, the relevant state is the pair $x$6, where $x$7 is the conditional law given the common noise. The conditional law solves a stochastic Fokker–Planck equation on $x$8, and the dynamics induce a semigroup on the product space

$x$9

Under polynomial growth and monotonicity assumptions, this semigroup is exponentially contractive, has a unique invariant measure, and yields uniform-in-time propagation of chaos with explicit Wasserstein rates (Chen et al., 21 Sep 2025).

6. Conceptual distinctions and recurrent misconceptions

A recurrent misconception is that “Measure Differential Equation” always refers to the same object. In fact, the classical measure-driven theory and the probability-measure theory are structurally different: in the first, the unknown is a function and the measure acts as a driver in time; in the second, the unknown is itself a measure evolving in a state space (Liu et al., 2017).

A second misconception is that weak solutions are automatically unique. In the PVF framework this is false: weak MDE solutions can be nonunique, and different approximation schemes may converge to different solutions. What is often unique is a semigroup selected by a prescribed approximation mechanism, such as LAS together with a compatible Dirac germ (Camilli et al., 2019).

A third misconception is that MDEs are merely conservative transport equations. The modern theory includes finite-speed diffusion and concentration phenomena, and the nonconservative extension includes source terms and nonlinear growth/decay that can change total mass while preserving nonnegativity (Düll et al., 2021).

Finally, the acronym “MDE” is itself ambiguous. In some fractional-calculus papers it denotes equations based on the M-fractional derivative $\mu_t$0, with characteristic-polynomial methods and variation of parameters resembling ordinary linear ODE theory. That usage is terminologically separate from measure differential equations in both the measure-driven and measure-valued senses (Padmapriya et al., 2019).

Taken together, these literatures show that “Measure Differential Equation” is not a single theory but a family of frameworks organized around a common idea: differential evolution in which measures appear either as coefficients, as drivers, or as the state itself. The modern arXiv literature has developed this idea in three especially active directions: distributional equations driven by Stieltjes measures, PVF-driven dynamics on Wasserstein spaces, and measure-valued formulations of mean-field and stochastic systems (Piccoli, 2017).