# Generalized gradient structures for measure-valued population dynamics and
their large-population limit

Jasper Hoeksema and Oliver Tse  Department of Mathematics and Computer
Science, Eindhoven University of Technology, 5600 MB Eindhoven,The
Netherlands; email addresses<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We consider the forward Kolmogorov equation corresponding to measure-valued
processes stemming from a class of interacting particle systems in population
dynamics, including variations of the Bolker-Pacala-Dieckmann-Law model. Under
the assumption of detailed balance, we provide a rigorous generalized gradient
structure, incorporating the fluxes arising from the birth and death of the
particles.

Moreover, in the large population limit, we show convergence of the forward
Kolmogorov equation to a Liouville equation, which is a transport equation
associated with the mean-field limit of the underlying process. In addition,
we show convergence of the corresponding gradient structures in the sense of
Energy-Dissipation Principles, from which we establish a propagation of chaos
result for the particle system and derive a generalized gradient-flow
formulation for the mean-field limit.

###### Contents

  1. 1 Introduction
    1. 1.1 Measure-valued population dynamics and mean-field limits
    2. 1.2 Gradient-flow formulation
    3. 1.3 Convergence results
    4. 1.4 Notation
  2. 2 Mean-field system
    1. 2.1 A priori estimates
    2. 2.2 Strong solutions
    3. 2.3 Variational characterization
  3. 3 Forward Kolmogorov equation
    1. 3.1 A priori estimates
    2. 3.2 Weak solutions
    3. 3.3 Variational characterization
  4. 4 Liouville equation and lifted dynamics
    1. 4.1 A priori estimates
    2. 4.2 Weak solutions
    3. 4.3 Superposition principle
    4. 4.4 Variational characterization
  5. 5 EDP convergence
    1. 5.1 $\varGamma$-convergence of $\mathcal{F}_{n}$
    2. 5.2 Uniform estimates
    3. 5.3 Proof of main results
  6. A Motivation from large deviations
  7. B Superposition principle in $\mathbb{R}^{\mathbb{N}}$
  8. C Non-continuous competition kernel

##  1\. Introduction

An important goal in theoretical biology and population dynamics is to derive
macroscopic equations from microscopic models [CFM06, FKK09]. For many
stochastic interacting particle systems involving birth, mutation, and death,
these connections have been made rigorous. One such class of particle systems
consists of spatially-structured models such as the Bolker-Pacala and
Dieckmann-Law (BPDL) model [BP97, LD00]. The dynamics of these particle
systems can be described by jump processes on the space of finite positive
measures and can be used to derive macroscopic models.

The convergence of such measure-valued jump processes under a mean-field
scaling to a large-population limit is shown for example in [FM04] via
martingale techniques, and in [FKK09], where an analytic approach to the
convergence of rescaled moment equations is used. In both approaches, the
limiting evolution is governed by a non-local evolution equation given by

(1.1)
$\partial_{t}u_{t}(x)=\int_{\mathbb{R}^{d}}m(y,x)\,u_{t}(y)\,\mathrm{d}y-u_{t}(x)\int_{\mathbb{R}^{d}}c(x,y)\,u_{t}(y)\,\mathrm{d}y.$

We will refer to (1.1) as the _mean-field equation_. Here, $u_{t}$ represents
the limiting density of particles at time $t$, and the parameter functions $m$
and $c$ are continuous and bounded functions stemming from birth, dispersal,
and competition in the BPDL model.

In recent years, there has been considerable activity in studying the mean-
field equation (1.1) and the BPDL model in more general spaces, allowing for
dynamics involving multiple species and combinations of discrete and
continuous traits. See for example [FKK21] for an overview of existing models,
where instead of $\mathbb{R}^{d}$ the underlying space is an arbitrary locally
compact Polish space. However, convergence in the large-population limit is
not considered.

Meanwhile, powerful variational tools have been developed in the last decade
for studying mean-field interacting jump processes and their limits under the
assumption of detailed balance. To highlight only a few: [EFLS16] studied
mean-field limits for measure-dependent jump processes; [Erb16] proved the
convergence of the spatially-homogeneous Kac-process to the Boltzmann
equation; [Sch19] investigated the macroscopic limit of Becker-Döring models;
[KJZ19] showed hydrodynamic limits for zero-range and exclusion processes;
[MM20] discussed convergence and higher-order approximations for chemical
reaction networks, an approach that was subsequently used in the setting of
discretized reaction-diffusion equations in [MSW22].

In this work, we extend and apply these variational techniques to prove the
mean-field limit for population dynamics over arbitrary compact Polish spaces,
with bounded measurable parameters $m,c$ satisfying a detailed balance
condition. In addition, we establish entropic propagation of chaos, which
controls the discrepancy between the microscopic and macroscopic models in a
precise sense. To the authors’ knowledge, this is the first convergence result
under such general assumptions.

To do so, we first introduce a new generalized gradient structure and rigorous
variational formulation for the forward Kolmogorov equation (FKE)
corresponding to the BPDL model, where the FKE describes the evolution of the
law of the measure-valued process. Our formulation incorporates not only the
equation itself but tracks the birth and death fluxes as well. This extends
the generalized gradient-flow framework of [PRST22] due to the unboundedness
of the underlying jump kernel, and the positivity of the fluxes.

We then show convergence of these generalized gradient structures under a
mean-field scaling and the large-population limit in the sense of Energy
Dissipation Principles (EDPs) (see [LMPR17]). The limiting gradient flow is
the Liouville equation corresponding to the mean-field equation, namely a
transport equation that describes the evolution of the law of a process that
follows deterministic dynamics described by (1.1) but for possibly random
initial conditions. This connection between the Liouville equation and the
mean-field equation is made rigorous with the help of a modification of the
superposition principle of [AC08].

In particular, we deduce that the laws determined by the FKE equation
concentrate around the solution of the mean-field equation (1.1), which due to
the convergence of the associated free energies translates into an entropic
propagation of chaos result, see Theorem 1.10.

##### Outline

The rest of this section is devoted to giving a brief overview of our setting
and presenting the main results. In Section 2 the mean-field equation and
corresponding gradient structure are introduced. We repeat this process in
Sections 3 and 4 for the forward Kolmogorov equation and the Liouville
equation respectively, with the proof of a modified superposition principle
delegated to Appendix B. Finally, in Section 5, we establish the EDP-
convergence of the gradient structures, and prove both the convergence to the
mean-field limit and the propagation of chaos.

###  1.1. Measure-valued population dynamics and mean-field limits

We consider the forward Kolmogorov equation that corresponds to a generalized
version of the BPDL model. In its classical form, the Bolker-Pacala model is a
purely spatially-structured microscopic model for a population of plants
involving the birth, dispersal, and either natural death or death by
competition for resources and can be modeled as a jump process in the space of
positive measures over $\mathbb{R}^{d}$. However, in certain models of
adaptive evolution it is the mutation of traits that play a role, instead of
spatial evolution (see [LD00, CFM06, CFM08]). Moreover, if one wants to model
multiple interacting species or marked configuration spaces, more general
spaces than $\mathbb{R}^{d}$ are needed [KLU99, FKK21]).

Therefore, let the trait space be an arbitrary Polish space, denoted
henceforth as $\mathcal{T}$. We model the BPDL-dynamics at any time $t$ as an
interacting particle system with particles
$A_{t}^{1},\dots,A_{t}^{N_{t}}\in\mathcal{T}$ at positions
$X_{t}^{1},\dots,X_{t}^{N_{t}}\in\mathcal{T}$, where the number of particles
$N_{t}$ at time $t$ is not fixed since particles can be removed from and added
to the system.

Moreover, let $b\in\mathcal{B}^{+}(\mathcal{T})$,
$d,c\in\mathcal{B}^{+}(\mathcal{T}\times\mathcal{T})$ be non-negative
measurable functions, $n>0$ a positive parameter, and
$\gamma\in\mathcal{M}^{+}_{loc}(\mathcal{T})$ a non-negative reference measure
such that

$\int_{\mathcal{T}}d(x,y)\,\gamma(\mathrm{d}y)=1,\qquad\mbox{ for all
}x\in\mathcal{T}.$

Then the BPDL-dynamics can be described as follows:

  * •

Each particle located at a position $x\in\mathcal{T}$ has two exponential
clocks: a _seed_ clock with rate $b(x)$ and a _death_ clock with rate
$\genfrac{}{}{}{2}{1}{n}\sum_{i=1}^{N_{t}}c(x,X_{t}^{i})$.

  * •

If the death clocks rings, the particle is deleted.

  * •

If the seed clock rings, a new particle is added at position $y\in\mathcal{T}$
with probability $d(x,y)\gamma(\mathrm{d}y)$.

Alternatively, we can describe these dynamics in the form of reacting
particles. Namely, setting $m(x,y):=b(x)d(x,y)$, then with a little of abuse
of notation we have

(1.2) $\displaystyle A^{i}_{t}$ $\displaystyle\to
A^{i}_{t}+A^{N_{t}+1}_{t}\quad$ with rate $\displaystyle
m\left(X_{t}^{i},X_{t}^{N_{1}+1}\right)\gamma\left(X_{t}^{N_{1}+1}\right),$
$\displaystyle A^{i}_{t}+A^{j}_{t}$ $\displaystyle\to A^{j}_{t}\quad$ with
rate $\displaystyle n^{-1}c\left(X^{i}_{t},X^{j}_{t}\right).$

We will refer to $m$ as the _mutation kernel_ , and $c$ as the _competition_
kernel. The parameter $n>0$ is called the _system size_ , in the sense that
that the scaling $n^{-1}c$ guarantees that if the amount of particles in the
system is of the order of $n$, the total rate of created or deleted particles
is of the same order.

Instead of looking at the individual positions of the particles, it is common
to only consider the measure-valued process $\nu_{t}$ determined by the
rescaled empirical measure

$\nu_{t}^{n}:=\frac{1}{n}\sum_{i=1}^{N(t)}\delta_{X_{t}^{i}}.$

Here, $\nu_{t}\in\Gamma:=\mathcal{M}^{+}(\mathcal{T})$ with
$\mathcal{M}^{+}(\mathcal{T})$ the space of finite non-negative measures. The
infinitesimal generator $Q_{n}$ of this process is given for all $F\in
C_{c}(\Gamma)$ by

$\qquad(Q_{n}F)(\nu)=n\int_{\mathcal{T}}\left(F\left(\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x}\right)-F(\nu)\right)\,\kappa^{+}[\nu](\mathrm{d}x)+n\int_{\mathcal{T}}\left(F\left(\nu-\genfrac{}{}{}{2}{1}{n}\delta_{x}\right)-F(\nu)\right)\,\kappa^{-}[\nu](\mathrm{d}x),$

where $\kappa^{\pm}[\nu]\in\Gamma$ are the measure-dependent birth/death-
kernels

$\kappa^{+}[\nu](\mathrm{d}x):=\left(\int_{y\in\mathcal{T}}m(y,x)\nu(\mathrm{d}y)\right)\gamma(\mathrm{d}x),\qquad\kappa^{-}[\nu](\mathrm{d}x):=\left(\int_{y\in\mathcal{T}}c(x,y)\nu(\mathrm{d}y)\right)\nu(\mathrm{d}x).$

The law of the process is now given by the corresponding forward Kolmogorov
equation

($\sf FKE_{n}$)
$\partial_{t}\mathsf{P}_{t}^{n}=Q_{n}^{*}\mathsf{P}_{t}^{n},\qquad\mathsf{P}^{n}_{t}\in\mathcal{P}(\Gamma).$

Depending on the setting, this formulation can be made rigorous in various
ways: for example via an analytical approach on configuration spaces as done
in [FKK09], which in fact models infinite configurations of particles over
$\mathbb{R}^{d}$, or via martingale techniques with $\mathcal{T}$ a closed
subset of $\mathbb{R}^{d}$ and $\gamma=\mathscr{L}^{d}|_{\mathcal{T}}$ (see
[FM04]). Moreover, in the latter, under the assumption of continuous, bounded,
and integrable mutation/competition kernels, it is also shown that the process
converges in the large-population limit $n\to\infty$ to the mean-field
equation (1.1), which can be rewritten as

($\sf MF$)
$\partial_{t}\nu_{t}=\kappa^{+}[\nu_{t}]-\kappa^{-}[\nu_{t}],\qquad\nu_{t}\in\Gamma.$

While different choices of scalings are possible, the mean-field equation
describes the macroscopic properties of the measure-valued process when the
population is large. An alternative way is to study the evolution of the
moments, which form a hierarchy similar to the BBGKY-hierarchy of correlation
functions, and under the so-called Vlasov scaling the first moment or
correlation function converges to ($\sf MF$). For the case of infinite
configurations over $\mathbb{R}^{d}$ this has been established, see [FKK10],
and both propagation of chaos in the Vlasov limit and the sub-Poissonian
property have been established as well [FKKK15].

In this work, we do not consider the measure-valued process itself, but take
the forward Kolmogorov equation ($\sf FKE_{n}$) as a starting point, and show
convergence to the mean-field equation in the sense that
$\mathsf{P}^{n}_{t}\to\delta_{\nu_{t}}$ narrowly on $\mathcal{P}(\Gamma)$
under suitable initial conditions. Throughout we assume the following:

######  Assumption 1.1.

The trait space $\mathcal{T}$ is a compact Polish space, and moreover

$\displaystyle\gamma$ $\displaystyle\in\Gamma\qquad$ (reference measure with
finite mass) $\displaystyle m,c$
$\displaystyle\in\mathcal{B}_{b}^{+}(\mathcal{T}\times\mathcal{T})\qquad$
(bounded rates) $\displaystyle c(x,x)$ $\displaystyle=0\quad$
$\displaystyle\mbox{ for all }x\in\mathcal{T}\qquad$ (no natural death)
$\displaystyle m(y,x)$ $\displaystyle=c(x,y)\quad$ $\displaystyle\mbox{ for
all }x,y\in\mathcal{T}\qquad$ (detailed balance)

Henceforth we equip the space $\Gamma$ with the narrow topology. Moreover, the
assumption of no natural death means that particles can only be deleted due to
competition with other particles. Together with the detailed balance condition
this guarantees that the jump kernel is reversible with respect to an
invariant measure $\Pi_{n}\in\mathcal{P}(\Gamma)$, which is obtained as a
push-forward of the Poisson measure $\pi_{n}$ with

$\mathcal{P}\left(\coprod_{N\geq
1}\mathcal{T}^{N}\right)\ni\pi_{n}:=\frac{1}{e^{n\gamma(\mathcal{T})}-1}\sum_{N=1}^{\infty}\frac{n^{N}}{N!}\gamma^{\otimes
N}.$

This allows us to write the forward Kolmogorov equation as a gradient flow of
the relative entropy with respect to $\Pi_{n}$, and equip it with a
corresponding variational structure, see Theorem 1.6.

In light of similar results in [EFLS16, MM20] for mean-field jump processes on
finite spaces and finite chemical reaction networks, one expects ($\sf
FKE_{n}$) to converge to the following _Liouville equation_

($\sf Li$)
$\partial_{t}\mathsf{P}_{t}+\mathrm{div}_{\Gamma}\left(\mathsf{P}_{t}\,\bigl{(}\kappa^{+}[\nu]-\kappa^{-}[\nu]\bigr{)}\right)=0,\qquad\mathsf{P}_{t}\in\mathcal{P}(\Gamma).$

It is a transport equation that can be interpreted as the lifting of mean-
field dynamics in $\Gamma$ to evolutions in $\mathcal{P}(\Gamma)$, and
describes the evolution of the law of random measures $\nu_{t}$ that all
satisfy ($\sf MF$). In particular, if $\nu_{t}$ a solution of ($\sf MF$) then
$\mathsf{P}_{t}:=\delta_{\nu_{t}}$ is itself a solution of ($\sf Li$).

It turns out that in our general setting this convergence holds as well, as
will be stated in Theorem 1.9. Letting
$V[\nu]=\kappa^{+}[\nu]-\kappa^{-}[\nu]$, we can therefore represent part of
our results in Figure 1.

${\eqref{eq:Forward
Kolmogorov}\quad\partial_{t}\mathsf{P}^{n}_{t}=Q_{n}^{*}\mathsf{P}^{n}_{t}\;}$${\;\eqref{eq:liouv}}$${\partial_{t}\mathsf{P}_{t}+\mathrm{div}_{\Gamma}\,(\mathsf{P}_{t}V[\nu])=0,}$${\mathsf{P}_{t}^{n},\mathsf{P}_{t}\in\mathcal{P}(\Gamma),}$${\;\eqref{eq:mf}}$${\partial_{t}\nu_{t}=V[\nu_{t}],}$${\nu_{t}\in\Gamma:=\mathcal{M}^{+}(\mathcal{T}).}$$\scriptstyle{n\to\infty}$
Figure 1. Convergence in the large-population limit

This convergence is a direct consequence of the convergence of the associated
gradient structures, which we will describe below.

###  1.2. Gradient-flow formulation

Our first main result concerns the variational formulation of the equations
($\sf FKE_{n}$), ($\sf MF$), ($\sf Li$) and their specific gradient structure.
Various gradient-flow formulations exist for jump processes, mean-field jump
processes, and chemical reaction networks [EFLS16, Erb16, KJZ19, MM20,
PRST22].

In these works a common starting point is to describe the relation between
$\rho_{t}$, representing either laws of some process or mean-field limits and
generalized fluxes $j_{t}$ in the form of an abstract continuity equation. For
example, in the case of independent particles following a common jump process
over a graph, $\rho_{t}$ corresponds to the number of particles on a node at
time $t$, and a choice of flux $j_{t}$ can be the so-called _net_ flux
$j_{t}$, which is related to the number of particles going through an edge.

However, we propose a slightly different structure, namely one that tracks the
effective _mass fluxes_ for both creation (arising from mutation) and
annihilation (arising from competition) separately. The use of mass fluxes
instead of usual particle fluxes ensures that in our convergence results as
$n\to\infty$ we have both convergences of laws and fluxes (see Theorem 1.8).

Moreover, separating the effects of creation and annihilation (henceforth
simply referred to as birth and death) instead of their combined contribution
allows us to incorporate more information in our variational formulation. The
downside is that we are forced to work with _positive fluxes_ , while the
framework in the aforementioned examples involves either quadratic or
generalized structures for signed net fluxes. In this sense we are closer to
the variational representations stemming from large deviations, involving so-
called one-way or unidirectional fluxes, see for example [MPR14, PR19, BBBO21,
PS22]. Indeed, our structure is motivated by large deviation theory, as we
will discuss briefly in Appendix A.

In all three cases, i.e. for ($\sf FKE_{n}$), ($\sf MF$) and ($\sf Li$), our
proposed structure is similar to the classical notion of a gradient flow in
the sense that they all satisfy an abstract Energy-Dissipation Balance. Since
we will repeat the same concept three times on different levels and for
different spaces, let us make the general and abstract concepts clear:

######  Formal Definition 1.2.

Given a free energy functional $\mathcal{F}(\rho)$, a dissipation potential
$\mathcal{R}(\rho,j)$, a Fisher information functional $\mathcal{D}(\rho)$,
and a linear operator $B$ with dual $B^{*}$, we consider pairs of curves
$(\rho,j)$ satisfying the continuity equation

($\mathsf{CE}$) $\partial_{t}\rho_{t}+B^{*}j_{t}=0,\quad\mbox{for a.e.\
$t\in[0,T]$,}$

and define the EDP-functional

$\mathcal{I}(\rho,j):=\int_{0}^{T}\mathcal{R}(\rho_{t},j_{t})\,\mathrm{d}t+\mathcal{F}(\rho_{T})-\mathcal{F}(\rho_{0})+\int_{0}^{T}\mathcal{D}(\rho_{t})\,\mathrm{d}t.$

Moreover, a _gradient-flow solution_ is a pair $(\hat{\rho},\hat{\jmath})$
satisfying ($\mathsf{CE}$) with $I(\hat{\rho},\hat{\jmath})=0$.

Throughout we require the non-negativity of $\mathcal{I}$. For a deeper look
at the mathematical basis of this sort of setting, especially for generalized
gradient systems incorporating net fluxes, see [PRST22].

In all three examples the generalized fluxes $j$ consist of two parts: $j^{+}$
and $j^{-}$, corresponding to birth and death. The continuity equations depend
on the setting and are summarized in Table 1, with $\mathcal{M}^{+}_{loc}$ as
the space of non-negative Radon measures.

######  Remark 1.3.

Note that the gradient-flow solution $(\hat{\rho},\hat{\jmath})$ is the null-
minimizer of $\mathcal{I}$, and satisfies the _energy-dissipation balance_

$\mathcal{F}(\hat{\rho}_{T})+\int_{0}^{T}\left(\mathcal{R}(\hat{\rho}_{t},\hat{\jmath}_{t})+\mathcal{D}(\hat{\rho}_{t})\right)\,\mathrm{d}t=\mathcal{F}(\hat{\rho}_{0}).$

Moreover, for small $T\ll 1$ one would expect

$I\approx\mathcal{R}(\hat{\rho},\hat{\jmath})+\langle\hat{\jmath},B\,\partial_{\rho}\mathcal{F}\rangle+\mathcal{D}(\hat{\rho}).$

In light of the generalized gradient-flow framework of [PRST22] and the
relation to minimizing movement schemes, a formal minimization procedure
provides the gradient-flow solution

$\displaystyle\partial_{t}\hat{\rho}+B^{*}\hat{\jmath}$ $\displaystyle=0$
$\displaystyle\hat{\jmath}$
$\displaystyle=(\,\partial_{2}\mathcal{R}^{*})(\hat{\rho},-B\,\partial_{\rho}\mathcal{F}),$

and that along the solution

(1.3)
$\mathcal{D}(\hat{\rho})=\mathcal{R}^{*}(\hat{\rho},-B\,\partial_{\rho}\mathcal{F}).$

where $\mathcal{R}^{*}(\rho,w)$ is the dual of the dissipation potential
$\mathcal{R}$. Finally, note that along the gradient-flow solution the free
energy $\mathcal{F}$ is non-increasing, i.e. $\mathcal{F}$ is a Lyapunov
functional.

These (in)equalities indeed hold in our setting. See also Appendix A, where we
compare the relation to generalized gradient flows for net fluxes, which
follow from the above after a contraction argument, and the connection to the
reversibility of the underlying process.

| $\mathsf{CE}$ | $\rho$ | $j=(j^{+},j^{-})$ | $BF$
---|---|---|---|---
($\sf MF$) | ($\mathscr{CE}$) | $\nu\in\Gamma$ | $(\lambda^{+},\lambda^{-})\in\Gamma^{2}$ | $(F,-F)$
($\sf FKE_{n}$) | ($\mathsf{CE}_{n}$) | $\mathsf{P}\in\mathcal{P}(\Gamma)$ | $(\mathsf{J}^{+},\mathsf{J}^{-})\in\mathcal{M}^{+}_{loc}(\Gamma\times\mathcal{T})^{2}$ |  $(\overline{\nabla}^{n,+}F,\overline{\nabla}^{n,-}F)$ (1.4)
($\sf Li$) | ($\mathsf{CE}_{\infty}$) | $\mathsf{P}\in\mathcal{P}(\Gamma)$ | $(\mathsf{J}^{+},\mathsf{J}^{-})\in\mathcal{M}^{+}_{loc}(\Gamma\times\mathcal{T})^{2}$ |  $(\mathrm{grad}_{\Gamma}F,-\mathrm{grad}_{\Gamma}F)$ (1.5)
Table 1. Continuity equations

Let $H(\mu_{1},\mu_{2})$ be the Hellinger distance, see (2.2), and
$\mathcal{E}\mathrm{nt}(\mu_{1}|\mu_{2})$ the relative entropy of $\mu_{1}$
with respect to $\mu_{2}$ for two (possible infinite) locally finite Borel
measures $\mu_{1},\mu_{2}$:

$\mathcal{E}\mathrm{nt}(\mu_{1}|\mu_{2}):=\left\\{\begin{aligned}
&\int\phi\left(\frac{\mathrm{d}\mu_{1}}{\mathrm{d}\mu_{2}}\right)\mathrm{d}\mu_{2},&&\mbox{if
$\mu_{1}\ll\mu_{2}$,}\\\
&+\infty,\qquad&&\mbox{otherwise,}\end{aligned}\right.$

where

$\phi(s)=s\log s-s+1.$

With the full technical details contained in Theorems 2.7, 3.8 and 4.7, we
then have the following triple of results below,

######  Theorem 1.4 (Mean-field, cf. Theorem 2.7).

Consider triples $(\nu,\lambda^{+},\lambda^{-})$, with
$\nu_{t},\lambda_{t}^{\pm}\in\Gamma$, satisfying the mean-field continuity
equation

($\mathscr{CE}$) $\partial_{t}\nu_{t}=\lambda_{t}^{+}-\lambda^{-}_{t}.$

Define the dissipation potential $\mathcal{R}_{MF}$, free energy
$\mathcal{F}_{MF}$ and Fisher information $\mathcal{D}_{MF}$ as

$\displaystyle\mathcal{R}_{MF}(\nu,\lambda^{+},\lambda^{-})$
$\displaystyle:=\mathcal{E}\mathrm{nt}(\lambda^{+}|\theta_{\nu})+\mathcal{E}\mathrm{nt}(\lambda^{-}|\theta_{\nu}),$
$\displaystyle\mathcal{F}_{MF}(\nu)$
$\displaystyle:=\genfrac{}{}{}{2}{1}{2}\mathcal{E}\mathrm{nt}(\nu|\gamma),$
$\displaystyle\mathcal{D}_{MF}(\nu)$ $\displaystyle:=\left\\{\begin{aligned}
&2H^{2}(\kappa^{+}[\nu],\kappa^{-}[\nu])\qquad&&\mbox{if $\nu\ll\gamma$,}\\\
&+\infty,\qquad&&\mbox{otherwise,}\end{aligned}\right.$

where $\theta_{\nu}$ is the geometric mean of the expected birth and death
fluxes, i.e.

$\theta_{\nu}:=\sqrt{\kappa^{+}[\nu]\kappa^{-}[\nu]}.$

Then the corresponding EDP-functional $I_{MF}$ given by

$\mathcal{I}_{MF}(\nu,\lambda^{+},\lambda^{-}):=\int_{0}^{T}\mathcal{R}_{MF}(\nu_{t},\lambda^{+}_{t},\lambda^{-}_{t})\,\mathrm{d}t+\mathcal{F}_{MF}(\nu_{T})-\mathcal{F}_{MF}(\nu_{0})+\int_{0}^{T}\mathcal{D}_{MF}(\nu_{t})\,\mathrm{d}t,$

is non-negative, and for any $\nu_{0}$ with $\mathcal{F}(\nu_{0})<\infty$ a
unique gradient-flow solution
$(\hat{\nu},\hat{\lambda}^{+},\hat{\lambda}^{-})$ exists, with $\hat{\nu}_{t}$
equal to the unique strong solution to ($\sf MF$) and
$\hat{\lambda}_{t}^{\pm}=\kappa^{\pm}[\hat{\nu}_{t}]$ for almost every
$t\in[0,T]$.

As mentioned, although treating birth and death separately provides us with
additional information, this prohibits the use of some of the previous works
for gradient structures because of the positivity of the fluxes. However,
there is still a strong connection to the variational formulations for jump
processes arising from the large deviations of fluxes as seen in [PR19] and
[BBBO21], see for example Appendix A on the equivalence of the EDP-functional
to the expected rate functional.

######  Remark 1.5.

It is straightforward to verify that if $\mathrm{d}\nu=u\mathrm{d}\gamma$

$\displaystyle\mathcal{R}^{*}_{MF}(\nu,\partial_{\nu}\mathcal{F}_{MF},-\partial_{\nu}\mathcal{F}_{MF})$
$\displaystyle=\int_{\mathcal{T}^{2}}1_{u(x)>0}c(x,y)\left(\sqrt{u(x)}-1\right)^{2}\gamma(\mathrm{d}x)\nu(\mathrm{d}y),$
$\displaystyle\mathcal{D}_{MF}(\nu)$
$\displaystyle=\int_{\mathcal{T}^{2}}c(x,y)\left(\sqrt{u(x)}-1\right)^{2}\gamma(\mathrm{d}x)\nu(\mathrm{d}y),$

and hence it is not directly clear that the relation (1.3) holds. However, as
will be shown for Theorem 2.7, at least along the solution $\hat{\nu}_{t}$ the
equivalence holds for a.e. $t\in[0,T]$.

######  Theorem 1.6 (Forward Kolmogorov, cf. Theorem 3.8).

Consider triples $(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})$, with
$\mathsf{P}_{t}\in\mathcal{P}(\Gamma)$ and
$J_{t}^{\pm}\in\mathcal{M}_{loc}(\Gamma\times\mathcal{T})$, satisfying the
continuity equation

($\mathsf{CE}_{n}$) $\displaystyle\langle
F,\partial_{t}\mathsf{P}_{t}\rangle=\langle\overline{\nabla}^{n,+}F,\mathsf{J}_{t}^{+}\rangle+\langle\overline{\nabla}^{n,-}F,\mathsf{J}_{t}^{-}\rangle,\qquad\forall
F\in C_{c}(\Gamma),$

where

(1.4)
$(\overline{\nabla}^{n,\pm}F)(\nu,x):=n\left(F(\nu\pm\genfrac{}{}{}{2}{1}{n}\delta_{x})-F(\nu)\right).$

Define the $n$-dependent Fisher information $\mathcal{D}_{n}$ as stated in
Definition 3.4, free energy

$\mathcal{F}_{n}(\mathsf{P}):=\frac{1}{2n}\mathcal{E}\mathrm{nt}(\mathsf{P}|\Pi_{n}),$

and dissipation potential

$\displaystyle\mathcal{R}_{n}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})$
$\displaystyle:=\mathcal{E}\mathrm{nt}(\mathsf{J}^{+}|\Theta_{\mathsf{P}}^{n,+})+\mathcal{E}\mathrm{nt}(\mathsf{J}^{-}|\Theta_{\mathsf{P}}^{n,-}),$

where, with a little abuse of notation (see (3.14)),

$\Theta_{\mathsf{P}}^{n,\pm}(\nu,x):=\sqrt{\Big{(}\mathsf{P}(\nu)\kappa^{\pm}[\nu]\Big{)}\left(\mathsf{P}(\nu\pm\genfrac{}{}{}{2}{1}{n}\delta_{x})\kappa^{\mp}[\nu\mp\genfrac{}{}{}{2}{1}{n}\delta_{x}]\right)}.$

Then the corresponding EDP-functional $I_{n}$ given by

$\mathcal{I}_{n}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-}):=\int_{0}^{T}\mathcal{R}_{n}(\mathsf{P}_{t},\mathsf{J}^{+}_{t},\mathsf{J}^{-}_{t})\,\mathrm{d}t+\mathcal{F}_{n}(\mathsf{P}_{T})-\mathcal{F}_{n}(\mathsf{P}_{0})+\int_{0}^{T}\mathcal{D}_{n}(\mathsf{P}_{t})\,\mathrm{d}t,$

is non-negative, and for any $\mathsf{P}_{0}$ with
$\mathcal{F}_{n}(\mathsf{P}_{0})<\infty$ a unique gradient-flow solution
$(\hat{\mathsf{P}},\hat{\mathsf{J}}^{\pm})$ exists, with
$\hat{\mathsf{P}}_{t}$ equal to a weak solution to ($\sf FKE_{n}$) and
$\hat{\mathsf{J}}_{t}^{\pm}=\hat{\mathsf{P}}_{t}\kappa_{\nu}^{\pm}$ for almost
every $t\in[0,T]$.

Similar to the mean-field case, the dissipation potential consists of relative
entropies with respect to geometric averages, now of forward and backward
rates along a transition $\nu\to\nu\pm\genfrac{}{}{}{2}{1}{n}\delta_{x}$.
Moreover, note that in contrast to the framework of [PRST22], we employ fluxes
$\mathsf{J}^{\pm}$ that are not finite measures. This is due to the
unboundedness of $\kappa_{\nu}$ as the mass of $\nu$ grows, which implies that
the underlying jump kernel over $\Gamma$ is itself unbounded as well, see
Section 3.

For the Liouville equation, let us define $\mathrm{Cyl}_{c}(\Gamma)$ as the
space of compactly supported smooth cylinder functions of the form

$F(\nu)=g\left(\langle 1,\nu\rangle,\langle f_{1},\nu\rangle,\dots,\langle
f_{m},\nu\rangle\right),\qquad g\in
C^{\infty}_{c}(\mathbb{R}^{m}),\;m\in\mathbb{N},$

where $f_{1},\dots,f_{m}\in C_{b}(\mathcal{T})$, and $\mathrm{grad}_{\Gamma}$
is the distributional gradient defined by

(1.5) $\mathrm{grad}_{\Gamma}\,F(\nu,x)=(\nabla g)\left(\langle
1,\nu\rangle,\langle f_{1},\nu\rangle,\dots,\langle
f_{m},\nu\rangle\right)\cdot(1,f_{1}(x),\dots,f_{m}(x))^{\top}.$

######  Theorem 1.7 (Liouville, cf. Theorem 4.7).

Consider triples $(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})$, with
$\mathsf{P}_{t}\in\mathcal{P}(\Gamma)$,
$\mathsf{J}^{\pm}\in\mathcal{M}_{loc}(\Gamma\times\mathcal{T})$, satisfying
the continuity equation

($\mathsf{CE}_{\infty}$) $\langle
F,\partial_{t}\mathsf{P}_{t}\rangle=\langle\mathsf{grad}_{\Gamma}F,\mathsf{J}^{+}_{t}\rangle-\langle\mathsf{grad}_{\Gamma}F,\mathsf{J}^{-}_{t}\rangle,\quad\forall
F\in\mathrm{Cyl}_{c}(\Gamma).$

Define the Fisher information $\mathcal{D}_{\infty}$ as stated in Definition
4.4, free energy

$\displaystyle\mathcal{F}_{\infty}(\mathsf{P})$
$\displaystyle:=\frac{1}{2}\int_{\Gamma}\mathcal{E}\mathrm{nt}(\nu|\gamma)\,\mathrm{d}\mathsf{P},$

and dissipation potential

$\mathcal{R}_{\infty}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-}):=\mathcal{E}\mathrm{nt}(\mathsf{J}^{+}|\Theta_{\mathsf{P}}^{\infty})+\mathcal{E}\mathrm{nt}(\mathsf{J}^{-}|\Theta_{\mathsf{P}}^{\infty}),\qquad\Theta_{\mathsf{P}}^{\infty}(\mathrm{d}\nu,\mathrm{d}x):=\theta_{\nu}(\mathrm{d}x)\mathsf{P}(\mathrm{d}\nu).$

Then the corresponding EDP-functional $I_{\infty}$ given by

$\mathcal{I}_{\infty}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-}):=\int_{0}^{T}\mathcal{R}_{\infty}(\mathsf{P}_{t},\mathsf{J}^{+}_{t},\mathsf{J}^{-}_{t})\,\mathrm{d}t+\mathcal{F}_{\infty}(\mathsf{P}_{T})-\mathcal{F}_{\infty}(\mathsf{P}_{0})+\int_{0}^{T}\mathcal{D}_{\infty}(\mathsf{P}_{t})\,\mathrm{d}t,$

is non-negative, and for any $\mathsf{P}_{0}$ with
$\mathcal{F}_{\infty}(\nu_{0})<\infty$ a unique gradient-flow solution
$(\hat{\mathsf{P}},\hat{\mathsf{J}}^{\pm})$ exists, with
$\hat{\mathsf{P}}_{t}$ a weak solution to ($\sf Li$) and
$\hat{\mathsf{J}}_{t}^{\pm}=\hat{\mathsf{P}}_{t}\kappa_{\nu}^{\pm}$ for almost
every $t\in[0,T]$.

Finally, for any $(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})$ such that
$I_{\infty}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})<\infty$, there exists
(with a little abuse of notation) a Borel probability measure $\Omega$ over
curves satisfying the mean-field continuity equation ($\mathscr{CE}$) such
that for all $t$ the time marginals $(e_{t})_{\\#}\Omega$ are equal to
$\mathsf{P}_{t}$, and

(1.6)
$\mathcal{I}_{\infty}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-}):=\int\mathcal{I}_{MF}(\nu,\lambda^{+},\lambda^{-})\,\mathrm{d}\Omega.$

The statement of (1.6) is the aforementioned superposition principle, which is
a modified version of the superposition principle [AT14] in metric measure
spaces, and the ones used in [EFLS16], [Erb16]. It allows one to essentially
jump back and forth between the Liouville equation and the mean-field
dynamics, and in particular, provides us with the non-negativity of
$\mathcal{I}_{\infty}$ and uniqueness of gradient-flow solutions.

###  1.3. Convergence results

Our final and most important result is that the above gradient structures
converge in the sense of _EDP-convergence_ (e.g. see [LMPR17, PS22]), a
generalization of the evolutionary $\Gamma$-convergence approach stated by
[SS04, Ser11] and expanded on in [Mie16], which implies convergence of the
gradient-flow solutions and their free energies.

We say that a sequence
$(\mathsf{P}^{n},\mathsf{J}^{n,+},\mathsf{J}^{n,-})\in\mathsf{CE}_{n}$
converges to some $(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})\in\mathsf{CE}$
if for all $t\in[0,T]$ the probability measures $\mathsf{P}_{t}^{n}$ converge
narrowly to $\mathsf{P}_{t}$ in $\mathcal{P}(\Gamma)$, and
$\mathsf{J}^{n,\pm}_{t}(\mathrm{d}\nu,\mathrm{d}x)\,\mathrm{d}t$ converge
vaguely to $\mathsf{J}^{\pm}_{t}(\mathrm{d}\nu,\mathrm{d}x)\,\mathrm{d}t$ in
$\mathcal{M}_{loc}([0,T]\times\Gamma\times\mathcal{T})$. Again postponing
technicalities, see Theorem 5.1, we have the following lower semi-continuity
and compactness result:

######  Theorem 1.8 (cf. Theorem 5.1).

The sequence of free energies $\mathcal{F}_{n}$ $\varGamma$-converges to
$\mathcal{F}_{\infty}$.

Moreover, the sequence of Fisher-information functionals and dissipation
potentials are all sequentially lower semicontinuous. In particular, for any
sequence
$(\mathsf{P}^{n},\mathsf{J}^{n,+},\mathsf{J}^{n,-})\in\mathsf{CE}_{n}$
converging to a
$(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})\in\mathsf{CE}_{\infty}$ such that
$\mathcal{F}_{n}(\mathsf{P}_{0}^{n})\to\mathcal{F}_{\infty}(\mathsf{P}_{0})$
as well, we have

$\liminf_{n\to\infty}\mathcal{I}_{n}(\mathsf{P}^{n},\mathsf{J}^{n,+},\mathsf{J}^{n,-})\geq\mathcal{I}_{\infty}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-}).$

Finally, for any sequence
$(\mathsf{P}^{n},\mathsf{J}^{n,+},\mathsf{J}^{n,-})\in\mathsf{CE}_{n}$ such
that

$\displaystyle\limsup_{n\to\infty}\mathcal{F}_{n}(\mathsf{P}_{0}^{n})<\infty,$
$\displaystyle\limsup_{n\to\infty}\mathcal{I}_{n}(\mathsf{P}^{n},\mathsf{J}^{n,+},\mathsf{J}^{n,-})<\infty,$

there exists a subsequence converging to some
$(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})\in\mathsf{CE}_{\infty}$.

Here the notion of EDP-convergence or evolutionary $\varGamma$-convergence
(where the $\varGamma$ is not to be confused with our space of positive
measures $\Gamma$) relates to the $\varGamma$-convergence of the free energies
$\mathcal{F}_{n}$ and suitable liminf-estimates for the dissipation potentials
and Fisher-information functionals (or local slopes in a metric setting).

In certain applications or for certain notions of convergence (e.g. see
[MMP21]) one also establishes $\varGamma$-convergence for the total
dissipation $\mathcal{R}_{n}+\mathcal{D}_{n}$ when written as functionals over
$C([0,T];\mathcal{P}(\Gamma))$. Moreover, $\varGamma$-convergence of the
functionals $\mathcal{I}_{n}$ over such path-spaces are related to the large
deviations of the underlying process [Kra19], as we briefly discuss in
Appendix A. In our framework this would require that for every
$(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})\in\mathsf{CE}_{\infty}$, we can
find a sequence
$(\mathsf{P}^{n},\mathsf{J}^{n,+},\mathsf{J}^{n,-})\in\mathsf{CE}_{n}$ that
converges to $(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})$ and satisfies the
limsup-estimate

$\limsup_{n\to\infty}\mathcal{I}_{n}(\mathsf{P}^{n},\mathsf{J}^{n,+},\mathsf{J}^{n,-})\leq\mathcal{I}_{\infty}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-}).$

However, in this paper we restrict ourselves only to the liminf-estimates,
which is sufficient to obtain convergence of the solutions, an approach also
taken in [EFLS16, Erb16, MM20]. Namely, by a lower semicontinuity and
compactness argument, Theorem 1.8 implies the convergence of both the
solutions and the free energies $\mathcal{F}_{n}$, if the initial data are
well prepared.

######  Theorem 1.9 (cf. Theorem 5.3).

Suppose that $\mathsf{P}_{0}^{n}\to\mathsf{P}$ with
$\mathcal{F}_{n}(\mathsf{P}_{0}^{n})\to\mathcal{F}_{\infty}(\mathsf{P}_{0})$
as well. Then for the sequence $\hat{\mathsf{P}}^{n}$ of gradient-flow
solutions to ($\sf FKE_{n}$), and $\hat{\mathsf{P}}$ the gradient-flow
solution to ($\sf Li$), we have that for all $t\in[0,T]$

$\hat{\mathsf{P}}_{t}^{n}\to\hat{\mathsf{P}}_{t}\mbox{\; narrowly,\quad
and}\quad\lim_{n\to\infty}\mathcal{F}_{n}(\hat{\mathsf{P}}^{n}_{t})=\mathcal{F}_{\infty}(\hat{\mathsf{P}}_{t}).$

In particular, if $\mathsf{P}_{0}=\delta_{\hat{\nu}_{0}}$ and $\hat{\nu}_{t}$
is the solution to the mean-field problem ($\sf MF$), then for all $t\in[0,T]$

$\displaystyle\hat{\mathsf{P}}_{t}^{n}\to\delta_{\hat{\nu}_{t}}\mbox{\;
narrowly,\quad
and}\quad\lim_{n\to\infty}\frac{1}{n}\mathcal{E}\mathrm{nt}(\hat{\mathsf{P}}^{n}_{t}|\Pi_{n})=\mathcal{E}\mathrm{nt}(\hat{\nu}_{t}|\gamma).$

The second half of Theorem 1.9, on the concentration around mean-field
solutions and convergence of entropies, follows directly from the definition
of $\mathcal{F}_{\infty}$ and uniqueness.

For interacting particle systems where the number of particles is fixed at
$n\in\mathbb{N}$ the narrow convergence
$\hat{\mathsf{P}}_{t}^{n}\to\delta_{\hat{\nu}_{t}}$ is equivalent to
propagation of chaos in the sense of Snitzman [Szn91], and would imply narrow
convergence of the $k$-particle marginals at time $t$ to $\nu_{t}^{\otimes
k}$. However, in our setting, this implies convergence of the $k$-correlation
functions, see [BGSRS20].

Moreover, the convergence of the free energies $\mathcal{F}_{n}$ implies the
stronger notion of entropic propagation of chaos if the initial condition is
sufficiently regular.

######  Theorem 1.10 (cf. Theorem 5.4).

Suppose that $\mathsf{P}^{n}_{0}\to\delta_{\hat{\nu}_{0}}$ with
$C^{-1}\leq\mathrm{d}\hat{\nu}_{0}/\mathrm{d}\gamma\leq C$ for some $C>0$. If

$\lim_{n\to\infty}\frac{1}{n}\mathcal{E}\mathrm{nt}(\hat{\mathsf{P}}_{0}^{n}|\Pi_{n,\hat{\nu}_{0}})=0,$

then

$\lim_{n\to\infty}\frac{1}{n}\mathcal{E}\mathrm{nt}(\hat{\mathsf{P}}_{t}^{n}|\Pi_{n,\hat{\nu}_{t}})=0,\qquad\mbox{
for all }t\geq 0,$

where $\Pi_{n,\nu}\in\mathcal{P}(\Gamma)$ stems from the Poisson measure
$\pi_{n,\nu}$with intensity measure $\nu$, i.e.

$\pi_{n,\nu}:=\frac{1}{e^{n\nu(\mathcal{T})}-1}\sum_{N=1}^{\infty}\frac{n^{N}}{N!}\nu^{\otimes
N}.$

To the authors’ knowledge, this is the first entropic propagation of chaos
result for bounded competition kernels over compact Polish spaces, under the
assumption of detailed balance.

#### Comments

We have given an overview of the generalized gradient structures that we
introduced for the forward Kolmogorov equation of our underlying interacting
particle system and eluded to how this sequence of structures converges to a
gradient structure induced by the mean-field limit. Throughout, we assumed
bounded measurable rates $m,c$ over a compact Polish space $\mathcal{T}$
satisfying the detailed balance condition $m(x,y)=c(x,y)$ and $c(x,x)=0$ for
all $x,y\in\mathcal{T}$, and we would like to briefly touch on possible
relaxations of these assumptions.

First, for the limit inferior in Theorem 5.1, there is a technical issue
concerning the possible non-continuity of the competition kernel $c$, which we
resolve by an approximation argument from large deviation theory [HHMT20], see
Appendix C. This argument can be straightforwardly extended to unbounded rates
$m$ and $c$ under certain exponential integrability estimates with respect to
the reference measure $\gamma$. However, the uniqueness of solutions and well-
posed of variational formulations would be less clear.

Moreover, it should be noted that although for brevity and clarity we chose
$\mathcal{T}$ to be compact, many of the listed results carry over to the case
of $\mathcal{T}$ Polish with finite $\gamma$, under suitable choices of
topologies and by bootstrapping from the tightness of $\gamma$. For
$\sigma$-finite $\gamma$, this is not necessarily the case and would depend
strongly on newly constructed estimates on the propagation of tightness.

A more fundamental restriction is the detailed balance assumption, which is
necessary to phrase the variational structures in terms of generalized
gradient systems and the evolution in terms of a gradient flow. However, there
exist possible extensions and decompositions of variational structures for
jump processes that do not assume detailed balance or even complex balance,
see for example [KJZ18] for an overview. Therefore, in future work, the
authors plan to generalize the variational methods outlined here to more
general evolutions.

#### Acknowledgments

The authors acknowledge support from NWO Vidi grant 016.Vidi.189.102 on
”Dynamical-Variational Transport Costs and Application to Variational
Evolution”.

###  1.4. Notation

Below we collect some of the notation used throughout this paper.

$\mathcal{T}$ | trait space, Assumption 1.1
---|---
$m,c$ | mutation/competition kernel, Assumption 1.1
$\gamma$ | reference measure, Assumption 1.1
$n$ | system size, Assumption 1.1
$\mathcal{E}\mathrm{nt}$ | relative entropy (2.3)
$H$ | Hellinger distance (2.2)
$\Psi,\Psi^{*}$ | dual pair (2.6),(2.5)
$\mathcal{M}^{+}$ | space of finite non-negative measures, with narrow topology
$\mathcal{M}^{+}_{loc}$ | space of non-negative Radon measures, with vague topology
$\Gamma:=\mathcal{M}^{+}(\mathcal{T})$ | state space of measure-valued process
$\Gamma_{n}\subset\Gamma$ | space of positive atomic measures with common mass $\genfrac{}{}{}{2}{1}{n}$ (3.3)
$\kappa^{\pm}_{\nu}=\kappa^{\pm}[\nu]$ | measure-dependent birth/death kernels (2.1)
$\theta_{\nu}$ | geometric mean of $\kappa^{+}_{\nu}$ and $\kappa^{-}_{\nu}$, Definition 2.4
$\mathscr{CE}$ | continuity equation for mean-field $\eqref{eq:mf1}$, Definition 2.1
$\mathcal{R}_{MF},\mathcal{F}_{MF},\mathcal{D}_{MF}$ | ingredients of EDP-functional $\mathcal{I}_{MF}$ for $\eqref{eq:mf1}$ , Definition 2.4
$Q_{n},Q_{n}^{*}$ | generator and dual generator (3.1) of ($\sf FKE_{n}$)
$\bar{\kappa}_{n}$ | jump kernel (3.4) corresponding to ($\sf FKE_{n}$)
$L_{n}$ | rescaled empirical measure map (3.2)
$\pi_{n},\Pi_{n}$ | invariant measures for particle system (3.5) and measure-valued process (3.6)
$\mathsf{T}^{n,\pm}$ | creation/annihilation mappings (3.8)
$\overline{\nabla}^{n,\pm},\mathrm{div}^{n,\pm}$ | discrete $\Gamma_{n}$-gradient (3.9) and divergence (3.10)
$\vartheta_{\mathsf{P}}^{\pm}$ | expected fluxes (3.12)
$\Theta_{\mathsf{P}}^{n,\pm}$ | geometric average $\vartheta_{\mathsf{P}}^{\pm}$ along transition, Definition (3.1)
$\mathsf{CE}_{n}$ | continuity equation for ($\sf FKE_{n}$), Definition (3.1)
$\mathcal{R}_{n},\mathcal{F}_{n},\mathcal{D}_{n}$ | ingredients of EDP-functional $\mathcal{I}_{n}$ for ($\sf FKE_{n}$), Definition 3.4
$d_{TV,w},W$ | weighted total variation metric (3.18)/transportation metric (4.11) over $\mathcal{P}(\Gamma)$
$\mathsf{CE}_{\infty}$ | continuity equation for $\eqref{eq:liouv2}$, Definition 4.3
$\mathcal{R}_{\infty},\mathcal{F}_{\infty},\mathcal{D}_{\infty}$ | ingredients of EDP-functional $\mathcal{I}_{\infty}$ for ($\sf Li$), Definition 4.4

##  2\. Mean-field system

In this section, we will discuss the gradient-flow formulation of the mean-
field equation under the detailed balance condition. Let us first make precise
the context of Theorem 1.4, and embed it within the more general statement of
Theorem 2.7 below.

Recall that the trait space $\mathcal{T}$ is a compact Polish space, and
$\Gamma:=\mathcal{M}^{+}(\mathcal{T})$ is the space of finite non-negative
measures over $\mathcal{T}$ equipped with the narrow topology. Fix a reference
measure $\gamma\in\Gamma$, and rates $m,c$ satisfying Assumption 1.1, i.e.
$m,c\in\mathcal{B}_{b}(\mathcal{T}\times\mathcal{T})$ with $m(x,y)=c(y,x)$ for
all $x,y\in\mathcal{T}$, and $c(x,x)=0$ for all $x\in\mathcal{T}$. The mean-
field equation then reads

($\sf MF$) $\partial_{t}\nu_{t}=\kappa^{+}[\nu_{t}]-\kappa^{-}[\nu_{t}],$

with measure-dependent birth and death kernels $\kappa^{\pm}:\Gamma\to\Gamma$
given by

(2.1)
$\kappa^{+}[\nu](\mathrm{d}x):=\int_{y\in\mathcal{T}}c(x,y)\gamma(\mathrm{d}x)\nu(\mathrm{d}y),\qquad\kappa^{-}[\nu](\mathrm{d}x):=\int_{y\in\mathcal{T}}c(x,y)\nu(\mathrm{d}x)\nu(\mathrm{d}y).$

Routinely, we will also adopt the shorthand notation
$\kappa_{\nu}^{\pm}:=\kappa^{\pm}[\nu]$. Now, setting
$c_{\nu}(x):=\int_{\mathcal{T}}c(x,y)\,\nu(\mathrm{d}y)$, it is clear that
that $\kappa^{+}_{\nu}=c_{\nu}\gamma$, $\kappa^{-}_{\nu}=c_{\nu}\nu$, and the
dynamics simplify to

$\partial_{t}\nu(\mathrm{d}x)=c_{\nu}(x)(\gamma(\mathrm{d}x)-\nu(\mathrm{d}x)).$

Strong solutions to ($\sf MF$) in either total variation or appropriate
$L^{1}$ spaces follow straightforwardly via classical methods, see Section
2.2.

The total variation norm $\|\cdot\|_{TV}$ on $\mathcal{M}(\mathcal{T})$ is
defined as

$\|\mu\|_{TV}:=\sup\left\\{\int_{\mathcal{T}}f\,\mathrm{d}\mu:f\in\mathcal{B}_{b}(\mathcal{T}),\,\|f\|_{\infty}\leq
1\right\\},\qquad\mu\in\mathcal{M}(\mathcal{T}),$

and the squared Hellinger distance $H^{2}$ is given by

(2.2)
$H^{2}(\nu,\eta):=\frac{1}{2}\int_{\mathcal{T}}\left(\sqrt{\frac{\mathrm{d}\nu}{\mathrm{d}\sigma}}-\sqrt{\frac{\mathrm{d}\mu}{\mathrm{d}\sigma}}\right)^{2}\mathrm{d}\sigma,$

with $\sigma$ a measure dominating both $\mu$ and $\nu$. Note that the
definition (2.2) is independent of the choice for the dominating measure
$\sigma$, and $\sigma=\nu+\eta$ is always admissible.

Moreover, recall the entropy function $\phi:\mathbb{R}_{\geq
0}\to\mathbb{R}_{\geq 0}$ and its Legendre dual
$\phi^{*}:\mathbb{R}\to\mathbb{R}$ by

$\phi(s):=s\log s-s+1,\qquad\phi^{*}(z):=e^{z}-1,$

and the relative entropy of $\nu$ with respect to $\mu$ as

(2.3) $\mathcal{E}\mathrm{nt}(\nu|\mu):=\left\\{\begin{aligned}
&\int_{\mathcal{T}}\phi\left(\frac{\mathrm{d}\nu}{\mathrm{d}\mu}\right)\mathrm{d}\mu,&&\quad\mbox{if
$\nu\ll\mu$},\\\ &+\infty&&\quad\mbox{otherwise.}\end{aligned}\right.$

We will consider curves satisfying the continuity equation

($\mathscr{CE}$) $\partial_{t}\nu_{t}=\lambda_{t}^{+}-\lambda_{t}^{-},$

in an appropriately weak sense.

######  Definition 2.1 (Mean-field continuity equation).

A triple $(\nu,\lambda^{+},\lambda^{-})$ satisfies the mean-field continuity
equation $\mathscr{CE}$ if

  1. (1)

the curve $[0,T]\ni t\mapsto\nu_{t}\in\Gamma$ is absolutely continuous with
respect to $\|\cdot\|_{TV}$,

  2. (2)

the Borel family $(\lambda_{t}^{\pm})_{t\in[0,T]}\subset\Gamma$ satisfies
$\int_{0}^{T}\|\lambda_{t}^{\pm}\|_{TV}\,\mathrm{d}t<\infty$,

  3. (3)

for every $s,t\in[0,T]$ and all $f\in C_{b}(\mathcal{T})$

$\int_{\mathcal{T}}f\mathrm{d}\nu_{t}-\int_{\mathcal{T}}f\mathrm{d}\nu_{s}=\int_{s}^{t}\left(\int_{\mathcal{T}}f\mathrm{d}\lambda_{r}^{+}-\int_{\mathcal{T}}f\mathrm{d}\lambda_{r}^{-}\right)\,\mathrm{d}r,\qquad\mbox{for
all $s,t$ with $0\leq s,t\leq T$.}$

We will refer to $\lambda^{\mathrm{net}}=\lambda^{+}-\lambda^{-}$ as the _net
flux_.

######  Remark 2.2.

When seen as approximations of particle systems the birth/death fluxes
$\lambda^{\pm}_{t}$ represent the _observed_ amount of mass being
created/annihilated around a certain point, and $\nu_{t}$ represents the
density of the particles, while $\kappa_{\nu}^{\pm}$ correspond to the
_expected_ birth and death fluxes of the BPDL model.

######  Remark 2.3 (Time-regularity).

As we will see in Lemma 2.11, if there exist a common dominating measure for
$\\{\nu_{t},\lambda^{+}_{t},\lambda_{t}^{-}\\}_{t\in[0,T]}$ then the
continuity equation holds in a strong sense: $\nu_{t}$ is an a.e.
differentiable map from $[0,T]$ to $(\Gamma,\|\cdot\|_{TV})$ and

$\partial_{t}\nu_{t}=\lambda_{t}^{+}-\lambda_{t}^{-},\qquad\mbox{for a.e.\
$t\in[0,T]$}.$

######  Definition 2.4.

Let $\theta_{\nu}$ be the geometric average of $\kappa_{\nu}^{+}$ and
$\kappa_{\nu}^{-}$, i.e.

$\mathrm{d}\theta_{\nu}:=\sqrt{\frac{\mathrm{d}\kappa_{\nu}^{+}}{\mathrm{d}\sigma}\frac{\mathrm{d}\kappa_{\nu}^{-}}{\mathrm{d}\sigma}}\mathrm{d}\sigma,$

for any dominating measure $\sigma$. We define the following objects:

  * •

The dissipation potential $\mathcal{R}_{MF}:\Gamma^{3}\to[0,+\infty]$,

$\mathcal{R}_{MF}(\nu,\lambda^{+},\lambda^{-}):=\mathcal{E}\mathrm{nt}(\lambda^{+}|\theta_{\nu})+\mathcal{E}\mathrm{nt}(\lambda^{-}|\theta_{\nu}),$

and the dual dissipation potential
$\mathcal{R}^{*}_{MF}:\Gamma\times\mathcal{B}_{b}(\mathcal{X})^{2}\to[0,+\infty]$,

$\mathcal{R}^{*}_{MF}(\nu,w^{+},w^{-}):=\int_{\mathcal{T}}(e^{w^{+}}-1)\,\mathrm{d}\theta_{\nu}+\int_{\mathcal{T}}(e^{w^{-}}-1)\,\mathrm{d}\theta_{\nu}.$

  * •

The free energy $\mathcal{F}_{MF}:\Gamma\to[0,+\infty]$,

$\mathcal{F}_{MF}(\nu):=\genfrac{}{}{}{2}{1}{2}\mathcal{E}\mathrm{nt}(\nu|\gamma),$

and Fisher information $\mathcal{D}_{MF}:\Gamma\to[0,+\infty]$,

$\mathcal{D}_{MF}(\nu):=\left\\{\begin{aligned}
&2H^{2}(\kappa_{\nu}^{+},\kappa_{\nu}^{-}),&&\qquad\mbox{if
$\nu\ll\gamma,$}\\\ &+\infty,&&\qquad\mbox{otherwise.}\end{aligned}\right.$

  * •

The _EDP-functional_ $\mathcal{I}_{MF}:\mathscr{CE}\to[0,+\infty]$ for all
curves with $\mathcal{F}_{MF}(\nu_{0})<\infty$

(2.4)
$\mathcal{I}_{MF}(\nu,\lambda^{+},\lambda^{-}):=\int_{0}^{T}\mathcal{R}_{MF}(\nu_{t},\lambda_{t},\lambda^{-}_{t})\,\mathrm{d}t+\mathcal{F}(\nu_{T})-\mathcal{F}(\nu_{0})+\int_{0}^{T}\mathcal{D}_{MF}(\nu_{t})\,\mathrm{d}t.$

######  Remark 2.5.

Since $\theta_{\nu}(\mathcal{T})<\infty$ by Lemma 2.9 all objects above are
well-defined, and it is straightforward to verify via the dual representation
of the entropy that $\mathcal{R}_{MF},\mathcal{R}^{*}_{MF}$ are truly dual
objects in the sense that

$\mathcal{R}(\nu,\lambda^{+},\lambda^{-}):=\sup_{w^{\pm}\in\mathcal{B}_{b}(\mathcal{T})}\left\\{\int_{\mathcal{T}}w^{+}\mathrm{d}\lambda^{+}+\int_{\mathcal{T}}w^{-}\mathrm{d}\lambda^{-}-\mathcal{R}^{*}(\nu,w^{+},w^{-})\right\\},$

and vice versa.

######  Remark 2.6.

If $\nu\ll\gamma$ with $\mathrm{d}\nu=u\mathrm{d}\gamma$, note that
$\mathrm{d}\theta_{\nu}=c_{\nu}\sqrt{u}\,\mathrm{d}\gamma$, and that the
Fisher information simplifies to

$\mathcal{D}_{MF}(\nu)=\int_{\mathcal{T}}c_{\nu}\left(\sqrt{u}-1\right)^{2}\mathrm{d}\gamma.$

We are now able to fully state the variational characterization of strong
solutions to the mean-field equation ($\sf MF$).

######  Theorem 2.7.

For any $(\nu,\lambda^{+},\lambda^{-})\in\mathscr{CE}$ with
$\mathcal{F}_{MF}(\nu_{0})<\infty$, we have
$\mathcal{I}_{MF}(\nu,\lambda^{+},\lambda^{-})\geq 0$ and

$\mathcal{I}_{MF}(\nu,\lambda^{+},\lambda^{-})=0\iff\left\\{\begin{aligned}
\quad&\mbox{$\nu_{t}$ is the unique strong solution to \eqref{eq:mf}},\quad\\\
\quad\lambda^{\pm}_{t}&=(\kappa^{\pm}_{\nu_{t}})\quad\mbox{for a.e.\
$t\in[0,T]$.}\quad\end{aligned}\right.$

Moreover, whenever $\mathcal{F}_{MF}(\nu_{0})<\infty$ and
$\mathcal{I}_{MF}(\nu,\lambda^{+},\lambda^{-})<\infty$ the chain rule for
$\mathcal{F}_{MF}$ holds: $\mathcal{F}_{MF}(\nu_{t})$ is absolutely continuous
and

$\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{F}_{MF}(\nu_{t})=\genfrac{}{}{}{2}{1}{2}\int_{\mathcal{T}}\log\frac{\mathrm{d}\nu_{t}}{\mathrm{d}\gamma}\,\mathrm{d}(\lambda^{+}_{t}-\lambda^{-}_{t})\,\mathrm{d}t,\qquad\mbox{for
a.e.\ $t\in[0,T]$}.$

The proof of Theorem 2.7 is postponed to Section 2.3, where we establish the
main technical ingredient, namely the chain rule for the entropy functional.

######  Remark 2.8.

The non-negativity of $\mathcal{I}_{MF}$ and the fact that null-minimizers are
solutions to ($\sf MF$) is related to the formal equivalence

$\mathcal{I}_{MF}(\nu,\lambda^{+},\lambda^{-})=\int_{0}^{T}\mathcal{L}(\nu_{t},\lambda_{t}^{+},\lambda^{-}_{t})\,\mathrm{d}t,$

where $\mathcal{L}$ is the so-called _Lagrangian_ given by

$\mathcal{L}(\nu,\lambda^{+},\lambda^{-}):=\mathcal{E}\mathrm{nt}(\lambda^{+}|\kappa_{\nu}^{+})+\mathcal{E}\mathrm{nt}(\lambda^{+}|\kappa_{\nu}^{-}).$

Note that $\mathcal{L}$ is non-negative and zero if only if
$\lambda^{\pm}=\kappa_{\nu}^{\pm}$. Although we do not prove the full
equivalence in this work, it does play a role in the intuition and motivation
behind the EDP-functional $\mathcal{I}_{MF}$ with the Lagrangian $\mathcal{L}$
stemming from a large deviation perspective, as seen in Appendix A.

###  2.1. A priori estimates

In this section, we will collect some elementary estimates and results that
are either necessary for the well-posedness of the mean-field equation and the
corresponding gradient structure, or necessary to do the same for the
Liouville equation in Section 4.

Let $\Psi^{*}$ be given as

(2.5) $\Psi^{*}(z):=2(\cosh(z)-1)=e^{z}+e^{-z}-2,$

and its dual $\Psi:=(\Psi^{*})^{*}$

(2.6) $\Psi(s)=s\log\left(\frac{s+\sqrt{s^{2}+4}}{2}\right)-\sqrt{s^{2}+4}+2$

######  Lemma 2.9.

Let $M:=\|c\|_{\infty}(1+\gamma(\mathcal{T}))$. Then the following estimates
hold:

  1. (i)

The measures $\kappa_{\nu}^{\pm}$ and $\theta_{\nu}$ are finite:

(2.7) $\kappa^{\pm}_{\nu}(\mathcal{T})\leq M(1+\nu(\mathcal{T})^{2}),$

and

(2.8) $\theta_{\nu}(\mathcal{T})\leq M(1+\nu(\mathcal{T})^{2})$

  2. (ii)

For any birth/death fluxes $\lambda^{\pm}\in\mathcal{M}^{+}(\mathcal{T})$, net
flux $\lambda^{\mathrm{net}}=\lambda^{+}-\lambda-$, and
$w^{\pm},w\in\mathcal{B}(\mathcal{T})$,

$\displaystyle\int_{\mathcal{T}}|w^{\pm}|\,\mathrm{d}\lambda^{\pm}$
$\displaystyle\leq\mathcal{E}\mathrm{nt}(\lambda^{\pm}|\theta_{\nu})+\int_{\mathcal{T}}\Psi^{*}(w)\,\mathrm{d}\theta_{\nu}+\theta_{\nu}(\mathcal{T}),$
$\displaystyle\int_{\mathcal{T}}|w|\,\mathrm{d}|\lambda^{\mathrm{net}}|$
$\displaystyle\leq\mathcal{R}_{MF}(\nu,\lambda^{+},\lambda^{-})+\int_{\mathcal{T}}\Psi^{*}(w)\,\mathrm{d}\theta_{\nu},$

  3. (iii)

For any birth/death fluxes $\lambda^{\pm}\in\Gamma$,

(2.9)
$\phi\left(\frac{\lambda^{\pm}(\mathcal{T})}{M(1+\nu(\mathcal{T})^{2})}\vee
1\right)M\leq\mathcal{R}_{MF}(\nu,\lambda^{+},\lambda^{-})$

######  Remark 2.10.

Although the estimate for $\theta_{\nu}$ can be made more precise, namely

$\theta_{\nu}(\mathcal{T})\leq\|c\|_{\infty}\gamma(\mathcal{T})^{1/2}\nu(\mathcal{T})^{3/2},$

we will not require it for our results.

###### Proof.

_(i)_ With
$\theta_{\nu}:=\sqrt{\mathrm{d}\kappa_{\nu}^{+}/\mathrm{d}\sigma\,\mathrm{d}\kappa_{\nu}^{-}/\mathrm{d}\sigma}\,\sigma$
for any dominating measure $\sigma$ we have by Hölder’s inequality

$\theta_{\nu}(\mathcal{T})\leq\sqrt{\kappa_{\nu}^{+}(\mathcal{T})\kappa_{\nu}^{-}(\mathcal{T})}.$

Note that
$\kappa^{+}_{\nu}(\mathcal{T})\leq\|c\|_{\infty}\gamma(\mathcal{T})\nu(\mathcal{T})$,
and $\kappa^{-}_{\nu}(\mathcal{T})\leq\|c\|_{\infty}\nu(\mathcal{T})^{2}$,
which provides (2.7). Since $z\leq 1+z^{2}$ for all $z\geq 0$ (2.8) follows
directly.

_(ii)_ First, suppose that $w\in\mathcal{B}_{b}(\mathcal{T})$. Using the
elementary inequality $e^{|a|}\leq e^{a}+e^{-a}$ we derive by duality of the
entropy

$\displaystyle\int_{\mathcal{T}}|w|\,\mathrm{d}\lambda^{\pm}$
$\displaystyle\leq\mathcal{E}\mathrm{nt}(\lambda^{\pm}|\theta_{\nu})+\int_{\mathcal{T}}(e^{|w|}-1)\,\mathrm{d}\theta_{\nu}$
$\displaystyle\leq\mathcal{E}\mathrm{nt}(\lambda^{\pm}|\theta_{\nu})+\int_{\mathcal{T}}\Psi^{*}(w)\,\mathrm{d}\theta_{\nu}+\theta_{\nu}(\mathcal{T}).$

Next, fix any measurable function $w\in\mathcal{B}(\mathcal{T})$ and set its
$k$-truncation $w_{k}:=\max\\{\min\\{w,k\\},-k\\}$. Since $\Psi^{*}$ is even
and monotone, by dominated convergence applied to the left-hand side and
monotone convergence to the right-hand side, the inequality holds for $w$ as
well.

_(iii)_ Without loss of generality, suppose that $\mathcal{R}_{MF}$ is finite.
Set $a(\nu):=(1+\nu(\mathcal{T})^{2})^{-1}$, and note that $0\leq a(\nu)\leq
1$. With $\tilde{\phi}(s):=\phi(s\vee 1)$ the monotone relaxation of $\phi$,
we then have the following chain of inequalities,

$\displaystyle\int_{\mathcal{T}}\phi\left(\frac{\mathrm{d}\lambda^{\pm}}{\mathrm{d}\theta_{\nu}}\right)\mathrm{d}\theta_{\nu}$
$\displaystyle\geq\int_{\mathcal{T}}\tilde{\phi}\left(\frac{\mathrm{d}\lambda^{\pm}}{\mathrm{d}\theta_{\nu}}\right)\mathrm{d}\theta_{\nu}$
$\displaystyle\geq\int_{\mathcal{T}}\tilde{\phi}\left(\frac{\mathrm{d}(a(\nu)\lambda^{\pm})}{\mathrm{d}(a(\nu)\theta_{\nu})}\right)\mathrm{d}(a(\nu)\theta_{\nu})$
$\displaystyle\geq\tilde{\phi}\left(\frac{a(\nu)\lambda^{\pm}(\mathcal{T})}{a(\nu)\theta_{\nu}(\mathcal{T})}\right)a(\nu)\theta_{\nu}(\mathcal{T}),$

where the last inequality follows from Jensen’s inequality. By convexity of
$\tilde{\phi}$ and $\tilde{\phi}(0)=0$ the latter expression is monotone in
$\theta_{\nu}(\mathcal{T})$, and hence by (2.8) we find

$\tilde{\phi}\left(\frac{\lambda^{\pm}(\mathcal{T})}{M(1+\nu(\mathcal{T})^{2})}\right)M\leq\mathcal{R}_{MF}(\nu,\lambda^{+},\lambda^{-}).$

∎

We will briefly state the improvement of regularity in time of $\nu_{t}$ if
there exists a common dominating measure. The proof is similar to Corollary
4.14 of [PRST22] and therefore omitted here.

######  Lemma 2.11.

Let $(\nu,\lambda^{+},\lambda^{-})\in\mathscr{CE}$ and suppose that there
exists a measure $\ell\in\Gamma$ such that $\nu_{t},\lambda_{t}^{\pm}\ll\ell$
for all $t\in[0,T]$.

Then there exists an absolutely continuous and a.e. differentiable map
$u:[0,T]\to L^{1}(\mathcal{T},\ell)$ and maps $g^{\pm}:[0,T]\to
L^{1}(\mathcal{T},\ell)$ such that $u_{t}=\mathrm{d}\nu_{t}/\mathrm{d}\ell$,
$g_{t}^{\pm}=\mathrm{d}\lambda_{t}^{\pm}/\mathrm{d}\ell$ and

$\partial_{t}u_{t}(x)=g_{t}^{+}(x)-g_{t}^{-}(x),\qquad\mbox{for a.e.\
$t\in[0,T]$}.$

In particular, the continuity equation holds in the strong sense, namely that
$\nu_{t}$ is an a.e. differentiable map from $[0,T]$ to
$(\Gamma,\|\cdot\|_{TV})$ and

$\partial_{t}\nu_{t}=\lambda_{t}^{+}-\lambda_{t}^{-},\qquad\mbox{for a.e.\
$t\in[0,T]$}.$

Next, we will list two results that are either necessary for the chain rule in
Section 3.3 or the superposition principle and well-posedness of the
continuity equation in Section 4.

######  Lemma 2.12.

For any $0\leq a\leq 1$, $z\in\mathbb{R}$

(2.10) $\Psi^{*}(az)\leq a^{2}\Psi^{*}(z).$

Moreover, for any net flux
$\lambda^{\mathrm{net}}\in\mathcal{M}(\mathcal{T})$,
$w\in\mathcal{B}(\mathcal{T})$

(2.11)
$\displaystyle\Psi\left(\frac{\|\lambda^{\mathrm{net}}\|_{TV}}{M(1+\nu(\mathcal{T}))}\right)M$
$\displaystyle\leq\mathcal{R}_{MF}(\nu,\lambda^{+},\lambda^{-}).$

###### Proof.

It is straightforward to check that $\Psi^{*}(z)/z^{2}$ is monotone increasing
for $z\geq 0$, from which the first statement follows.

Now, for the net flux, it is convenient to go through the dual representation.
Set $a(\nu):=(1+\nu(\mathcal{T}))^{-1}$. By duality, for any
$w\in\mathcal{B}_{b}(\mathcal{T})$

(2.12) $\displaystyle\mathcal{R}_{MF}(\nu,\lambda^{+},\lambda^{-})\geq
a(\nu)\int_{\mathcal{T}}w(x)\,\mathrm{d}\lambda^{\mathrm{net}}-\int_{\mathcal{T}}\Psi^{*}\big{(}a(\nu)w(x)\big{)}\,\mathrm{d}\theta_{\nu}.$

However, by (2.10),

$\int_{\mathcal{T}}\Psi^{*}\big{(}a(\nu)w(x)\big{)}\,\mathrm{d}\theta_{\nu}\leq\int_{\mathcal{T}}\Psi^{*}\big{(}w(x)\big{)}a(\nu)^{2}\,\mathrm{d}\theta_{\nu}\leq
M\Psi^{*}(\|w\|_{\infty}).$

Taking the supremum over all $w\in\mathcal{B}_{b}(\mathcal{T})$ in (2.12) we
find (2.11). ∎

######  Lemma 2.13.

Let $\\{f_{i}\\}_{i\in\mathbb{N}}\subset C_{b}(\mathcal{T})$ be a countable
and dense set of bounded continuous functions. Suppose
$(\nu,\lambda^{+},\lambda^{-})$ is such that

  1. (i)

the curve $[0,T]\ni t\mapsto\nu_{t}\in\Gamma$ is narrowly continuous

  2. (ii)

$(\lambda_{t}^{\pm})_{t\in[0,T]}\subset\Gamma$ is a Borel family with

$\int_{0}^{T}\mathcal{R}_{MF}(\nu_{t},\lambda^{+}_{t},\lambda^{-}_{t})\,\mathrm{d}t<\infty$

  3. (iii)

For all $i\in\mathbb{N}$

$\int_{\mathcal{T}}f_{i}\,\mathrm{d}\nu_{t}-\int_{\mathcal{T}}f_{i}\,\mathrm{d}\nu_{s}=\int_{s}^{t}\left(\int_{\mathcal{T}}f_{i}\,\mathrm{d}\lambda_{r}^{+}-\int_{\mathcal{T}}f_{i}\,\mathrm{d}\lambda_{r}^{-}\right)\mathrm{d}r,\qquad\mbox{for
all $s,t$ with $0\leq s,t\leq T$.}$

Then $(\nu,\lambda^{+},\lambda^{-})\in\mathscr{CE}$, i.e. the triple satisfies
the mean-field continuity equation.

###### Proof.

Since $\nu_{t}$ is narrowly continuous its mass is uniformly bounded in time,
hence let $C:=\sup_{t\in[0,T]}\nu_{t}(\mathcal{T})$. By (2.9) and monotonicity
of $\phi(\cdot\vee 1)$ we have for a.e. $t\in[0,T]$,

$\phi\left(\frac{\lambda_{t}^{\pm}(\mathcal{T})}{M(1+C^{2})}\vee
1\right)M\leq\mathcal{R}_{MF}(\nu_{t},\lambda^{+}_{t},\lambda^{-}_{t}),$

and therefore by convexity of $\phi(\cdot\vee 1)$

$\int_{0}^{T}\lambda_{t}^{\pm}(\mathcal{T})<\infty.$

Since the measures
$\lambda_{t}^{\pm}(\mathrm{d}x)\,\mathrm{d}t\in\mathcal{M}^{+}([0,T]\times\Gamma)$
are finite, by density of $f_{i}$ in $C_{b}(\mathcal{T})$ it is clear that for
all $f\in C_{b}(\mathcal{T})$

$\int_{\mathcal{T}}f\,\mathrm{d}\nu_{t}-\int_{\mathcal{T}}f\,\mathrm{d}\nu_{s}=\int_{s}^{t}\left(\int_{\mathcal{T}}f\,\mathrm{d}\lambda_{r}^{+}-\int_{\mathcal{T}}f\,\mathrm{d}\lambda_{r}^{-}\right)\,\mathrm{d}r,\qquad\mbox{for
all $s,t$ with $0\leq s,t\leq T$.}$

By a monotone class argument this can be extended to all
$f\in\mathcal{B}_{b}(\mathcal{T})$ and we derive that $\nu_{t}$ is indeed TV-
absolutely continuous and $(\nu,\lambda^{+},\lambda^{-})\in\mathscr{CE}$. ∎

###  2.2. Strong solutions

Strong solutions to ($\sf MF$) exist and are unique, and we list the most
important properties here. It should be noted that these arguments apply even
without the detailed balance condition $m(x,y)=c(y,x)$ and only require both
$\|m\|_{\infty}$ and $\|c\|_{\infty}<\infty$ to be finite, but for simplicity,
we will restrict ourselves to our framework. Moreover, in all results the time
window $T>0$ is arbitrary.

######  Definition 2.14.

A _strong_ solution to ($\sf MF$) is any TV-absolutely continuous and a.e.
differentiable mapping $\nu:[0,T]\to(\Gamma,\|\cdot\|_{TV})$ satisfying

(2.13) $\displaystyle\partial_{t}\nu_{t}(\mathrm{d}x)$
$\displaystyle=\kappa_{\nu_{t}}^{+}(\mathrm{d}x)-\kappa_{\nu_{t}}^{-}(\mathrm{d}x)$

Recall that $\kappa^{+}_{\nu}(\mathrm{d}x)=\langle
c(x,\cdot),\nu\rangle\gamma(\mathrm{d}x)$ and
$\kappa^{-}_{\nu}(\mathrm{d}x)=\langle c(x,\cdot),\nu\rangle\nu(\mathrm{d}x)$.

######  Remark 2.15.

Note that if $\nu$ is a strong solution to ($\sf MF$) automatically
$(\nu,\kappa^{+}_{\nu},\kappa_{\nu}^{-})\in\mathscr{CE}$.

Vice versa, if $(\nu,\kappa^{+}_{\nu},\kappa_{\nu}^{-})\in\mathscr{CE}$ then
$\nu_{t}$ is a strong solution. Namely, any TV-absolutely continuous curve
$\nu_{t}$ possesses a common dominating measure $\ell\in\Gamma$, which implies
$\kappa_{\nu_{t}}^{\pm}\ll\ell+\gamma$. By Lemma 2.11 the curve $\nu$ is
indeed a a.e. differentiable mapping to $(\Gamma,\|\cdot\|_{TV})$

######  Lemma 2.16.

For any $\bar{\nu}\in\Gamma$ there exist a unique strong solution $\nu_{t}$ to
($\sf MF$) such that $\nu_{0}=\bar{\nu}$.

Moreover, if $\bar{\nu}\ll\gamma$, then also $\nu_{t}\ll\gamma$ for all
$t\in[0,T]$.

The proof is an adaptation from [FM04, Proposition 7.2], which is stated for
Lebesgue absolutely continuous measures over $\mathcal{T}=\mathbb{R}^{d}$. In
short, the linear dependence of the birth flux on the mass of $\nu$ gives a
bound on this mass uniform in time, in which case both $\kappa^{\pm}_{\nu}$
are Lipschitz in $\nu$ on $(\Gamma,\|\cdot\|)$, and classical existence theory
can be applied.

###### Proof.

First, note that for the linear case of

$\partial_{t}\nu_{t}(\mathrm{d}x)=b_{t}(\mathrm{d}x)-c_{t}(x)\nu_{t}(\mathrm{d}x),$

with $c_{t}\in\mathcal{B}_{b}$ uniformly bounded and $b_{s}\in\Gamma$ with
$\int_{0}^{T}\|b_{s}\|_{TV}\,\mathrm{d}t<\infty$ with a common dominating
measure, it is easy to verify that a unique strong non-negative solution
exists and is given by

$\nu_{t}:=e^{-\int_{0}^{t}c_{s}(x)\mathrm{d}s}\left(\int_{0}^{t}b_{s}e^{\int_{0}^{s}c_{r}\,\mathrm{d}r}\mathrm{d}s+\nu_{0}\right).$

We now set $\nu^{0}_{t}:=\bar{\nu}$ for all $t\in[0,T]$, and perform the
implicit Picard iteration

$\partial_{t}\nu^{k+1}_{t}(\mathrm{d}x)=\langle
c(x,\cdot),\nu^{k})\rangle\gamma(\mathrm{d}x)-\langle
c(x,\cdot),\nu^{k}\rangle\nu^{k+1}(\mathrm{d}x),\qquad\nu^{k+1}_{0}:=\bar{\nu},$

i.e. $\nu^{k+1}=(\mathcal{G}\nu^{k})$ with

$(\mathcal{G}\nu)_{t}(\mathrm{d}x):=e^{-\int_{0}^{t}\langle
c(x,\cdot),\nu_{r}\rangle\mathrm{d}s}\left(\int_{0}^{t}\langle
c(x,\cdot),\nu_{s})\gamma(\mathrm{d}x)\rangle e^{\int_{0}^{s}\langle
c(x,\cdot),\nu_{r}\rangle\,\mathrm{d}r}\mathrm{d}s+\bar{\nu}(\mathrm{d}x)\right).$

It is straightforward to check that for all $t\in[0,T]$

$\sup_{k\geq 1}\nu^{k}_{t}(\mathcal{T})\leq
e^{\|c\|_{\infty}\gamma(\mathcal{T})t}\bar{\nu}(\mathcal{T})\leq
e^{\|c\|_{\infty}\gamma(\mathcal{T})T}\bar{\nu}(\mathcal{T})=:C.$

We will show that $\mathcal{G}$ is contractive under a suitable metric on the
space of curves with initial data $\bar{\nu}$ and mass bounded by $C$. This
implies there exists a $TV$-absolutely continuous curve $\nu$ such that

$\nu_{t}-\nu_{s}=\int_{s}^{t}\left(\kappa_{\nu_{r}}+\kappa_{\nu_{r}}^{-}\right)\,\mathrm{d}r,\qquad\mbox{for
all $s,t$ with $0\leq s,t\leq T$.}$

Moreover, since in the iterations $\nu^{k}\ll\bar{\nu}+\gamma$ for all $\nu$
it is clear that we obtain strong solutions in $L^{1}(\bar{\nu}+\gamma)$. In
particular, for $\bar{\nu}\ll\gamma$ we have $\nu_{t}\ll\gamma$ for all
$t\in[0,T]$ as well.

Now, note that $\langle c(x,\cdot),\nu\rangle$ depends Lipschitz on $\nu$ in
$(\Gamma,\|\cdot\|_{TV})$ due to the uniform bound on mass. This implies that
there exists a constant $K$ such that for any two admissible curves
$\nu,\tilde{\nu}$:

$\|(\mathcal{G}\nu)_{t}-(\mathcal{G}\tilde{\nu})_{t}\|_{TV}\leq
K\int_{0}^{t}\|\nu_{s}-\tilde{\nu}_{s}\|_{TV}\,\mathrm{d}s,\qquad\mbox{for all
$t\in[0,T]$.}$

Hence, by a Gronwall-type argument, we find that for any $\varepsilon>0$ for
all $t\in[0,T]$

$\|(\mathcal{G}\nu)_{t}-\mathcal{G}(\tilde{\nu})_{t}\|_{TV}e^{-(K+\varepsilon)t}\leq\frac{K}{K+\varepsilon}\left(\sup_{s\in[0,T]}\|\nu_{s}-\tilde{\nu}_{s}\|_{TV}e^{-(K+\varepsilon)s}\right),$

thus yielding the contraction required to apply the Banach fixed-point
theorem. ∎

Finally, for the use in entropic propagation chaos of Theorem 5.4, it is
convenient to characterize the conditions for which $u_{t}$ is bounded from
above and below. The following statement follows directly from a Gronwall-type
argument.

######  Lemma 2.17.

Suppose $\nu_{0}$ is such that
$C^{-1}\leq\mathrm{d}\nu_{0}/\mathrm{d}\gamma(x)<C$ for some constant $C>0$
and all $x\in\mathcal{T}$. Then there exist a constant $C_{T}>0$ such that for
the corresponding solution

$C_{T}^{-1}\leq\frac{\mathrm{d}\nu_{0}}{\mathrm{d}\gamma}(x)<C_{T},\qquad\mbox{
for all }x\in\mathcal{T},\mbox{ for all }t\in[0,T].$

###  2.3. Variational characterization

We will now prove the non-negativity of our EDP-functional $\mathcal{I}_{MF}$
and the characterization of strong solutions to ($\sf MF$) as minimizers of
$\mathcal{I}_{MF}$. To do so we first need the prove the chain rule for the
free energy $\mathcal{F}_{MF}$ along curves with finite $\mathcal{I}_{MF}$.

There is an important technical issue concerning the Fisher information, in
the sense that on curves with finite $\mathcal{I}_{MF}$ the chain rule
inequality holds for the following replacement:

$\mathcal{D}^{-}_{MF}(\nu):=\int_{\mathcal{T}}\Psi^{*}\left(\frac{1}{2}\log
u\right)\mathrm{d}\theta_{\nu}=\int_{u>0}c_{\nu}(x)\left(\sqrt{u}-1\right)^{2}\mathrm{d}\gamma,$

for any $\nu\ll\gamma$ with $u:=\mathrm{d}\nu/\mathrm{d}\gamma$. Note that
$0\leq\mathcal{D}^{-}_{MF}(\nu)\leq\mathcal{D}_{MF}(\nu)$ and
$\mathcal{D}^{-}_{MF}=\mathcal{R}_{MF}^{*}(\partial_{\nu}\mathcal{F}_{MF})$.

We will see the same principle arise in Section 3 for the variational
characterization of the forward Kolmogorov equation, which is also observed in
[PRST22, Section 5].

######  Lemma 2.18.

For any curve $(\nu,\lambda^{+},\lambda^{-})\in\mathscr{CE}$ with
$\mathcal{F}_{MF}(\nu_{0})<\infty$ and
$\mathcal{I}_{MF}(\nu,\lambda^{+},\lambda^{-})<\infty$ it holds that $[0,T]\ni
t\mapsto\mathcal{F}_{MF}(\nu_{t})$ is absolutely continuous and a.e.
differentiable with

$\frac{\mathrm{d}\,}{\mathrm{d}\,t}\mathcal{F}_{MF}(\nu_{t})=\frac{1}{2}\int_{\mathcal{T}}\log\left(\frac{\mathrm{d}\nu_{t}}{\mathrm{d}\gamma}\right)\,\mathrm{d}\lambda^{\mathrm{net}}_{t},\qquad\mbox{for
a.e.\ $t\in[0,T]$}.$

Moreover, for such a curve

$\mathcal{I}_{MF}(\nu,\lambda^{+},\lambda^{-})\geq\mathcal{I}_{MF}^{-}:=\int_{0}^{T}\left(\mathcal{R}_{MF}(\nu_{t},\lambda^{+}_{t},\lambda^{-}_{t})+\frac{1}{2}\int_{\mathcal{T}}\log\frac{\mathrm{d}\nu_{t}}{\mathrm{d}\gamma}\,\mathrm{d}\lambda^{\mathrm{net}}_{t}+\mathcal{D}^{-}_{MF}(\nu_{t})\right)\,\mathrm{d}t\geq
0.$

######  Remark 2.19.

In fact, for such curves, for a.e. $t$ both the terms

$\int_{\mathcal{T}}\log\left(\frac{\mathrm{d}\nu_{t}}{\mathrm{d}\gamma}\right)\,\mathrm{d}\lambda^{\pm}_{t},$

will be finite, and hence

$\int_{\mathcal{T}}\log\left(\frac{\mathrm{d}\nu_{t}}{\mathrm{d}\gamma}\right)\,\mathrm{d}\lambda_{t}=\int_{\mathcal{T}}\log\left(\frac{\mathrm{d}\nu_{t}}{\mathrm{d}\gamma}\right)\,\mathrm{d}\lambda_{t}^{+}-\int_{\mathcal{T}}\log\left(\frac{\mathrm{d}\nu_{t}}{\mathrm{d}\gamma}\right)\,\mathrm{d}\lambda_{t}^{-},$

###### Proof.

Fix any curve $(\nu,\lambda^{+},\lambda^{-})\in\mathscr{CE}$ with
$\mathcal{F}_{MF}(\nu_{0})<\infty$. We will show that whenever
$\mathcal{I}_{MF}<\infty$ the mapping
$t\mapsto\mathcal{E}\mathrm{nt}(\nu_{t}|\gamma)$ is absolutely continuous and
satisfies the chain rule, i.e.

$\frac{\mathrm{d}\,\mathcal{E}\mathrm{nt}(\nu_{t}|\gamma)}{\mathrm{d}\,t}=\int_{\mathcal{T}}\log\left(\frac{\mathrm{d}\nu_{t}}{\mathrm{d}\gamma}\right)\mathrm{d}(\lambda^{+}_{t}-\lambda^{-}_{t})\,\mathrm{d}t,\qquad\mbox{for
a.e.\ $t\in[0,T]$}.$

Suppose that $\mathcal{I}_{MF}<\infty$. Since $\mathcal{E}\mathrm{nt}$ is
bounded from below, $\mathcal{E}\mathrm{nt}(\nu_{0}|\gamma)<\infty$ implies
that

$\int_{0}^{T}\mathcal{R}_{MF}(\nu_{t},\lambda^{+}_{t},\lambda^{-}_{t})\,\mathrm{d}t<\infty,\qquad\int_{0}^{T}\mathcal{D}_{MF}(\nu_{t})\,\mathrm{d}t<\infty.$

In particular for a.e. $t\in[0,T]$ it holds that $\nu_{t}\ll\gamma$,
$\lambda_{t}^{\pm}\ll\theta_{\nu_{t}}$, and in turn
$\theta_{\nu_{t}}\ll\gamma$. In fact, due to TV-continuity of $\nu_{t}$, we
have $\nu_{t}\ll\gamma$ for all $t\in[0,T]$. Moreover,
$\int_{0}^{T}\lambda^{\pm}_{t}(\mathcal{T})<\infty$ and
$\sup_{t}\nu_{t}(\mathcal{T})<\infty$.

Setting $u_{t}:=\mathrm{d}\nu_{t}/\mathrm{d}\gamma$, we have

$\mathrm{d}\theta_{\nu_{t}}=c_{\nu_{t}}\sqrt{u_{t}}\,\mathrm{d}\gamma,$

and in particular $\theta_{\nu_{t}}(\\{u_{t}=0\\})=0$. Similarly,
$\lambda_{t}^{\pm}\ll\theta_{\nu_{t}}$ for a.e. $t$ and hence $u_{t}>0$ for
$\lambda^{\pm}_{t},\lambda^{\mathrm{net}}_{t}$-a.e. $x$ for such $t$ as well.
Furthermore, since for a.e. $t$ we have
$\lambda_{t}^{\pm}\ll\theta_{\nu_{t}}\ll\gamma$ we find by Lemma 2.11 that
$u:[0,T]\to L^{1}(\mathcal{T},\gamma)$ is absolutely continuous and
differentiable at a.e. $r\in[0,T]$.

Consider any such $r$ with
$\mathcal{R}_{MF}(\nu_{r},\lambda_{r}^{+},\lambda_{r}^{-}),\mathcal{D}(\nu_{r})<\infty$.
By Lemma 2.9, for any $w\in\mathcal{B}_{b}(\mathcal{T})$,

(2.14)
$\left|\int_{\mathcal{T}}w\,\mathrm{d}\lambda^{\mathrm{net}}_{r}\right|\leq\int_{\mathcal{T}}|w|\,\mathrm{d}|\lambda_{r}|\leq\mathcal{R}_{MF}(\nu_{r},\lambda^{+}_{r},\lambda^{-}_{r})+\int_{\mathcal{T}}\Psi^{*}(w)\mathrm{d}\theta_{\nu_{r}}.$

Now let $\phi_{m}$ be the convex and uniformly Lipschitz regularizations of
$\phi$ constructed by using the truncations
$\phi_{m}^{\prime}:=[\phi^{\prime}]_{m}=\max\\{\min\\{\phi,m\\},-m\\}$ and
$\phi(s):=\int_{1}^{s}\phi_{m}^{\prime}(z)\,\mathrm{d}z$. Note that
$\phi_{m}^{\prime}$ converges pointwise to $\phi^{\prime}$, and both
$\phi_{m}$ and $|\phi^{\prime}_{m}|$ converge monotonically to $\phi$ and
$|\phi^{\prime}|$ respectively.

Moreover, note that $\phi^{\prime}(u_{r})=\log u_{r}$ is
$\theta_{\nu_{r}}$-a.e. finite, and similarly $\lambda^{\pm}_{r}$-a.e. as
well. Therefore, since $\Psi^{*}$ is even and monotone on $\mathbb{R}_{\geq
0}$ we derive

$\displaystyle\int_{\mathcal{T}}\Psi^{*}(\genfrac{}{}{}{2}{1}{2}\phi^{\prime}_{m}(u_{r}))\mathrm{d}\theta_{\nu_{r}}$
$\displaystyle\leq\int_{\mathcal{T}}\Psi^{*}(\genfrac{}{}{}{2}{1}{2}\phi^{\prime}(u_{r}))\mathrm{d}\theta_{\nu_{r}}=\mathcal{D}^{-}_{MF}(\nu_{r}).$

Recall that $\mathcal{D}^{-}_{MF}(\nu_{r})\leq\mathcal{D}_{MF}(\nu_{r})$. By
substituting $w=\genfrac{}{}{}{2}{1}{2}\phi_{m}^{\prime}$ in (2.14) we find

$\frac{1}{2}\int_{\mathcal{T}}\phi_{m}^{\prime}(u_{r})\,\mathrm{d}\lambda^{\mathrm{net}}_{r}\leq\frac{1}{2}\int_{\mathcal{T}}|\phi_{m}^{\prime}(u_{r})|\,\mathrm{d}|\lambda^{\mathrm{net}}_{r}|\leq\mathcal{R}_{MF}(\nu_{r},\lambda^{+}_{r},\lambda^{-}_{r})+\mathcal{D}^{-}_{MF}(\nu_{r}),$

and after a monotone convergence argument

(2.15)
$\frac{1}{2}\int_{\mathcal{T}}\phi^{\prime}(u_{r})\,\mathrm{d}\lambda^{\mathrm{net}}_{r}\leq\frac{1}{2}\int_{\mathcal{T}}|\phi^{\prime}(u_{r})|\,\mathrm{d}|\lambda^{\mathrm{net}}_{r}|\leq\mathcal{R}_{MF}(\nu_{r},\lambda^{+}_{r},\lambda^{-}_{r})+\mathcal{D}^{-}_{MF}(\nu_{r}).$

Note that for every $m$ the function $\phi_{m}$ is smooth and uniformly
Lipschitz, thus the functional $\int\phi_{m}(u_{r})\,\mathrm{d}\gamma$ is
$\|\cdot\|_{TV}$-Lipschitz continuous and hence absolutely continuous by TV-
regularity of $\nu_{r}$. Moreover, since $\lambda_{r}^{\pm}\ll\gamma$ and
$u_{r}$ is a.e. differentiable in $L^{1}(X,\gamma)$ it is straightforward to
check that

$\int_{\mathcal{T}}\phi_{m}(u_{t})\,\mathrm{d}\gamma-\int_{\mathcal{T}}\phi_{m}(u_{s})\,\mathrm{d}\gamma=\int_{s}^{t}\int_{\mathcal{T}}\phi_{m}^{\prime}(u_{r})\,\mathrm{d}(\lambda^{+}_{r}-\lambda^{-}_{r})\,\mathrm{d}r,\qquad\mbox{for
all $s,t\in[0,T]$}.$

Therefore, since $\mathcal{E}\mathrm{nt}(\nu_{0}|\gamma)$ is finite by
assumption and the functionals $\int\phi_{m}(u_{t})\,\mathrm{d}\gamma$
converge monotonically to $\mathcal{E}\mathrm{nt}(\nu_{t}|\gamma)$, we find

$\displaystyle\frac{1}{2}\left|\int_{\mathcal{T}}\phi(u_{t})\,\mathrm{d}\gamma-\int_{\mathcal{T}}\phi(u_{0})\,\mathrm{d}\gamma\right|$
$\displaystyle\leq\frac{1}{2}\limsup_{m\to\infty}\int_{0}^{T}\int_{\mathcal{T}}|\phi_{m}^{\prime}(u_{r})|\,\mathrm{d}|\lambda_{r}^{+}-\lambda_{r}^{-}|\,\mathrm{d}r$
$\displaystyle\leq\int_{0}^{T}\left(\mathcal{R}_{MF}(\nu_{r},\lambda_{r}^{+},\lambda_{r}^{-})+\mathcal{D}^{-}_{MF}(\nu_{r})\right)\,\mathrm{d}r.$

In particular $\mathcal{E}\mathrm{nt}(\nu_{t}|\gamma)$ is finite for all
$t\in[0,T]$, and after repeating the argument for $s,t\in[0,T]$ we conclude by
a dominated convergence argument that

$\int_{\mathcal{T}}\phi(u_{t})\,\mathrm{d}\gamma-\int_{\mathcal{T}}\phi(u_{s})\,\mathrm{d}\gamma=\int_{s}^{t}\int_{\mathcal{T}}\phi^{\prime}(u_{r})\,\mathrm{d}\lambda^{\mathrm{net}}_{r}\,\mathrm{d}r,$

and

$\displaystyle\mathcal{I}_{MF}$
$\displaystyle=\int_{0}^{T}\left(\mathcal{R}_{MF}(\nu_{t},\lambda^{+}_{t},\lambda_{t}^{-})+\frac{1}{2}\int_{\mathcal{T}}\phi^{\prime}(u_{t})\,\mathrm{d}\lambda^{\mathrm{net}}_{t}+\mathcal{D}_{MF}(\nu_{t})\right)\,\mathrm{d}t$
$\displaystyle\geq\int_{0}^{T}\left(\mathcal{R}_{MF}(\nu_{t},\lambda^{+}_{t},\lambda_{t}^{-})+\frac{1}{2}\int_{\mathcal{T}}\phi^{\prime}(u_{t})\,\mathrm{d}\lambda^{\mathrm{net}}_{t}+\mathcal{D}^{-}_{MF}(\nu_{t})\right)\,\mathrm{d}t\geq
0.$

∎

We are now finally in a position to prove Theorem 2.7. With the chain rule
above all that remains is on one hand showing that
$\mathcal{I}^{-}_{MF}(\nu,\lambda^{+},\lambda_{t}^{-})=0$ implies that
$\lambda_{t}^{\pm}=\kappa_{\nu_{t}}^{\pm}$ for a.e. $t$, and on the other
hand, showing that if $\nu$ is a strong solution it holds that
$\mathcal{I}^{-}_{MF}(\nu,\kappa_{\nu}^{+},\kappa_{\nu}^{-})=0$ and
$\mathcal{D}^{-}_{MF}=\mathcal{D}_{MF}$ for a.e. $t\in[0,T]$. The second part
again involves proving a chain rule, but now along the solution curve.

###### Proof of Theorem 2.7.

First, consider any $(\nu,\lambda^{+},\lambda^{-})\in\mathscr{CE}$ with
$\mathcal{F}_{MF}(\nu_{0})<\infty$, and $\mathcal{I}_{MF}=0$. By Lemma 2.18,

$\int_{0}^{T}\left(\mathcal{R}_{MF}(\nu_{t},\lambda^{+}_{t},\lambda_{t}^{-})+\frac{1}{2}\int_{\mathcal{T}}\phi^{\prime}(u_{t})\,\mathrm{d}\lambda^{\mathrm{net}}_{t}+\mathcal{D}^{-}_{MF}(\nu_{t})\right)\,\mathrm{d}t=0.$

Now, recall that $\mathrm{d}\theta_{\nu}=c_{\nu}\sqrt{u}\,\mathrm{d}\gamma$.
Setting $g_{t}^{\pm}:=\mathrm{d}\lambda^{\pm}_{t}/\mathrm{d}\theta_{\nu}$, it
holds that $\log(u_{t})\,g_{t}^{\pm}<\infty$ for $\theta_{\nu_{t}}$-a.e. $x$
and a.e. $t$, and by the inequality (2.15) that $|\log
u_{t}|\,|g_{t}^{+}-g_{t}^{-}|$ is $\theta_{\nu_{t}}$-integrable. Therefore, by
straightforward algebraic manipulations we find that for a.e. $t$,

$\displaystyle\mathcal{R}_{MF}(\nu_{t},\lambda^{+}_{t},\lambda_{t}^{-})+\frac{1}{2}\int_{\mathcal{T}}\phi^{\prime}(u_{t})\,\mathrm{d}\lambda^{\mathrm{net}}_{t}+\mathcal{D}_{MF}^{-}(\nu_{t})$
$\displaystyle=\int_{\mathcal{T}}\left(\phi(g^{+}_{t})+\genfrac{}{}{}{2}{1}{2}\log(u_{t})g^{+}_{t}+\phi^{*}(\genfrac{}{}{}{2}{1}{2}\log
u_{t})+\phi(g^{-}_{t})-\genfrac{}{}{}{2}{1}{2}\log(u_{t})g^{-}_{t}+\phi^{*}(-\genfrac{}{}{}{2}{1}{2}\log
u_{t})\right)\mathrm{d}\theta_{\nu_{t}}.$

Due to the duality between $\phi$ and $\phi^{*}$ this expression is zero if
only if $\theta_{\nu_{t}}$-a.e.

$g_{t}^{\pm}=(\phi^{\prime})^{-1}(\mp\genfrac{}{}{}{2}{1}{2}\log u_{t}).$

Recalling that $\theta_{\nu}=c_{\nu}\sqrt{u}\gamma$,
$\kappa_{\nu}^{+}=c_{\nu}\gamma$ and $\kappa^{-}_{\nu}=c_{\nu}u\gamma$ we find
that indeed for a.e. $t$,

$\lambda^{\pm}_{t}=\kappa_{\nu_{t}}^{\pm}.$

Vice versa, assume that $\nu_{t}$ is a strong solution with
$\mathcal{F}_{MF}(\nu_{0})<\infty$. Recall that $\nu_{t}\ll\gamma$ for all
$t\in[0,T]$ by Lemma 2.16, and hence $\kappa_{\nu_{t}}^{\pm}\ll\gamma$ as
well. Therefore we can again write
$u_{t}:=\mathrm{d}\nu_{t}/\mathrm{d}\gamma$, $\kappa_{\nu}^{+}=c_{\nu}\gamma$,
$\kappa^{-}_{\nu}=c_{\nu}u\gamma$ and $\theta_{\nu}=c_{\nu}\sqrt{u}\gamma$.
Moreover, $u:[0,T]\to L^{1}(\mathcal{T},\gamma)$ is absolutely continuous and
a.e. differentiable, and thus for every regularized entropy function:

$\int_{\mathcal{T}}\phi_{m}(u_{T})\,\mathrm{d}\gamma-\int_{\mathcal{T}}\phi_{m}(u_{0})\,\mathrm{d}\gamma=\int_{0}^{T}\int_{\mathcal{T}}c_{\nu_{t}}\phi_{m}^{\prime}(u_{t})(1-u_{t})\,\mathrm{d}\gamma\,\mathrm{d}t.$

Note that the latter expression is non-positive since $\phi^{\prime}(z)(z-1)$
is non-negative, due to the convexity of $\phi_{m}$ and $\phi_{m}(1)=0$.
Moreover, recall that the regularized entropies converge for every $\nu$, are
non-negative, and $\mathcal{E}\mathrm{nt}(\nu_{0}|\gamma)<0$ by assumption.
Therefore

$\limsup_{m\to\infty}\int_{0}^{T}\int_{\mathcal{T}}c_{\nu_{t}}\phi_{m}^{\prime}(u_{t})(u_{t}-1)\,\mathrm{d}\gamma\,\mathrm{d}t\leq\mathcal{E}\mathrm{nt}(\nu_{0}|\gamma)<\infty.$

It is clear that to obtain $\mathcal{I}_{MF}=0$ it is sufficient to prove that
for any $\nu$ with $\nu\ll\gamma$,

$\frac{1}{2}\lim_{m\to\infty}\int_{\mathcal{T}}c_{\nu}\phi_{m}^{\prime}(u)(u-1)\,\mathrm{d}\gamma=\mathcal{R}(\nu,\kappa_{\nu}^{+},\kappa_{\nu}^{-})+\mathcal{D}(\nu).$

By non-negativity of the integrand both

$\lim_{m\to\infty}\int_{u=0}c_{\nu}\phi_{m}^{\prime}(u)(u-1)\,\mathrm{d}\gamma<\infty.$

and

$\lim_{m\to\infty}\int_{u>0}c_{\nu}\phi_{m}^{\prime}(u)(u-1)\,\mathrm{d}\gamma<\infty.$

Since $\phi^{\prime}(0)=-m$ this implies that in fact for all $m$

$\int_{u=0}c_{\nu}\phi_{m}^{\prime}(u)(u-1)\,\mathrm{d}\gamma=m\int_{u=0}c_{\nu}\,\mathrm{d}\gamma,$

but since the former is finite after taking the limit $m\to\infty$, we deduce
that

$\int_{u=0}c_{\nu}\,\mathrm{d}\gamma=0,$

and hence $\gamma(\\{u=0,c_{\nu}>0)=0$. Moreover, by monotone convergence we
have

$\int_{u>0}c_{\nu}\log(u)(u-1)\,\mathrm{d}\gamma=\lim_{m\to\infty}\int_{u>0}c_{\nu}\phi_{m}^{\prime}(u)(u-1)\,\mathrm{d}\gamma.$

Note by straightforward algebraic manipulation that

$\genfrac{}{}{}{2}{1}{2}\log(z)(z-1)=\phi(\sqrt{z})\sqrt{z}+\phi(1/\sqrt{z})\sqrt{z}+(\sqrt{z}-1)^{2}\,\qquad\mbox{
for all }z>0.$

Therefore

$\displaystyle\frac{1}{2}\int_{u>0}c_{\nu}\log(u)(u-1)\mathrm{d}\gamma$
$\displaystyle=\int_{u>0,\,c_{\nu}>0}c_{\nu}\left(\phi\left(\sqrt{u}\right)\sqrt{u}+\phi\left(1/\sqrt{u}\right)\sqrt{u(x)}+(\sqrt{u}-1)^{2}\right)\mathrm{d}\gamma$
$\displaystyle=\int_{u>0,\,c_{\nu}>0}\left(\phi\left(\frac{\mathrm{d}\kappa_{\nu}^{+}}{\mathrm{d}\theta_{\nu}}\right)\frac{\mathrm{d}\theta_{\nu}}{\mathrm{d}\gamma}+\phi\left(\frac{\mathrm{d}\kappa_{\nu}^{-}}{\mathrm{d}\theta_{\nu}}\right)\frac{\mathrm{d}\theta_{\nu}}{\mathrm{d}\gamma}+c_{\nu}(\sqrt{u}-1)^{2}\right)\mathrm{d}\gamma.$

Since all terms are non-negative we can separate terms and reduce the
expression to

$\genfrac{}{}{}{2}{1}{2}\int_{u>0}c_{\nu}\log(u)(u-1)\mathrm{d}\gamma=\mathcal{R}_{MF}(\nu,\kappa_{\nu}^{+},\kappa_{\nu}^{-})+\mathcal{D}_{MF}(\nu).$

Here the equality follows from the fact that $\gamma(\\{u=0,c_{\nu}>0)=0$ and
hence

$\int_{u>0,\,c_{\nu}>0}c_{\nu}(\sqrt{u}-1)^{2}\mathrm{d}\gamma=\int_{\mathcal{T}}c_{\nu}(\sqrt{u}-1)^{2}\mathrm{d}\gamma=\mathcal{D}_{MF}(\nu),$

i.e. $\mathcal{D}_{MF}^{-}(\nu)=\mathcal{D}_{MF}(\nu)$, and

$\int_{u>0,\,c_{\nu}>0}\phi\left(\frac{\mathrm{d}\kappa_{\nu}^{\pm}}{\mathrm{d}\theta_{\nu}}\right)\frac{\mathrm{d}\theta_{\nu}}{\mathrm{d}\gamma}\mathrm{d}\gamma=\mathcal{E}\mathrm{nt}(\kappa^{\pm}_{\nu}|\theta_{\nu}).$

∎

##  3\. Forward Kolmogorov equation

In the introduction, we discussed how the BPDL model describes a measure-
valued process $\nu^{n}_{t}$ in $\Gamma$ involving particles being created and
annihilated, with the corresponding Forward Kolmogorov equation

($\sf FKE_{n}$) $\partial_{t}\mathsf{P}_{t}=Q_{n}^{*}\mathsf{P}_{t},$

where $\mathsf{P}_{t}\in\mathcal{P}(\Gamma)$ for all $t\in[0,T]$ and
$Q_{n}^{*}$ is the dual of the infinitesimal generator $Q_{n}$ with

(3.1)
$\qquad(Q_{n}F)(\nu)=n\int_{\mathcal{T}}\big{(}F(\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x})-F(\nu)\big{)}\,\kappa^{+}_{\nu}(\mathrm{d}x)+n\int_{\mathcal{T}}\big{(}F(\nu-\genfrac{}{}{}{2}{1}{n}\delta_{x})-F(\nu)\big{)}\,\kappa^{-}_{\nu}(\mathrm{d}x),$

for all $F\in C_{c}(\Gamma)$. Throughout this section the parameter $n>0$ will
be fixed.

In the case of $\mathcal{T}=\mathbb{R}^{d}$ it is shown in [FM04] that a
measure-valued process with generator $Q_{n}$ exists, and is in fact a jump
process in $\Gamma$ corresponding to the jump kernel $\bar{\kappa}_{n}$ shown
below. However, for our general setting with $\mathcal{T}$ a compact Polish
space, we will take ($\sf FKE_{n}$) simply as a starting point, and do not
consider the existence or convergence of the measure-valued process
$\nu_{t}^{n}$ itself—even though we will sometimes borrow the language of jump
processes for illustration purposes.

In this section, we will state the general version of Theorem 1.6, by showing
that a detailed balance condition holds, establishing a generalized gradient
structure for the Forward-Kolmogorov equation, and characterizing the
solutions as minimizers of corresponding EDP-functionals. Similar to Section 2
we first give an overview of the ingredients to state the main results and
then leave the proofs for the existence of solutions and the variational
characterization to Sections 3.2 and 3.3.

Note that due to the fact that

$\sup_{\nu\in\Gamma}\kappa^{\pm}_{\nu}(\mathcal{T})=+\infty,$

the operator $Q_{n}$ is not bounded on $\mathcal{B}_{b}(\Gamma)$. If it were,
suitable solutions and possible variational formulation would fall into the
framework of [PRST22], where triples $(V,\pi,\kappa)$ are considered, with $V$
a Polish space, $\pi$ a finite measure, and $\kappa(x,\mathrm{d}y)$ a jump
kernel satisfying a detailed balance condition with respect to $\pi$ and the
boundedness condition

$\sup_{x}\int_{V}k(x,\mathrm{d}y)<\infty.$

They construct solutions to the forward Kolmogorov equation that are
absolutely continuous to $\pi$ and characterize them as minimizers of a
suitable EDP functional involving the net flux. In this section, we generalize
part of this framework to unbounded kernels and so-called one-way or uni-
directional fluxes and tailor it to our setting of interacting particle
systems.

Namely, let the rescaled empirical measure mapping $L_{n}:\coprod_{N\geq
1}\mathcal{T}^{N}\to\Gamma$ be given as

(3.2) $L_{n}(x_{1},\dots,x_{N}):=\frac{1}{n}\sum_{i=1}^{N}\delta_{x_{i}}.$

and let $\Gamma_{n}\subset\Gamma$ be the space of finite _positive_ discrete
measures with common unit weight $\genfrac{}{}{}{2}{1}{n}$, i.e.

(3.3) $\Gamma_{n}:=L_{n}\left(\coprod_{N\geq 1}\mathcal{T}^{N}\right).$

Note that the operators $Q_{n},Q_{n}^{*}$ can be represented as

$\displaystyle(Q_{n}F)(\nu)$
$\displaystyle=\int_{\Gamma_{n}}\left(F(\eta)-F(\nu)\right)\bar{\kappa}(\nu,\mathrm{d}\eta),$
$\displaystyle(Q_{n}^{*}\mathsf{P})(\mathrm{d}\nu)$
$\displaystyle=\int_{\eta\in\Gamma_{n}}\mathsf{P}(\mathrm{d}\eta)\bar{\kappa}_{n}(\eta,\mathrm{d}\nu)-\mathsf{P}(\mathrm{d}\nu)\int_{\eta\in\Gamma_{n}}\bar{\kappa}_{n}(\nu,\mathrm{d}\eta),$

where $\bar{\kappa}_{n}(\nu,\cdot)\in\mathcal{M}^{+}(\Gamma_{n})$ for all
$\nu\in\Gamma_{n}$ is a jump kernel over $\Gamma_{n}$ given by

(3.4)
$\bar{\kappa}_{n}(\nu,\mathrm{d}\eta):=n\int_{\mathcal{T}}\delta_{\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x}}(\mathrm{d}\eta)\,\kappa^{+}_{\nu}(\mathrm{d}x)+n\int_{\mathcal{T}}\delta_{\nu-\genfrac{}{}{}{2}{1}{n}\delta_{x}}(\mathrm{d}\eta)\,\kappa^{-}_{\nu}(\mathrm{d}x).$

Moreover, we consider Poisson measures $\Pi_{n}\in\mathcal{P}(\Gamma_{n})$
induced by the reference measure $\gamma$. Namely, with the measure
$\pi_{n}\in\mathcal{P}(\coprod_{N\geq 1}\mathcal{T}^{N})$ given by

(3.5)
$\pi_{n}:=\frac{1}{e^{n\gamma(\mathcal{T})}-1}\sum_{N=1}^{\infty}\frac{n^{N}}{N!}\gamma^{\otimes
N},$

we define

(3.6) $\Pi_{n}:=(L_{n})_{\\#}\pi.$

We will show in Lemma 3.12 that the measures $\Pi_{n}$ are invariant measures
of ($\sf FKE_{n}$) and that $\bar{\kappa}_{n}$ satisfies the detailed balance
condition with respect to $\Pi_{n}$, i.e. we have the symmetry

(3.7)
$\Pi_{n}(\mathrm{d}\nu)\bar{\kappa}_{n}(\nu,\mathrm{d}\eta)=\Pi_{n}(\mathrm{d}\nu)\bar{\kappa}_{n}(\nu,\mathrm{d}\eta).$

It is straightforward to check that even though $\bar{\kappa}_{n}$ is
unbounded, we still have the weighted integrability condition

$\sup_{\nu\in\Gamma_{n}}\left\\{(1+\nu(\mathcal{T})^{-2})\int_{\Gamma_{n}}\bar{\kappa}_{\nu}(\nu,\mathrm{d}\eta)\right\\}<+\infty.$

Therefore we can still bootstrap from gradient-flow solutions in the sense of
[PRST22] for regularized triples
$(\Gamma_{n},\Pi_{n},\bar{\kappa}^{\varepsilon}_{n})$, after passing from a
net flux to a one-way flux formulation, see Appendix A, to obtain unique
gradient-flow solutions as defined in Section 3.2.

To discuss the continuity equation and the dissipation potentials properly, we
need to introduce some additional notation. We define the following creation
and annihilation operators:

(3.8) $\displaystyle\mathsf{T}^{n,+}$
$\displaystyle:\Gamma_{n}\times\mathcal{T}\to\Gamma_{n}\times\mathcal{T},\qquad\mathsf{T}^{n,+}(\nu,x)=(\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x},x)=:(\mathsf{T}_{x}^{n,+}\nu,x),$
$\displaystyle\mathsf{T}^{n,-}$
$\displaystyle:\Gamma_{n}\times\mathcal{T}\to\Gamma_{n}\times\mathcal{T},\qquad\mathsf{T}^{n,-}(\nu,x)=(\nu-\genfrac{}{}{}{2}{1}{n}\delta_{x},x)=:(\mathsf{T}_{x}^{n,-}\nu,x),$

with the convention that $\mathsf{T}^{n,-}(\nu,x)=\nu$ if
$x\notin\mathrm{supp}(\nu)$. Note that
$\mathsf{T}^{n,-}\circ\mathsf{T}^{n,+}=\mathsf{Id}$ always holds, and
$\mathsf{T}^{n,+}\circ\mathsf{T}^{n,-}(\nu,x)=(\nu,x)$ whenever
$x\in\mathrm{supp}(\nu)$.

We further define the discrete $\Gamma_{n}$-gradients
$\overline{\nabla}^{n,\pm}:C_{c}(\Gamma_{n})\to
C_{c}(\Gamma_{n}\times\mathcal{T})$:

(3.9)
$(\overline{\nabla}^{n,\pm}F)(\nu,x):=n(F(\mathsf{T}_{x}^{n,\pm}\nu)-F(\nu)),$

and the corresponding $\Gamma_{n}$-divergence
$\overline{\text{div}}^{n,\pm}:\mathcal{M}_{loc}^{+}(\Gamma_{n}\times\mathcal{T})\to\mathcal{M}_{loc}(\Gamma_{n})$,
dual to $\overline{\nabla}^{n,\pm}$, given by

(3.10)
$(\overline{\text{div}}^{n,\pm}\mathsf{J})=n\left(\mathsf{p}^{\Gamma_{n}}_{\\#}\mathsf{J}-(\mathsf{p}^{\Gamma_{n}}\circ\mathsf{T}^{n,\pm})_{\\#}\mathsf{J}\right),$

where $\mathsf{p}^{\Gamma_{n}}:{\Gamma_{n}}\times\mathcal{T}\to\Gamma_{n}$
denotes the projection to the first variable.

We consider the familes of curves satisfying

($\mathsf{CE}_{n}$)
$\partial_{t}\mathsf{P}_{t}+(\overline{\text{div}}^{n,+}\mathsf{J}_{t}^{+})+(\overline{\text{div}}^{n,-}\mathsf{J}_{t}^{-})=0$

in the following appropriate distributional sense.

######  Definition 3.1 (Continuity equation).

A triple $(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})$ satisfies the continuity
equation $\mathsf{CE}_{n}$, if

  1. (1)

the curve $[0,T]\ni t\mapsto\mathsf{P}_{t}\in\mathcal{P}(\Gamma_{n})$ is
narrowly continuous,

  2. (2)

the Borel family
$(\mathsf{J}^{\pm}_{t})_{t\in[0,T]}\in\mathcal{M}^{+}_{loc}(\Gamma_{n}\times\mathcal{T})$
satisfies

$\mathrm{supp}(\mathsf{J}^{-}_{t})\subseteq\left\\{(\nu,x)\,:\,\nu(\mathcal{T})\geq\genfrac{}{}{}{2}{2}{n},\,x\in\mathrm{supp}(\nu)\right\\},$

  3. (3)

$\int_{0}^{T}\int_{\Gamma_{n}\times\mathcal{T}}(1+\nu(\mathcal{T})^{2})^{-1}\,\mathrm{d}\mathsf{J}^{\pm}_{t}\,\mathrm{d}t<\infty$,

  4. (4)

for every $s,t\in[0,T]$ and all $F\in C_{c}(\Gamma_{n})$

(3.11)
$\int_{\Gamma_{n}}F(\nu)\,\mathrm{d}\mathsf{P}_{t}-\int_{\Gamma_{n}}F(\nu)\,\mathrm{d}\mathsf{P}_{s}=\int_{s}^{t}\int_{\Gamma_{n}\times\mathcal{T}}\left((\overline{\nabla}^{n,+}F)\,\mathrm{d}\mathsf{J}_{r}^{+}+(\overline{\nabla}^{n,-}F)\,\mathrm{d}\mathsf{J}_{r}^{-}\right)\,\mathrm{d}r.$

Throughout we will call arbitrary measures
$\mathsf{J}^{\pm}\in\mathcal{M}^{+}_{loc}(\Gamma_{n}\times\mathcal{T})$
_admissible_ if

$\mathrm{supp}(\mathsf{J}^{-})\subseteq\left\\{(\nu,x)\,:\,\nu(\mathcal{T})\geq\genfrac{}{}{}{2}{2}{n},\,x\in\mathrm{supp}(\nu)\right\\}$

and

$\int_{\Gamma_{n}\times\mathcal{T}}(1+\nu(\mathcal{T})^{2})^{-1}\,\mathrm{d}\mathsf{J}^{\pm}<\infty.$

Moreover, since $\Gamma_{n}$ is a closed subspace of the Polish space
$\Gamma$, the extension of $\mathsf{P}$ to $\mathcal{P}(\Gamma)$ and the
extension of $\mathsf{J}^{\pm}$ to
$\mathcal{M}^{+}_{loc}(\Gamma\times\mathcal{T})$ are well-defined. For
simplicity we will simply refer to them as $\mathsf{P}$, $\mathsf{J}^{\pm}$ as
well, and drop the $n$-dependence in most arguments.

It is also clear that for any admissible $\mathsf{J}^{\pm}$

$(\overline{\nabla}^{n,\pm}F)(\nu,x):=n\left(F(\nu\pm\genfrac{}{}{}{2}{1}{n}\delta_{x})-F(\nu)\right),\qquad(\nu,x)\in\mathrm{supp}(\mathsf{J}^{\pm})$

and in particular (3.11) is equivalent to

$\int_{\Gamma}F(\nu)\,\mathrm{d}\mathsf{P}_{t}-\int_{\Gamma}F(\nu)\,\mathrm{d}\mathsf{P}_{s}=\int_{s}^{t}\int_{\Gamma\times\mathcal{T}}\Big{(}n\big{(}F(\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x})-F(\nu)\big{)}\,\mathrm{d}\mathsf{J}_{r}^{+}+n\big{(}F(\nu-\genfrac{}{}{}{2}{1}{n}\delta_{x})-F(\nu)\big{)}\,\mathrm{d}\mathsf{J}_{r}^{-}\Big{)}\,\mathrm{d}r.$

for all $F\in C_{c}(\Gamma)$. Note that this can again be extended to all
$F\in\mathcal{B}_{c}(\Gamma)$ via a monotone class argument.

######  Remark 3.2.

Condition (2) represents the restriction that particles can only be deleted if
there are at least two particles in the system, consistent with the fact that
$\mathsf{P}\in\mathcal{P}(\Gamma_{n})$ and hence the underlying process never
attains $\nu=0$.

Moreover, condition (3) reflects the unboundedness of the observed fluxes
$\mathsf{J}^{\pm}$, which stems from the unboundedness of the birth/death
kernels $\kappa^{\pm}_{\nu}$ in $\nu$.

######  Remark 3.3.

Whenever $\mathsf{J}^{\pm}$ are of the form

$\mathsf{J}_{t}^{\pm}(\mathrm{d}\nu,\mathrm{d}x)=\mathsf{P}_{t}(\mathrm{d}\nu)\lambda^{\pm}[t,\nu](\mathrm{d}x)$

with $\lambda^{\pm}[t,\nu]\in\mathcal{M}^{+}(\mathcal{T})$ for all
$\nu\in\Gamma$ and $t\in[0,T]$, the continuity equation (3.11) describes the
forward Kolmogorov equation corresponding to an interacting birth/death
process with the birth/death kernels $\lambda^{\pm}[t,\nu]$ depending on both
time and the empirical measure of the particles $\nu$. The time-dependent jump
kernel is then given by

$\bar{\kappa}_{n,t}(\mathrm{d}\nu,\mathrm{d}\eta)=n\left(\int_{\mathcal{T}}\delta_{\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x}}(\mathrm{d}\eta)\,\lambda^{+}[t,\nu](\mathrm{d}x)+\int_{\mathcal{T}}\delta_{\nu-\genfrac{}{}{}{2}{1}{n}\delta_{x}}(\mathrm{d}\eta)\,\lambda^{-}[t,\nu](\mathrm{d}x)\right).$

In order to define the dissipation potentials, let us introduce the measures
$\vartheta_{\mathsf{P}}^{\pm}\in\mathcal{M}_{loc}^{+}(\Gamma\times\mathcal{T})$

(3.12)
$\vartheta_{\mathsf{P}}^{\pm}(\mathrm{d}\nu\,\mathrm{d}x):=\mathsf{P}(\mathrm{d}\nu)\kappa^{\pm}_{\nu}(\mathrm{d}x).$

Note that for any curve $(\mathsf{P}_{t})_{t\in[0,T]}$ the measures
$\mathsf{J}_{t}^{\pm}:=\vartheta_{\mathsf{P}_{t}}^{\pm}$ satisfy the
conditions (2) and (3), where the latter holds because $c(x,x)=0$.

Moreover, as will be shown in Lemma 3.12, we have the following symmetry

(3.13)
$\vartheta_{\Pi_{n}}^{\pm}=\mathsf{T}^{n,\mp}_{\\#}\vartheta^{\mp}_{\Pi_{n}}.$

from which the detailed balance condition $\eqref{eq:fke_rev1}$ directly
follows.

######  Definition 3.4.

Let $\Theta^{n,\pm}_{\mathsf{P}}\in\mathcal{M}_{loc}(\Gamma\times\mathcal{T})$
be the geometric average of $\vartheta^{\pm}_{\mathsf{P}}$ and
$\mathsf{T}^{n,\mp}_{\\#}\vartheta^{\mp}_{\mathsf{P}}$, i.e.

(3.14)
$\Theta^{n,\pm}_{\mathsf{P}}(\mathrm{d}\nu,\mathrm{d}x):=\sqrt{\frac{\mathrm{d}\vartheta^{\pm}_{\mathsf{P}}}{\mathrm{d}\Sigma}\frac{\mathrm{d}(\mathsf{T}^{n,\mp}_{\\#}\vartheta^{\mp}_{\mathsf{P}})}{\mathrm{d}\Sigma}}\,\,\mathrm{d}\Sigma,$

for any dominating measure $\Sigma$.

The dissipation potential
$\mathcal{R}_{n}:\mathcal{P}(\Gamma)\times\mathcal{M}_{loc}^{+}(\Gamma\times\mathcal{T})^{2}\to[0,+\infty]$
and dual dissipation potential
$\mathcal{R}^{*}_{n}:\mathcal{P}(\Gamma)\times\mathcal{B}_{c}(\Gamma\times\mathcal{T})^{2}$
are given by

$\displaystyle\mathcal{R}_{n}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})$
$\displaystyle:=\mathcal{E}\mathrm{nt}(\mathsf{J}^{+}|\Theta^{n,+}_{\mathsf{P}})+\mathcal{E}\mathrm{nt}(\mathsf{J}^{-}|\Theta^{n,-}_{\mathsf{P}}),$
$\displaystyle\mathcal{R}_{n}^{*}(\mathsf{P},\omega^{+},\omega^{-})$
$\displaystyle:=\int_{\Gamma\times\mathcal{T}}(e^{\omega^{+}}-1)\,\mathrm{d}\Theta^{n,+}_{\mathsf{P}}+\int_{\Gamma\times\mathcal{T}}(e^{\omega^{-}}-1)\,\mathrm{d}\Theta^{n,-}_{\mathsf{P}}$

For the free energy $\mathcal{F}_{n}:\mathcal{P}(\Gamma)\to[0,+\infty]$ and
Fisher information $\mathcal{D}_{n}:\mathcal{P}(\Gamma)\to[0,+\infty]$

$\displaystyle\mathcal{F}_{n}(\mathsf{P})$
$\displaystyle:=\genfrac{}{}{}{2}{1}{2n}\mathcal{E}\mathrm{nt}(\mathsf{P}|\Pi_{n})$
$\displaystyle\mathcal{D}_{n}(\mathsf{P})$
$\displaystyle:=\left\\{\begin{aligned}
&H^{2}(\vartheta_{\mathsf{P}}^{+},\mathsf{T}_{\\#}^{n,-}\vartheta_{\mathsf{P}}^{-})+H^{2}(\vartheta_{\mathsf{P}}^{-},\mathsf{T}_{\\#}^{n,+}\vartheta_{\mathsf{P}}^{+})\qquad&&\mbox{if
$\mathsf{P}\ll\Pi_{n}$,}\\\ &+\infty&&\mbox{otherwise}.\\\
\end{aligned}\right.$

For the _EDP-functional_ $\mathcal{I}_{n}:\mathsf{CE}_{n}\to[0,+\infty]$ for
all curves with $\mathcal{F}_{n}(\mathsf{P}_{0})<\infty$

$\mathcal{I}_{n}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-}):=\int_{0}^{T}\mathcal{R}_{n}(\mathsf{P}_{t},\mathsf{J}_{t}^{+},\mathsf{J}^{-}_{t})\,\mathrm{d}t+\mathcal{F}_{n}(\nu_{T})-\mathcal{F}_{n}(\nu_{0})+\int_{0}^{T}\mathcal{D}_{n}(\mathsf{P}_{t})\,\mathrm{d}t.$

######  Remark 3.5.

The definition of $\Theta_{\mathsf{P}}^{n,\pm}$ is independent of the
dominating measure $\Sigma$. Moreover, formally

$\Theta_{\mathsf{P}}^{n,+}(\nu,x)=\sqrt{(\mathsf{P}(\nu)\kappa^{+}[\nu])(\mathsf{P}(\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x})\kappa^{-}[\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x}}]),$

i.e. it represents the geometric mean of the expected fluxes going forwards
and backwards along the transition
$\nu\leftrightarrow\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x}$.

In addition, due to the symmetry (3.13) the measures
$\Theta_{\mathsf{P}}^{n,\pm}$ simplify whenever $\mathsf{P}\ll\Pi_{n}$, i.e.
if $\mathrm{d}\mathsf{P}=U\mathrm{d}\Pi_{n}$ we have

$\Theta^{n,\pm}_{\mathsf{P}}(\mathrm{d}\nu,\mathrm{d}x)=\sqrt{U(\nu)U(\nu\pm\genfrac{}{}{}{2}{1}{n}\delta_{x})}\,\vartheta_{\Pi_{n}}^{\pm}(\mathrm{d}\nu,\mathrm{d}x).$

######  Remark 3.6.

Note that $\mathcal{D}_{n}$ is a jointly convex function in
$(\vartheta^{\pm}_{\mathsf{P}},\mathsf{T}_{\\#}^{n,\mp}\vartheta_{\mathsf{P}}^{\mp})$,
and lower semicontinuous if $\mathcal{F}_{n}$ is bounded. Moreover, it is
straightforward to check that whenever $\mathsf{P}\ll\Pi_{n}$ with
$\mathrm{d}\mathsf{P}=U\Pi_{n}$ it holds

$\displaystyle\mathcal{D}_{n}(\mathsf{P})$
$\displaystyle=\frac{1}{2}\int_{\Gamma\times\mathcal{T}}\left(\sqrt{U(\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x})}-\sqrt{U(\nu)}\right)^{2}\mathrm{d}\vartheta^{+}_{\Pi_{n}}+\frac{1}{2}\int_{\Gamma\times\mathcal{T}}\left(\sqrt{U(\nu-\genfrac{}{}{}{2}{1}{n}\delta_{x})}-\sqrt{U(\nu)}\right)^{2}\mathrm{d}\vartheta^{-}_{\Pi_{n}}$
$\displaystyle=\int_{\Gamma\times\mathcal{T}}\left(\sqrt{U(\nu\pm\genfrac{}{}{}{2}{1}{n}\delta_{x})}-\sqrt{U(\nu)}\right)^{2}\mathrm{d}\vartheta^{\pm}_{\Pi_{n}}.$

Finally, for technical purposes, we also introduce a version for net fluxes.

######  Definition 3.7.

The _upward_ net flux $\mathsf{J}^{\mathrm{net}}$ is defined as

$\mathsf{J}^{\mathrm{net}}:=\mathsf{J}^{+}-\mathsf{T}^{n,-}_{\\#}\mathsf{J}^{-}$

Note that $\mathsf{J}^{\mathrm{net}}(\nu,x)$ can be interpreted as the net
flux along the jump $\nu\leftrightarrow\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x}$.

The continuity equation for the net flux reduces to

$\int_{\Gamma}F(\nu)\,\mathrm{d}\mathsf{P}_{t}-\int_{\Gamma}F(\nu)\,\mathrm{d}\mathsf{P}_{s}=\int_{s}^{t}\int_{\Gamma\times\mathcal{T}}n\big{(}F(\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x})-F(\nu)\big{)}\,\mathrm{d}\mathsf{J}^{\mathrm{net}}_{r}\,\mathrm{d}r$

We are now in a position to give the general version of Theorem 1.6.

######  Theorem 3.8.

For any $(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})\in\mathsf{CE}_{n}$ with
$\mathcal{F}_{n}(\mathsf{P}_{0})<\infty$ we have
$\mathcal{I}_{n}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})\geq 0$,

(3.15)
$\mathcal{I}_{n}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})=0\implies\left\\{\begin{aligned}
\quad&\mbox{$\mathsf{P}_{t}$ is a weak solution to \eqref{eq:FKEn}}\quad\\\
\quad&\mathsf{J}^{\pm}_{t}=\mathsf{P}_{t}\kappa_{\nu}^{\pm}\quad\mbox{for
a.e.\ $t\in[0,T]$},\quad\end{aligned}\right.$

and there exist a unique gradient-flow solution, i.e. a curve $(\mathsf{P})$
such that
$\mathcal{I}_{n}(\mathsf{P},\mathsf{P}_{t}\kappa_{\nu}^{+},\mathsf{P}_{t}\kappa_{\nu}^{-})=0$.

Moreover, whenever $\mathcal{F}_{n}(\mathsf{P}_{0})<\infty$ and
$\mathcal{I}_{n}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})<\infty$, the chain
rule for $\mathcal{F}_{n}$ and the net flux holds holds:
$\mathcal{F}_{n}(\mathsf{P}_{t})$ is absolutely continuous and

$\frac{\mathrm{d}\,}{\mathrm{d}t}\mathcal{F}_{n}(\mathsf{P}_{t})=\frac{n}{2}\int_{\Gamma\times\mathcal{T}}(\log
U(\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x})-\log
U(\nu))\,\mathrm{d}\mathsf{J}^{\mathrm{net}}_{t}\,\mathrm{d}t,\qquad\mbox{for
a.e.\ $t\in[0,T]$.}$

The proof of Theorem 3.8 is postponed to Section 3.3, and follows from the
existence of a gradient-flow solution via EDP-convergence of a sequence of
regularized problems established in Section 3.2, and its uniqueness via a
convexity argument.

######  Remark 3.9.

Similar to the mean-field case, the non-negativity of $\mathcal{I}_{n}$ and
the identification of solutions to ($\sf MF$) as null-minimizers of
$\mathcal{I}_{n}$ is related to the formal equivalence

$\mathcal{I}_{n}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})=\int_{0}^{T}\mathcal{L}_{n}(\mathsf{P}_{t},\mathsf{J}^{+}_{t},\mathsf{J}^{-}_{t})\,\mathrm{d}t,$

where $\mathcal{L}_{n}$ is the so-called _Lagrangian_ given by

$\mathcal{L}_{n}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-}):=\mathcal{E}\mathrm{nt}(\mathsf{J}^{+}|\mathsf{P}\kappa_{\nu}^{+})+\mathcal{E}\mathrm{nt}(\mathsf{J}^{-}|\mathsf{P}\kappa_{\nu}^{-}).$

We discuss the implication of this relation in Appendix A.

######  Remark 3.10 (Net flux).

To show the existence of gradient-flow solutions in the sense of null-
minimizers of $\mathcal{I}_{n}$ we will have to jump from gradient-flow
solutions in the sense of [PRST22], see Theorem 3.20. The expressions for net-
fluxes are in fact contractions of those for one-way or uni-directional
fluxes, as discussed in Section A, which we use to show that the two notions
of gradient-flow solutions are equivalent.

###  3.1. A priori estimates

Below we will state the estimates and identities necessary to prove the chain
rule and establish the existence of solutions.

Recall that $\vartheta_{\mathsf{P}}^{\pm}$ satisfies the same restrictions
(Conditions (2) and (3)) as the fluxes $\mathsf{J}^{\pm}$. This is easily
verified, but since we will use it repeatedly let us state it here precisely.

######  Lemma 3.11.

For any $\mathsf{P}\in\mathcal{P}(\Gamma_{n})$

$\mathrm{supp}(\vartheta_{\mathsf{P}}^{-})\subseteq\left\\{(\nu,x)\,:\,\nu(\mathcal{T})\geq\genfrac{}{}{}{2}{2}{n},\,x\in\mathrm{supp}(\nu)\right\\}.$

In particular, for any $\omega\in C_{c}(\Gamma\times\mathcal{T})$

$\displaystyle\int_{\Gamma\times\mathcal{T}}\omega(\nu,x)\,\mathrm{d}(\mathsf{T}^{n,\pm}_{\\#}\vartheta_{\mathsf{P}}^{\pm})$
$\displaystyle=\int_{\Gamma\times\mathcal{T}}\omega(\nu\pm\genfrac{}{}{}{2}{1}{n}\delta_{x},x)\,\mathrm{d}\vartheta_{\mathsf{P}}^{\pm},$

and

$\mathsf{T}^{n,\mp}_{\\#}\circ\mathsf{T}^{n,\pm}_{\\#}\vartheta_{\mathsf{P}}^{\pm}=\vartheta_{\mathsf{P}}^{\pm}.$

Finally,

$\mathsf{T}^{n,\pm}_{\\#}\Theta^{n,\pm}_{\mathsf{P}}=\Theta^{n,\mp}_{\mathsf{P}}.$

The above identities allow us to prove the symmetry condition that implies the
detailed balance condition (3.7).

######  Lemma 3.12 (Detailed balance).

$\vartheta_{\Pi_{n}}^{\pm}=\mathsf{T}^{n,\mp}_{\\#}\vartheta^{\mp}_{\Pi_{n}}.$

###### Proof.

Fix an arbitrary $\omega\in C_{c}(\Gamma\times\mathcal{T})$, and for any
ordered collection of $N$ variables in $\mathcal{T}$ set
$\mathbf{x}^{N}:=(x_{1},\dots,x_{N})\in X^{N}$. We then have the following.

$\displaystyle\int_{\Gamma\times\mathcal{T}}\omega(\nu,x)\,\mathrm{d}\vartheta^{+}_{\Pi_{n}}$
$\displaystyle=\frac{1}{e^{n\gamma(\mathcal{T})}-1}\sum_{N=1}^{\infty}\frac{n^{N}}{N!}\int_{\mathcal{T}^{N}}\left(\int_{\mathcal{T}}\omega\left(L_{n}(\mathbf{x}^{N}),x\right)\kappa^{+}\left[L_{n}(\mathbf{x^{N}})\right](\mathrm{d}x)\right)\gamma^{\otimes
N}(\mathrm{d}\mathbf{x}^{N}),$
$\displaystyle\int_{\Gamma\times\mathcal{T}}\omega(\nu,x)\,\mathrm{d}(\mathsf{T}^{n,-}_{\\#}\vartheta^{-}_{\Pi_{n}})$
$\displaystyle=\frac{1}{e^{n\gamma(\mathcal{T})}-1}\sum_{N=1}^{\infty}\frac{n^{N}}{N!}\int_{\mathcal{T}^{N}}\left(\int_{\mathcal{T}}\omega\left(L_{n}(\mathbf{x}^{N})-\genfrac{}{}{}{2}{1}{n}\delta_{x},x\right)\kappa^{-}\left[L_{n}(\mathbf{x}^{N})\right](\mathrm{d}x)\right)\gamma^{\otimes
N}(\mathrm{d}\mathbf{x}^{N}).$

Since $\kappa^{-}[\genfrac{}{}{}{2}{1}{n}\delta_{y}]=0$ for any
$y\in\mathcal{T}$, the sum in the right-hand side of the last expression
starts from $N=2$, thus reducing the expression to

$\frac{1}{e^{n\gamma(\mathcal{T})}-1}\sum_{N=1}^{\infty}\frac{n^{N+1}}{(N+1)!}\int_{\mathcal{T}^{N+1}}\left(\int_{\mathcal{T}}\omega\left(L(\mathbf{x}^{N+1})-\genfrac{}{}{}{2}{1}{n}\delta_{x},x\right)\kappa^{-}\left[L_{n}(\mathbf{x}^{N+1})\right](\mathrm{d}x)\right)\gamma^{\otimes(N+1)}(\mathrm{d}\mathbf{x}^{N+1}),$

It is clear that, for our desired equality, it is enough to show that for
every $N$,

$\displaystyle n\int_{\mathcal{T}^{N+1}}$
$\displaystyle\left(\int_{\mathcal{T}^{2}}\omega\left(L_{n}(\mathbf{x}^{N+1})-\genfrac{}{}{}{2}{1}{n}\delta_{x},x\right)c(x,y)\,L_{n}(\mathbf{x}^{N+1})^{\otimes
2}(\mathrm{d}x,\mathrm{d}y)\right)\,\gamma^{\otimes(N+1)}(\mathrm{d}\mathbf{x}^{N+1})$
$\displaystyle=(N+1)\int_{\mathcal{T}^{N}}\left(\int_{X^{2}}\omega\left(L_{n}(\mathbf{x}^{N}),x\right)c(x,y)\gamma(\mathrm{d}x)L_{n}(\mathbf{x}^{N})(\mathrm{d}y)\right)\,\gamma^{\otimes
N}(\mathrm{d}\mathbf{x}^{N}).$

To do so, note that since $c(x,x)=0$,

$\displaystyle
n\int_{X^{2}}\omega\left(L_{n}(\mathbf{x}^{N+1})-\genfrac{}{}{}{2}{1}{n}\delta_{x},x\right)c(x,y)\,L_{n}(\mathbf{x}^{N+1})^{\otimes
2}(\mathrm{d}x,\mathrm{d}y)$ $\displaystyle\hskip
80.00012pt=\frac{1}{n}\sum_{i=1}^{N+1}\sum_{j\neq
i}\omega\left(L_{n}(\mathbf{x}^{N+1})-\genfrac{}{}{}{2}{1}{n}\delta_{x_{i}},x_{i}\right)c(x_{i},x_{j})$
$\displaystyle\hskip
80.00012pt=\sum_{i=1}^{N+1}\int_{\mathcal{T}}\omega\left(L_{n}(\mathbf{x}^{N+1})-\genfrac{}{}{}{2}{1}{n}\delta_{x_{i}},x_{i}\right)c(x_{i},y)\left(L_{n}(\mathbf{x}^{N+1})-\genfrac{}{}{}{2}{1}{n}\delta_{x_{i}}\right)(\mathrm{d}y).$

Hence, by symmetry of $\gamma^{\otimes(N+1)}$, we obtain

$\displaystyle\int_{\mathcal{T}^{N+1}}$
$\displaystyle\left(\sum_{i=1}^{N+1}\int_{\mathcal{T}}\omega\left(L_{n}(\mathbf{x}^{N+1})-\genfrac{}{}{}{2}{1}{n}\delta_{x_{i}},x_{i}\right)c(x_{i},y)\left(L_{n}(\mathbf{x}^{N+1})-\genfrac{}{}{}{2}{1}{n}\delta_{x_{i}}\right)(\mathrm{d}y)\right)\gamma^{\otimes(N+1)}(\mathrm{d}\mathbf{x}^{N+1})$
$\displaystyle=(N+1)\int_{\mathcal{T}^{N}}\left(\int_{\mathcal{T}^{2}}\omega\left(L_{n}(\mathbf{x}^{N}),x\right)c(x,y)\gamma(\mathrm{d}x)L_{n}(\mathbf{x}^{N})(\mathrm{d}y)\right)\gamma^{\otimes
N}(\mathrm{d}\mathbf{x}^{N}),$

as desired. ∎

Recall from Lemma 2.9 that that

$\kappa^{\pm}_{\nu}(\mathcal{T})\leq M(1+\nu(\mathcal{T})^{2}),$

where $M:=(1+\gamma(\mathcal{T}))\|c\|_{\infty}$. Now let

$M_{n}:=\max\bigl{\\{}1+2/n^{2},2\bigr{\\}}M,$

and the jointly convex and lower semicontinuous function
$\Upsilon:\mathbb{R}_{\geq 0}^{3}\to[0,+\infty]$ given by

$\Upsilon(w,u,v):=\begin{cases}\sqrt{uv}&\quad\mbox{if $w=0$,}\\\
\phi\left(\frac{w}{\sqrt{uv}}\right)\sqrt{uv}&\quad\mbox{if $u,v>0$,}\\\
+\infty&\quad\mbox{if $w>0$, and either $u=0$ or $v=0$.}\end{cases}$

We then have the following result.

######  Lemma 3.13.

The following statements hold:

  1. (i)

For all $\mathsf{P}$

$\int_{\Gamma\times\mathcal{T}}(1+\nu(\mathcal{T}))^{-2}\,\Theta_{\mathsf{P}}^{n,\pm}(\mathrm{d}\nu\,\mathrm{d}y)\leq\int_{\Gamma\times\mathcal{T}}(1+\nu(\mathcal{T})^{2})^{-1}\,\Theta_{\mathsf{P}}^{n,\pm}(\mathrm{d}\nu\,\mathrm{d}y)\leq
M_{n}.$

  2. (ii)

For any $\mathsf{P}$, admissible $\mathsf{J}^{\pm}$, and net flux
$\mathsf{J}^{\mathrm{net}}=\mathsf{J}^{+}-\mathsf{T}^{n,-}_{\\#}\mathsf{J}^{-}$,
$\omega\in\mathcal{B}(\Gamma\times\mathcal{T})$, we have

$\int_{\Gamma\times\mathcal{T}}|\omega|\,\mathrm{d}|\mathsf{J}^{\mathrm{net}}|\,\leq\mathcal{R}_{n}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})+\int_{\Gamma\times\mathcal{T}}\Psi^{*}(\omega)\,\mathrm{d}\Theta_{\mathsf{P}}^{n,+}.$

Moreover,

(3.16a)
$\displaystyle\phi\left(1\vee\frac{1}{M_{n}}\int_{\Gamma\times\mathcal{T}}(1+\nu(X)^{2})^{-1}\,\mathsf{J}^{\pm}(\mathrm{d}\nu,\mathrm{d}x)\right)M_{n}\,\leq\mathcal{R}_{n}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-}),$
(3.16b)
$\displaystyle\Psi\left(\frac{1}{M_{n}}\int_{\Gamma\times\mathcal{T}}(1+\nu(X))^{-1}\,|\mathsf{J}^{\mathrm{net}}|(\mathrm{d}\nu,\mathrm{d}x)\right)M_{n}\,\leq\mathcal{R}_{n}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-}).$

  3. (iii)

For all admissible $\mathsf{P},\mathsf{J}^{\pm}$,

(3.17)
$\mathcal{E}\mathrm{nt}(\mathsf{J}^{\pm}|\Theta^{n,\pm})=\int_{\Gamma\times\mathcal{T}}\Upsilon\left(\frac{\mathrm{d}\mathsf{J}^{\pm}}{\mathrm{d}\Sigma},\frac{\mathrm{d}\vartheta_{\mathsf{P}}^{\pm}}{\mathrm{d}\Sigma},\frac{\mathrm{d}(\mathsf{T}^{n,\mp}_{\\#}\vartheta_{\mathsf{P}}^{\mp})}{\mathrm{d}\Sigma}\right)\mathrm{d}\Sigma,$

for any common dominating measure $\Sigma$. Moreover, if
$\mathrm{d}\mathsf{P}=U\mathrm{d}\Pi_{n}$,

$\mathcal{E}\mathrm{nt}(\mathsf{J}^{\pm}|\Theta^{n,\pm})=\int_{\Gamma\times\mathcal{T}}\Upsilon\left(\frac{\mathrm{d}\mathsf{J}^{\pm}}{\mathrm{d}\vartheta_{\mathsf{P}}^{\pm}},U(\nu),U(\nu\pm\genfrac{}{}{}{2}{1}{n}\delta_{x})\right)\mathrm{d}\vartheta^{\pm}_{\mathsf{P}}.$

######  Remark 3.14.

Since $M_{n}\leq 3M$ for all $n\geq 1$ the estimates (3.16) are uniform in
$n$, which we will use in the EDP-convergence to establish tightness of
sequences $\mathsf{J}^{n,\pm}$ under bound on $\mathcal{I}_{n}$. Moreover, the
representation (3.17) is used to deduce the lower-semicontinuity of
$\mathcal{I}_{n}$ for sequences of curves.

###### Proof.

_(i)_ For any $x^{*}\in\mathcal{T}$, $\nu\in\Gamma$, we have

$\displaystyle\max\\{\kappa^{\pm}(\mathcal{T}),\kappa^{\pm}[\mathsf{T}^{n,\pm}_{x^{*}}(\nu)](\mathcal{T})\\}$
$\displaystyle\leq
M\max\left\\{1+\nu(\mathcal{T})^{2},\,1+(\mathsf{T}^{n,+}_{x^{*}}(\nu))(\mathcal{T})^{2},\,1+(\mathsf{T}^{n,-}_{x^{*}}(\nu))(\mathcal{T})^{2}\right\\}$
$\displaystyle\leq M_{n}(1+\nu(\mathcal{T})^{2})$

due to the inequality

$1+(\genfrac{}{}{}{2}{1}{n}+z)^{2}\leq
1+\genfrac{}{}{}{2}{2}{n^{2}}+2z^{2},\qquad\mbox{ for all }z\geq 0.$

In particular,

$\max\left\\{\int_{\Gamma\times\mathcal{T}}(1+\nu(\mathcal{T})^{2})^{-1}\mathrm{d}\vartheta_{\mathsf{P}}^{\pm},\int_{\Gamma\times\mathcal{T}}(1+\nu(\mathcal{T})^{2})^{-1}\mathrm{d}\mathsf{T}^{n,\mp}_{\\#}\vartheta_{\mathsf{P}}^{\pm}\right\\}\leq
M_{n},$

and hence the desired statement follows after applying Jensen’s inequality.

_(ii)_ By duality we have for any
$\omega\in\mathcal{B}_{c}(\Gamma\times\mathcal{T})$,

$\displaystyle\int_{\Gamma\times\mathcal{T}}\omega^{+}\,\mathrm{d}\mathsf{J}^{+}+\int_{\Gamma\times\mathcal{T}}\omega^{-}\,\mathrm{d}\mathsf{J}^{-}\leq\mathcal{R}_{n}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})+\int_{\Gamma\times\mathcal{T}}(e^{\omega^{+}}-1)\,\mathrm{d}\Theta_{\mathsf{P}}^{n,+}+\int_{\Gamma\times\mathcal{T}}(e^{\omega^{-}}-1)\,\mathrm{d}\Theta_{\mathsf{P}}^{n,-}.$

Substituting $\omega^{+}=\omega$, $\omega^{-}=-\omega\circ\mathsf{T}^{n,-}$
and using the fact that
$\mathsf{T}^{n,-}_{\\#}\Theta^{n,-}_{\mathsf{P}}=\Theta^{n,+}_{\mathsf{P}}$ we
derive

$\displaystyle\int_{\Gamma\times\mathcal{T}}\omega\,\mathrm{d}\mathsf{J}^{\mathrm{net}}\leq\mathcal{R}_{n}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})+\int_{\Gamma\times\mathcal{T}}\Psi^{*}(\omega)\,\mathrm{d}\Theta_{\mathsf{P}}^{n,+}.$

Since $\Psi^{*}$ is even we can replace $\omega$ and $\mathsf{J}$ by their
absolutes in the inequality, after substituting for $\omega$ appropriately,
and we conclude with a monotone convergence argument. The inequalities (3.16a)
and (3.16b) now follow similarly as in Lemma 2.9 via respectively Jensen’s
inequality and a dual approach.

_(iii)_ Let us only consider $\mathsf{J}^{+}$, $\Theta^{n,+}$ (the case for
$\mathsf{J}^{-}$, $\Theta^{n,-}$ is similar). Suppose
$\mathcal{E}\mathrm{nt}(\mathsf{J}^{+}|\Theta^{n,+})<\infty$ and recall that

$\Theta^{n,+}_{\mathsf{P}}(\mathrm{d}\nu,\mathrm{d}x):=\sqrt{\frac{\mathrm{d}\vartheta^{+}_{\mathsf{P}}}{\mathrm{d}\Sigma}\frac{\mathrm{d}(\mathsf{T}^{n,-}_{\\#}\vartheta^{-}_{\mathsf{P}})}{\mathrm{d}\Sigma}}\,\,\mathrm{d}\Sigma,$

where $\Sigma$ is a dominating measure, e.g
$\Sigma=\vartheta^{+}_{\mathsf{P}}+\mathsf{T}^{n,-}_{\\#}\vartheta^{-}_{\mathsf{P}}$.
Then $\mathsf{J}^{+}\ll\Theta_{\mathsf{P}}^{n,+}\ll\Sigma$, and it follows
that $\mathsf{J}^{+}$-a.e.
$\mathrm{d}\vartheta_{\mathsf{P}}^{+}/\mathrm{d}\Sigma$,
$\mathrm{d}(\mathsf{T}^{n,-}_{\\#}\vartheta_{\mathsf{P}}^{-})/\mathrm{d}\Sigma>0$,
from which one can easily verifies (3.17).

Vice versa, suppose that

$\int_{\Gamma\times\mathcal{T}}\Upsilon\left(\frac{\mathrm{d}\mathsf{J}^{+}}{\mathrm{d}\Sigma},\frac{\mathrm{d}\vartheta_{\mathsf{P}}^{+}}{\mathrm{d}\Sigma},\frac{\mathrm{d}(\mathsf{T}^{n,-}_{\\#}\vartheta_{\mathsf{P}}^{-})}{\mathrm{d}\Sigma}\right)\mathrm{d}\Sigma<\infty,$

for some dominating measure $\Sigma$. Then again $\mathsf{J}^{+}$-a.e. we have
that $\mathrm{d}\vartheta_{\mathsf{P}}^{+}/\mathrm{d}\Sigma$,
$\mathrm{d}(\mathsf{T}^{n,-}_{\\#}\vartheta_{\mathsf{P}}^{-})/\mathrm{d}\Sigma>0$,
and by super-linearity of $\phi$ deduce that in fact
$\mathsf{J}^{+}\ll\tilde{\Sigma}$ for any dominating measure of
$\vartheta_{\mathsf{P}}^{+}$ and
$\mathsf{T}^{n,-}_{\\#}\vartheta_{\mathsf{P}}^{-}$, which together implies
$\mathsf{J}^{+}\ll\Theta_{\mathsf{P}}^{n,+}$ and the result follows similarly
as above. ∎

Finally, we discuss the time-regularity of $\mathsf{P}_{t}$ for admissible
curves and state the analog of Lemma 2.11. Let the weighted total variation
metric $d_{TV,w}$ be given as

(3.18) $\displaystyle d_{TV,w}(\mathsf{P}^{1},\mathsf{P}^{2})$
$\displaystyle:=\int_{\Gamma}(1+\nu(\mathcal{T})^{2})^{-1}\,\mathrm{d}|\mathsf{P}^{1}-\mathsf{P}^{2}|.$

Note that $d_{TV,w}$ is lower semicontinuous with respect to the narrow
topology, and while convergence in $d_{TV,w}$ does not directly imply narrow
convergence, it does so on narrowly pre-compact sets.

######  Lemma 3.15.

For any $(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})\in\mathsf{CE}_{n}$ we have
$\mbox{ for all }s,t\in[0,T]$:

(3.19) $d_{TV,w}(\mathsf{P}_{s},\mathsf{P}_{t})\leq
4n\max\Bigl{\\{}1+\genfrac{}{}{}{2}{2}{n^{2}},2\Bigr{\\}}\int_{s}^{t}\int_{\Gamma\times\mathcal{T}}(1+\nu(\mathcal{T})^{2})^{-1}\mathrm{d}(\mathsf{J}_{r}^{+}+\mathsf{J}^{-}_{r})\,\mathrm{d}r.$

Suppose in addition that
$\mathsf{P}_{t}\ll\Pi_{n},\mathsf{J}^{\pm}_{t}\ll\vartheta_{\Pi_{n}}^{\pm}$
for all $t\in[0,T]$ and set

$\displaystyle\ell:=(1+\nu(\mathcal{T})^{2})^{-1}\Pi_{n},\qquad\Sigma^{\pm}:=(1+\nu(\mathcal{T})^{2})^{-1}\vartheta^{\pm}_{\Pi_{n}}.$

Then there exists an absolutely continuous and a.e. differentiable map
$U:[a,b]\to L^{1}(\mathcal{P}(\Gamma),\ell)$ and maps $G^{\pm}:[0,T]\to
L^{1}(\Sigma^{\pm})$ such that
$U_{t}=\mathrm{d}\mathsf{P}_{t}/\mathrm{d}\Pi_{n}$,
$G_{t}^{\pm}=\mathrm{d}\mathsf{J}^{\pm}/\mathrm{d}\vartheta_{\Pi_{n}}^{\pm}$,
and

(3.20) $\displaystyle\partial_{t}U_{t}(\nu)$
$\displaystyle=n\int_{\mathcal{T}}(G_{t}^{-}(\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x},x)-G_{t}^{+}(\nu,x))\,\kappa^{+}_{\nu}(\mathrm{d}x)$
$\displaystyle\hskip
40.00006pt+n\int_{\mathcal{T}}(G_{t}^{+}(\nu-\genfrac{}{}{}{2}{1}{n}\delta_{x},x)-G_{t}^{-}(\nu,x))\,\kappa^{-}_{\nu}(\mathrm{d}x).$

Alternatively, in terms of the net-flux
$\mathsf{J}^{\mathrm{net}}=G\vartheta_{\mathsf{P}}^{+}$ with
$G^{\mathrm{net}}:=G^{+}-G^{-}\circ\mathsf{T}^{n,+}$,

$\displaystyle\partial_{t}U_{t}(\nu)=n\int_{\mathcal{T}}G^{\mathrm{net}}_{t}(\nu-\genfrac{}{}{}{2}{1}{n}\delta_{x},x)\,\kappa^{-}_{\nu}(\mathrm{d}x)-n\int_{\mathcal{T}}G^{\mathrm{net}}_{t}(\nu,x)\,\kappa^{+}_{\nu}(\mathrm{d}x).$

######  Remark 3.16.

Note that the estimate (3.19) for the weighted total variation metric blows up
as $n\to\infty$. For the proof of EDP-convergence we instead use a weaker
metric, the transportation-like metric $W$ defined by (4.4), which does behave
uniform-in-$n$ for a sequence of curves with finite
$\limsup_{n\to\infty}\mathcal{I}_{n}$.

###### Proof.

Due to the continuity equation and after a monotone class argument, we have
the crude estimate

$\left|\int_{\Gamma}F\mathrm{d}(\mathsf{P}_{t}-\mathsf{P}_{s})\right|\leq
n\int_{s}^{t}\left[\int_{\Gamma\times\mathcal{T}}(|F(\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x})|+|F(\nu)|)\,\mathrm{d}\mathsf{J}_{r}^{+}+\int_{\Gamma\times\mathcal{T}}(|F(\nu-\genfrac{}{}{}{2}{1}{n}\delta_{x})|+|F(\nu)|)\,\mathrm{d}\mathsf{J}_{r}^{-}\right]\mathrm{d}r,$

for any $F\in\mathcal{B}_{c}(\Gamma)$. Now fix $F\in\mathcal{B}_{c}(\Gamma)$,
and let $K:=\sup_{\nu\in\Gamma}F(\nu)(1+\nu(\mathcal{T})^{2})$. Note that by
the bounds of Lemma 3.13 for any $\nu\in\Gamma_{n}$, we have the following
estimates

$\displaystyle|F|(\nu)$ $\displaystyle\leq K(1+\nu(\mathcal{T})^{2})^{-1}$
$\displaystyle|F|(\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x})$ $\displaystyle\leq
K\big{(}1+(\genfrac{}{}{}{2}{1}{n}+\nu(\mathcal{T}))^{2}\big{)}^{-1}\leq
K(1+\nu(\mathcal{T})^{2})^{-1},$
$\displaystyle|F|(\nu-\genfrac{}{}{}{2}{1}{n}\delta_{x})$ $\displaystyle\leq
K\big{(}1+(\genfrac{}{}{}{2}{-1}{n}+\nu(\mathcal{T}))^{2}\big{)}^{-1}\leq
K\max\\{1+\genfrac{}{}{}{2}{2}{n^{2}},2\\}(1+\nu(\mathcal{T})^{2})^{-1},$

and therefore

$\left|\int_{\Gamma}F\mathrm{d}(\mathsf{P}_{t}-\mathsf{P}_{s})\right|\leq
4nK\max\Bigl{\\{}1+\genfrac{}{}{}{2}{2}{n^{2}},2\Bigr{\\}}\int_{s}^{t}\int_{\Gamma\times\mathcal{T}}(1+\nu(\mathcal{T})^{2})^{-1}\mathrm{d}(\mathsf{J}^{+}_{r}+\mathsf{J}^{-}_{r})\,\mathrm{d}r.$

Taking the supremum over all $F\in\mathcal{B}_{c}(\Gamma)$ with
$\sup_{\nu\in\Gamma}F(\nu)(1+\nu(\mathcal{T})^{2})\leq 1$ we conclude that

$\displaystyle d_{TV,w}(\mathsf{P}_{s},\mathsf{P}_{t})$
$\displaystyle=\int_{\Gamma}(1+\nu(\mathcal{T})^{2})^{-1}\,\mathrm{d}|\mathsf{P}_{t}^{1}-\mathsf{P}_{s}^{2}|$
$\displaystyle\leq
4n\max\Bigl{\\{}1+\genfrac{}{}{}{2}{2}{n^{2}},2\Bigr{\\}}\int_{s}^{t}(1+\nu(\mathcal{T})^{2})^{-1}(\mathrm{d}\mathsf{J}^{+}_{r}+\mathsf{J}^{-}_{r})\,\mathrm{d}r.$

Next, suppose that
$\mathsf{P}_{t}\ll\Pi_{n},\mathsf{J}^{\pm}_{t}\ll\vartheta_{\Pi_{n}}^{\pm}$
for all $t\in[0,T]$. Let $U_{t}=\mathrm{d}\mathsf{P}_{t}/\mathrm{d}\Pi_{n}$,
$G_{t}^{\pm}=\mathrm{d}\mathsf{J}^{\pm}/\mathrm{d}\vartheta_{\Pi_{n}}^{\pm}$.
Note that by the absolutely continuity of $\mathsf{P}_{t}$ with respect to
$d_{TV,w}$, the map $t\mapsto U_{t}$ is absolutely continuous in
$L^{1}(\ell)$. Moreover, for every $F\in\mathcal{B}_{c}(\Gamma)$ the
continuity equation reads as

$\displaystyle\int_{\Gamma}F(U_{t}-U_{s})\,\mathrm{d}\Pi_{n}$
$\displaystyle=\int_{s}^{t}\int_{\Gamma\times\mathcal{T}}(F(\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x})-F(\nu))G_{r}^{+}(\nu,x)\,\mathrm{d}\vartheta_{\Pi_{n}}^{+}\,\mathrm{d}r$
$\displaystyle\hskip
20.00003pt+\int_{s}^{t}\int_{\Gamma\times\mathcal{T}}(F(\nu-\genfrac{}{}{}{2}{1}{n}\delta_{x})-F(\nu))G_{r}^{-}(\nu,x)\,\mathrm{d}\vartheta_{\Pi_{n}}^{-}\,\mathrm{d}r.$

But due to Lemma 3.12, the integrands can be rewritten as follows

$\displaystyle\int_{\Gamma\times\mathcal{T}}F(\nu\pm\genfrac{}{}{}{2}{1}{n}\delta_{x})G_{r}^{\pm}(\nu,x)\,\mathrm{d}\vartheta_{\Pi_{n}}^{\pm}$
$\displaystyle=\int_{\Gamma\times\mathcal{T}}F(\nu)G_{r}^{\pm}(\nu\mp\genfrac{}{}{}{2}{1}{n}\delta_{x},x)\,\mathrm{d}(\mathsf{T}^{n,\pm}_{\\#}\vartheta_{\Pi_{n}}^{\pm})$
$\displaystyle=\int_{\Gamma\times\mathcal{T}}F(\nu)G_{r}^{\pm}(\nu\mp\genfrac{}{}{}{2}{1}{n}\delta_{x},x)\,\mathrm{d}\vartheta_{\Pi_{n}}^{\mp},$

and therefore

$\displaystyle\int_{\Gamma}F(U_{t}-U_{s})\,\mathrm{d}\Pi_{n}$
$\displaystyle=\int_{s}^{t}\int_{\Gamma\times\mathcal{T}}F(\nu)(G_{r}^{-}(\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x},x)-G_{r}^{+}(\nu,x))\kappa^{+}_{\nu}(\mathrm{d}x)\,\mathrm{d}\Pi_{n}(\mathrm{d}\nu)\,\,\mathrm{d}r$
$\displaystyle\hskip
20.00003pt+\int_{s}^{t}\int_{\Gamma\times\mathcal{T}}F(\nu)(G_{r}^{+}(\nu-\genfrac{}{}{}{2}{1}{n}\delta_{x},x)-G_{r}^{-}(\nu,x))\,\kappa^{-}_{\nu}(\mathrm{d}x)\,\mathrm{d}\Pi_{n}(\mathrm{d}\nu)\,\mathrm{d}r,$

which is the weak formulation of (3.20). Putting in the pre-factors
$(1+\nu(\mathcal{T})^{2})^{-1}$ to state the expression in terms of the finite
measures $\ell$ and $\Sigma$, and noting that due to time-regularity
$(1+\nu(\mathcal{T})^{2})^{-1}\mathsf{P}_{t}$ is TV-regular, we can proceed as
in Corollary 4.14 of [PRST22] and conclude the proof after redefining
$U,G^{\pm}$ on negligible sets. ∎

###  3.2. Weak solutions

In this section we will discuss the existence of weak solutions to ($\sf
FKE_{n}$), i.e. solutions to

$\partial_{t}\mathsf{P}=\overline{\text{div}}^{n,+}\vartheta_{\mathsf{P}}^{+}+\overline{\text{div}}^{n,-}\vartheta_{\mathsf{P}}^{-},$

in appropriate weak form, but with the property that
$\mathcal{I}_{n}(\mathsf{P},\vartheta_{\mathsf{P}}^{+},\vartheta_{\mathsf{P}}^{-})\leq
0$. In the next section we will show that $\mathcal{I}_{n}\geq 0$ and that
gradient-flow solutions, i.e. those with $\mathcal{I}_{n}=0$, are in fact
unique.

######  Definition 3.17.

A curve $(\mathsf{P}_{t})_{t\in[0,T]}$ is a weak solution to ($\sf FKE_{n}$)
if $\mathrm{supp}\,\mathsf{P}_{t}\in\Gamma_{n}$ for all $t\in[0,T]$,
$\mathsf{P}_{t}$ is continuous in the narrow topology and for all
$s,t\in[0,T]$, and all $F\in C_{c}(\Gamma)$,

$\int_{\Gamma}F(\nu)\,\mathrm{d}\mathsf{P}_{t}-\int_{\Gamma}F(\nu)\,\mathrm{d}\mathsf{P}_{s}=\int_{s}^{t}\int_{\Gamma\times\mathcal{T}}\left((\overline{\nabla}^{n,+}F)\,\mathrm{d}\vartheta_{\mathsf{P}_{t}}^{+}+(\overline{\nabla}^{n,-}F)\,\mathrm{d}\vartheta_{\mathsf{P}_{t}}^{-}\right)\,\mathrm{d}r.$

######  Remark 3.18.

Recall that
$\int(1+\nu(X)^{2})\,\mathrm{d}\vartheta_{\mathsf{P}_{t}}^{\pm}\leq M_{n}$
independently of $\mathsf{P}_{t}$. Hence it is easy to check that
$(\mathsf{P})$ is a weak solution if and only if
$(\mathsf{P},\vartheta_{\mathsf{P}}^{+},\vartheta_{\mathsf{P}}^{-})\in\mathsf{CE}_{n}$.

Moreover, solutions turn out to inherit polynomial mass-estimates from the
initial condition, see e.g. Theorem 3.1 of [FM04] for the case in
$\mathbb{R}^{d}$. While throughout we do not assume more from the initial
condition than having finite entropy with respect to $\Pi_{n}$ (which does
imply the finiteness of the first moment) and unfortunately arbitrary curves
$(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})$ with finite $\mathcal{I}_{n}$ do
not preserve moment estimates, we will include the statement for completeness.

######  Lemma 3.19.

Fix any $p\geq 0$, and assume that $(\mathsf{P})$ is a weak solution with
initial datum satisfying

$\int_{\Gamma}\nu(X)^{p}\,\mathsf{P}_{0}(\mathrm{d}\nu)<\infty.$

Then

$\sup_{t\in[0,T]}\int_{\Gamma}\nu(X)^{p}\,\mathsf{P}_{t}(\mathrm{d}\nu)<\infty.$

###### Proof.

Set $F(\nu):=f(\nu(X))$ with $f(z):=z^{p}$ and let $F_{k}(\nu):=f_{k}(\nu(X))$
be its sequence of truncations. Then for every $k$,

$\displaystyle\int_{\Gamma}F_{k}\,\mathrm{d}(\mathsf{P}_{t}-\mathsf{P}_{0})$
$\displaystyle=\int_{0}^{t}\left(\int_{\Gamma\times\mathcal{T}}\overline{\nabla}^{n,+}F_{k}\,\mathrm{d}\vartheta_{\mathsf{P}_{r}}^{+}+\int_{\Gamma\times\mathcal{T}}\overline{\nabla}^{n,-}F\mathrm{d}\vartheta_{\mathsf{P}_{r}}^{-}\right)\,\mathrm{d}r$
$\displaystyle\leq\int_{0}^{t}\int_{\Gamma\times\mathcal{T}}\big{(}f_{k}(\nu(X)+\genfrac{}{}{}{2}{1}{n})-f_{k}(\nu(X))\big{)}\,\mathsf{P}_{r}(\mathrm{d}\nu)\,\kappa^{+}_{\nu}(\mathrm{d}x)\,\mathrm{d}r,$

since $f_{k}$ is non-decreasing and hence $\overline{\nabla}^{n,-}F_{k}\leq
0$. Moreover, note that $z(f(z+\genfrac{}{}{}{2}{1}{n})-f(z))\leq
C_{p,n}\,f(z)$ for a suitable constant $C_{p,n}$, and by monotonicity and non-
negativeness of $f$ the same inequality holds for the truncations $f_{k}$. By
a Gronwall-type argument we then obtain

$\int_{\Gamma}F_{k}(\nu)\,\mathrm{d}\mathsf{P}_{t}\leq
e^{tK_{p,n}}\int_{\Gamma}F_{k}(\nu)\,\mathrm{d}\mathsf{P}_{0}\leq
e^{tK}\int_{\Gamma}F(\nu)\,\mathrm{d}\mathsf{P}_{0},$

with the constant $K_{p,n}=C_{p,n}\|c\|_{\infty}\gamma(X)$ independent of $k$.
Taking $k\to\infty$ we derive the desired inequality by monotone convergence.
∎

We can now state the existence result of a weak solution satisfying one-half
of the Energy-Dissipation principle, which is complemented by the chain rule
proved in Section 3.3. The existence proof is one of EDP-convergence (see also
Section 5), bootstrapping from problems with bounded kernels and the results
of [PRST22].

######  Theorem 3.20.

Suppose that

$\mathcal{E}\mathrm{nt}(\bar{\mathsf{P}}|\Pi_{n})<\infty.$

Then there exist a weak solution $(\mathsf{P})$ with initial datum
$\bar{\mathsf{P}}$ such that

$\int_{0}^{T}\mathcal{R}_{n}(\mathsf{P}_{t},\vartheta_{\mathsf{P}_{t}}^{+},\vartheta_{\mathsf{P}_{t}}^{-})\,\mathrm{d}t+\mathcal{F}_{n}(\mathsf{P}_{T})-\mathcal{F}_{n}(\mathsf{P}_{0})+\int_{0}^{T}\mathcal{D}_{n}(\mathsf{P}_{t})\,\mathrm{d}t\leq
0.$

###### Proof.

Fix any $\bar{\mathsf{P}}$ with
$\mathcal{E}\mathrm{nt}(\bar{\mathsf{P}}|\Pi_{n})<\infty$. We proceed by
approximating the unbounded kernel $\bar{\kappa}_{n}$ with bounded ones. For
every $\varepsilon>0$, we introduce the regularized jump kernel
$\bar{\kappa}_{n}^{\varepsilon}(\nu,\mathrm{d}\eta)$ over $\Gamma$ defined by

$\bar{\kappa}_{n}^{\varepsilon}(\nu,\mathrm{d}\eta):=\frac{1}{1+\varepsilon\nu(\mathcal{T})\eta(\mathcal{T})}\bar{\kappa}_{n}(\nu,\mathrm{d}\eta).$

In terms of birth/death kernels this can be rewritten as

$\bar{\kappa}_{n}^{\varepsilon}(\nu,\mathrm{d}\eta)=n\int_{\mathcal{T}}\delta_{\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x}}(\mathrm{d}\eta)\,\kappa^{+,\varepsilon}_{\nu}(\mathrm{d}x)+n\int_{\mathcal{T}}\delta_{\nu-\genfrac{}{}{}{2}{1}{n}\delta_{x}}(\mathrm{d}\eta)\,\kappa^{-,\varepsilon}_{\nu}(\mathrm{d}x),$

where

$\kappa^{\pm,\varepsilon}_{\nu}:=\frac{1}{1+\varepsilon\nu(\mathcal{T})(\nu(\mathcal{T})\pm\genfrac{}{}{}{2}{1}{n})}\kappa^{\pm}_{\nu}.$

Note that

(3.21) $\sup_{\nu\in\Gamma}\kappa^{\pm,\varepsilon}_{\nu}<\infty\qquad\mbox{
for all }\varepsilon>0.$

Correspondingly, we denote $\vartheta_{\mathsf{P}}^{\pm,\varepsilon}$,
$\Theta^{n,\pm,\varepsilon}_{\mathsf{P}}$, $Q_{n,\varepsilon}$,
$Q_{n,\varepsilon}^{*}$, $\mathcal{R}_{n,\varepsilon}$,
$\mathcal{D}_{n,\varepsilon}$, $\mathcal{I}_{n,\varepsilon}$,
$(\mathsf{FKE}_{n,\varepsilon})$ as the relevant quantities, operators,
functionals and forward Kolmogorov equations induced by
$\kappa^{\pm,\varepsilon}_{\nu}$. We will first show existence of gradient-
flow solutions for the regularized problems, i.e. curves such that
$\mathcal{I}_{n,\varepsilon}=0$, and then construct an appropriate limit curve
as $\varepsilon\to 0$.

Thus, fix any $\varepsilon>0$. Due to the bound (3.21) it is clear that
$Q_{n,\varepsilon}$ is a bounded operator since

$\sup_{\nu\in\Gamma}\int_{\Gamma}\bar{\kappa}_{n}^{\varepsilon}(\nu,\mathrm{d}\eta)<\infty.$

Moreover, since the prefactor $\nu(\mathcal{T})\eta(\mathcal{T})$ is symmetric
under swapping of $\nu$ and $\eta$, it straightforward to verify that
$\bar{\kappa}_{n}^{\varepsilon}$ is still reversible with respect to the same
invariant measure $\Pi_{n}$, i.e. we have

$\Pi_{n}(\mathrm{d}\nu)\bar{\kappa}_{n}^{\varepsilon}(\nu,\mathrm{d}\eta)=\Pi_{n}(\mathrm{d}\eta)\bar{\kappa}_{n}^{\varepsilon}(\eta,\mathrm{d}\nu).$

The triple $(\Gamma,\Pi_{n},\bar{\kappa}_{n}^{\varepsilon})$ therefore
satisfies the assumptions of [PRST22]. Keeping in mind the difference in
definitions of $\Psi^{*}$ due to extra the factor $2$, by [PRST22, Theoren
6.6] there exist a unique curve $U^{\varepsilon}\in
C^{1}([0,T],L^{1}(\Gamma,\Pi_{n}))$ such that
$U_{0}=\mathrm{d}\bar{\mathsf{P}}/\mathrm{d}\Pi_{n}$, and

$\left\\{\begin{aligned}
\partial_{t}U_{t}(\nu)&=\int_{\Gamma}(U_{t}(\eta)-U_{t}(\nu))\bar{\kappa}_{n}^{\varepsilon}(\nu,\mathrm{d}\eta),\qquad\mbox{for
a.e.\ $t\in[0,T]$},\\\
\mathcal{E}\mathrm{nt}(\mathsf{P}_{0}|\Pi_{n})-\mathcal{E}\mathrm{nt}(\mathsf{P}_{T}|\Pi_{n})&=\int_{0}^{T}\int_{\Gamma\times\Gamma}\Psi\left(U_{t}(\eta)-U_{t}(\nu)\right)\sqrt{U_{t}(\nu)U_{t}(\eta)}\,\Pi_{n}(\mathrm{d}\nu)\,\bar{\kappa}_{n}^{\varepsilon}(\nu,\mathrm{d}\eta)\,\mathrm{d}t\\\
&\quad+\int_{\Gamma\times\Gamma}\left(\sqrt{U_{t}(\eta)}-\sqrt{U_{t}(\nu)}\right)^{2}\Pi_{n}(\mathrm{d}\nu)\,\bar{\kappa}_{n}^{\varepsilon}(\nu,\mathrm{d}\eta)\,\mathrm{d}t,\end{aligned}\right.$

with $\mathsf{P}_{t}:=U_{t}\Pi_{n}$ as usual. In particular the entropy
$\mathcal{E}\mathrm{nt}(\mathsf{P}|\Pi_{n})$ decreases along the solution and
hence

$\sup_{t\in[0,T]}\mathcal{E}\mathrm{nt}(\mathsf{P}_{t}|\Pi_{n})\leq\mathcal{E}\mathrm{nt}(\bar{\mathsf{P}}|\Pi_{n}).$

By evenness of $\Psi$, symmetry of $\Pi_{n}\bar{\kappa}_{n}^{\varepsilon}$ and
the identity (A.2), we can express for any $U$ after substituting for
$\bar{\kappa}_{n}^{\varepsilon}(\nu,\mathrm{d}\eta)$

$\displaystyle\frac{1}{2}\int_{\Gamma\times\Gamma}$
$\displaystyle\Psi\left(U(\eta)-U(\nu)\right)\sqrt{U(\nu)U(\eta)}\,\Pi_{n}(\mathrm{d}\nu)\,\bar{\kappa}_{n}^{\varepsilon}(\nu,\mathrm{d}\eta)$
$\displaystyle=\int_{U(\eta)>0,U(\nu)>0}\phi\left(\sqrt{U(\nu)/U(\eta)}\right)\sqrt{U(\eta)U(\nu)}\,\Pi_{n}(\mathrm{d}\nu)\,\bar{\kappa}_{n}^{\varepsilon}(\nu,\mathrm{d}\eta)$
$\displaystyle=\int_{U(\nu+n^{-1}\delta_{x})>0,U(\nu)>0}\phi\left(\frac{\mathrm{d}\vartheta_{\mathsf{P}}^{+,\varepsilon}}{\mathrm{d}\Theta^{+,n,\varepsilon}_{\mathsf{P}}}\right)\sqrt{U(\nu+\genfrac{}{}{}{2}{1}{n}\delta_{x})U(\nu)}\,\Pi_{n}(\mathrm{d}\nu)\,\kappa^{+,\varepsilon}_{\nu}(\mathrm{d}x)$
$\displaystyle\qquad+\int_{U(\nu-n^{-1}\delta_{x})>0,U(\nu)>0}\phi\left(\frac{\mathrm{d}\vartheta_{\mathsf{P}}^{-,\varepsilon}}{\mathrm{d}\Theta^{-,n,\varepsilon}_{\mathsf{P}}}\right)\sqrt{U(\nu-\genfrac{}{}{}{2}{1}{n}\delta_{x})U(\nu)}\,\Pi_{n}(\mathrm{d}\nu)\,\kappa^{-,\varepsilon}_{\nu}(\mathrm{d}x)$
$\displaystyle=\mathcal{R}_{n,\varepsilon}\left(\mathsf{P},\vartheta_{\mathsf{P}}^{+,\varepsilon},\vartheta_{\mathsf{P}}^{-,\varepsilon}\right).$

Moreover, it is straightforward to check that

$\int_{\Gamma\times\Gamma}\left(\sqrt{U_{t}(\eta)}-\sqrt{U_{t}(\nu)}\right)^{2}\Pi_{n}(\mathrm{d}\nu)\,\bar{\kappa}_{n}^{\varepsilon}(\nu,\mathrm{d}\eta)=\mathcal{D}_{n,\varepsilon}(\mathsf{P}),$

and therefore with $J^{\pm}:=\vartheta_{\mathsf{P}}^{\pm,\varepsilon}$ we
conclude

$\mathcal{I}_{n,\varepsilon}(\mathsf{P},\mathsf{J}^{+},\mathsf{J}^{-})=0.$

Finally, note that by Lemma 3.13 and Remark 3.6

$\displaystyle\mathcal{E}\mathrm{nt}\left(\mathsf{J}^{\pm}|\Theta_{\mathsf{P}}^{n,+,\varepsilon}\right)$