Reversible birth-and-death dynamics in continuum: free-energy dissipation and attractor properties

TL;DR

This paper studies reversible birth-and-death processes in continuous space, demonstrating entropy decay and convergence to Gibbs measures, revealing attractor properties and energy dissipation mechanisms.

Contribution

It introduces a framework for analyzing entropy dissipation and long-term behavior of birth-and-death dynamics with Gibbs measures in continuum.

Findings

Entropy decreases along trajectories for a broad class of initial measures.

Long-time limits of the process are Gibbs point processes with the same interaction.

The proof uses a novel representation of entropy dissipation via Palm measures.

Abstract

We consider continuous-time birth-and-death dynamics in $\mathbb{R}^d$ that admit at least one infinite-volume Gibbs point process based on area interactions as a reversible measure. For a large class of starting measures, we show that the specific relative entropy decays along trajectories, and that all possible long-time weak limit points are also Gibbs point processes with respect to the same interaction. Our proof rests on a representation of the entropy dissipation in terms of the Palm version of the propagated measure.

Equations

$$h(\nu |\mu )=\sum _{x\in E}\nu (x)\log \frac{\nu (x)}{\mu (x)},$$

$$\partial {\nu }_t={\nu }_t\mathcal{L}.$$

$$\eta =\sum _{i\ge 1}{\delta }_{x_i}\equiv \{x_1,x_2,\dots \},x_i\in {\mathbb{R}}^d.$$

$${\nu }_{\Lambda }^{(\omega )}(\mathrm{d}{\eta }_{\Lambda }):=Z_{\Lambda }(\omega {)}^{-1}{\mathrm{e}}^{-H_{\Lambda }({\eta }_{\Lambda }{\omega }_{{\Lambda }^c})}{\pi }_{\Lambda }(\mathrm{d}{\eta }_{\Lambda }),$$

$$H_{\Lambda}(\eta):=\lim_{n\uparrow\infty}\big{(}H(\eta_{\Lambda_{n}})-H(\eta_{\Lambda_{n}\setminus\Lambda})\big{)}.$$

$$H(\eta)=|B_{R}(\eta)|:=\Big{\lvert}\bigcup_{x\in\eta}B_{R}(x)\Big{\rvert},$$

$$H_{\Lambda }(\eta )=H_{\Lambda }({\eta }_{\Lambda \oplus B(0,2R)}),\text{where }\Lambda \oplus B(0,2R):=\{x\in {\mathbb{R}}^d:\mathrm{dist}(x,\Lambda )\le 2R\}.$$

$$\int \nu (\mathrm{d}\omega )f(\omega )=\int \nu (\mathrm{d}\omega )\int {\nu }_{\Lambda }^{(\omega )}(\mathrm{d}{\eta }_{\Lambda })f({\eta }_{\Lambda }{\omega }_{{\Lambda }^c}),\Lambda ⋐{\mathbb{R}}^d,f\ge 0\text{ and measurable}.$$

$$h(x,\eta):=\lim_{n\uparrow\infty}\big{(}H(\eta_{\Lambda_{n}}+\delta_{x})-H(\eta_{\Lambda_{n}})\big{)}$$

$$b(x,\eta ):={\mathrm{e}}^{-h(x,\eta )}.$$

$$(\mathscr{L}f)(\eta):=\int_{\mathbb{R}^{d}}\mathrm{d}x\ b(x,\eta)\big{(}f(\eta+\delta_{x})-f(\eta)\big{)}+\sum_{x\in\eta}\big{(}f(\eta-\delta_{x})-f(\eta)\big{)}.$$

$$\begin{array}{rl}{X}_t^{(\eta )}(B)={\int }_{B\times [0,\infty {)}^2\times [0,t]}\mathcal{N}(\mathrm{d}x,\mathrm{d}u,\mathrm{d}r,\mathrm{d}s)& \text{mathds}{1}_{[0,b(x,X_{s-}^{(\eta )})]}(u)\text{mathds}{1}_{(t-s,\infty )}(r)\\ & +{\int }_{B\times [0,\infty )}\hat{\eta }(\mathrm{d}x,\mathrm{d}r)\text{mathds}{1}_{(t,\infty )}(r).\end{array}$$

$$(T_{t}f)(\eta ):=\mathbb{E}[f(X_t^{(\eta )})],t\ge 0,$$

$$\displaystyle\frac{\mathrm{d}}{\mathrm{d}s}T_{s}f\Big{|}_{s=t}=T_{t}\mathscr{L}f=\mathscr{L}T_{t}f,\quad f\in\mathcal{L},$$

$$z^{-N_{\Lambda }}\le \frac{\mathrm{d}{\mu }_{\Lambda }}{\mathrm{d}{\pi }_{\Lambda }}\le z^{N_{\Lambda }}.$$

$$I_{\Lambda }(\mu |\nu ):=I({\mu }_{\Lambda }|{\nu }_{\Lambda }):=\int \mu (\mathrm{d}\eta )\log \frac{\mathrm{d}{\mu }_{\Lambda }}{\mathrm{d}{\nu }_{\Lambda }}({\eta }_{\Lambda }),$$

$$\mathcal{I}(\mu |\nu ):=\underset{n\uparrow \infty }{\mathrm{liminf}}\frac{I_{{\Lambda }_n}(\mu |\nu )}{\left|{\Lambda }_n\right|}.$$

$$\int \mu (\mathrm{d}\eta )\sum _{x\in \eta }f(x,\eta -x)=\lambda \int \mathrm{d}x{\mu }_o^{!}[f(x,\cdot )],f:{\mathbb{R}}^d\times \Omega \to [0,\infty ),$$

$${\mathcal{J}}_{\Lambda }({\mu }_{\Lambda }|\nu ):=\int \mathrm{d}x\int {\nu }_{\Lambda }(\mathrm{d}\eta )b_{\Lambda }^{\nu }(x,{\eta }_{\Lambda })D_x\frac{\mathrm{d}{\mu }_{\Lambda }}{\mathrm{d}{\nu }_{\Lambda }}({\eta }_{\Lambda })D_x\log \frac{\mathrm{d}{\mu }_{\Lambda }}{\mathrm{d}{\nu }_{\Lambda }}({\eta }_{\Lambda }),$$

$$D_{x}F(\eta ):=F(\eta +{\delta }_x)-F(\eta )$$

$$\mathcal{I}(\mu |\nu )-\mathcal{I}(\mu T_t|\nu )$$

$$\displaystyle\xi^{\mu}(s):=I\big{(}b(o,\cdot)\mu T_{s}\,\big{|}\,(\mu T_{s})_{o}^{!}\big{)}+I\big{(}(\mu T_{s})_{o}^{!}\,\big{|}\,b(o,\cdot)\mu T_{s}\big{)}\geq 0.$$

$$C_{\mu }^{!}(B\times F):={\int }_{\Omega }\mu (d\eta ){\int }_{{\mathbb{R}}^d}\eta (dx)\text{mathds}{1}_B(x)\text{mathds}{1}_F(\eta -{\delta }_x),B\in \mathfrak{B}({\mathbb{R}}^d),F\in \mathcal{F}.$$

$$\frac{\mathrm{d}{C}_{\mu }^{!}}{\mathrm{d}x\otimes \mathrm{d}\mu }(x,\eta )=b(x,\eta )$$

$$\displaystyle\nu\bigl{[}\mathds{1}_{H}(\eta)(\mathscr{L}\mathds{1}_{F})(\eta)\bigr{]}=\nu\Big{[}\int_{B}\mathrm{d}x\,b(x,\eta)\mathds{1}_{H}(\eta)\mathds{1}_{F}(\eta+\delta_{x})\Big{]}\text{ and }$$

$$\displaystyle\nu\bigl{[}\mathds{1}_{F}(\eta)(\mathscr{L}\mathds{1}_{H})(\eta)\bigr{]}=\int C_{\nu}^{!}(\mathrm{d}x,\mathrm{d}\eta)\,\mathds{1}_{B}(x)\mathds{1}_{F}(\eta+\delta_{x})\mathds{1}_{H}(\eta).$$

$$\displaystyle\nu\big{[}\mathds{1}_{H}(\eta)(\mathscr{L}\mathds{1}_{F})(\eta)\big{]}$$

$$\displaystyle=\nu\Big{[}\int_{B}\mathrm{d}x\,b(x,\eta)\mathds{1}_{H}(\eta)\mathds{1}_{F}(\eta+\delta_{x})\Big{]}.$$

$$\displaystyle\nu\big{[}\mathds{1}_{F}(\eta)(\mathscr{L}\mathds{1}_{H})(\eta)\big{]}$$

$$b(o,\cdot )\mu ={\mu }_o^{!}.$$

Full text

1. Introduction

The theory of Gibbs point processes, describing interacting systems of particles in continuous space, is by now a classical topic in probability and statistical mechanics and for various questions, somewhat satisfactory results have been obtained: existence [DV20], phase transitions [GLM95, CCK95], and the equivalence of various ensembles [Geo95], to name a few examples that are of course not exhaustive. On the lattice, these results on the equilibrium behavior of large systems of interacting particles are complemented by a detailed study of associated dynamics, see for example the classical reference [Lig05]. In contrast, very little is known about dynamical aspects in continuum, in particular out-of-equilibrium dynamics and convergence to equilibrium.

1.1. Well-defined dynamics

The first issue lies in the fact that the unbounded nature of point processes in continuum means that already establishing a well-defined dynamics proves to be a major challenge, and cannot be recovered by applying Liggett’s classical results, cf. [Pen08]. Inspired by biologically motivated models (e.g., locally regulated populations), the study of birth-and-death processes in the continuum, with birth and death rates depending on the configuration of the system, was initiated in [Pre75] for bounded systems. Building on this, the seminal work [HS78], shows, for a one-dimensional system, the well-posedness of the dynamics and a characterization of its reversible measures, using a martingale-problem approach. Extending their results to more general cases is then an intrinsically non-trivial matter, and a few approaches have been explored in the literature: the first uses Dirichlet forms to characterize stationary measures as Gibbs states via their correlation functions, see e.g. [KS06, KKP08, FKKO14]; the second characterizes the Markov process as a solution of a martingale problem, see e.g.  [Kur80, Gar95, GK06]; finally, it is worth noting that the lookdown-construction approach presented in [EK19] covers a broad class of spatial birth-and-death processes, with the aim of capturing the genealogy of a population.

1.2. Out-of-equilibrium dynamics

While the above works are able to construct well-defined dynamics and investigate key properties of the processes such as characterizations of the stationary measures, the proofs do not make use of the actual dynamics, and yield little information about its long-time behavior, in particular on its weak limit points. Again, also in this case the situation is much better understood on the lattice, see below, while it has so far not been explored for the type of systems we consider here.

One possible starting point for investigating the long-time behavior of Markov processes out of equilibrium is the following simple observation for continuous-time Markov chains on finite state spaces. For concreteness, let us denote the state space by $E$ and consider an irreducible generator $\mathscr{L}$. In this situation, it is well known that there exists a unique measure $\mu$ that is time-stationary with respect to the Markov semigroup $(P_{t})_{t\geq 0}$ generated by $\mathscr{L}$. Because of the irreducibility, $\mu$ puts strictly positive mass on every state $x\in E$ and we can define the relative entropy of another probability measure $\nu$ with respect to $\mu$ by

[TABLE]

where we use the convention that $0\log 0=0$. For an initial distribution $\nu$ let us denote the distribution at time $t\geq 0$ by $\nu_{t}$. By Jensen’s inequality one can then easily check that for any $\nu$ the map $t\mapsto h(\nu_{t}\lvert\mu)$ is non-increasing and is only constant if $\nu=\mu$. This tells us that the functional $\nu\mapsto h(\nu\lvert\mu)$ is a strict Lyapunov function for the unique fixed point $\mu$ of the measure-valued ODE

[TABLE]

This directly implies that $(\nu_{t})_{t\geq 0}$ converges to $\mu$ as $t$ tends to infinity for any initial distribution $\nu$. Of course, this is nothing but the classical ergodic theorem for finite-state Markov processes, but this proof shows that the convergence to the unique time-stationary measure also fits precisely into the physical picture of convergence to equilibrium via relative-entropy dissipation.

Moreover, it can also be adapted to more complex situations. In the context of classical interacting particle systems on the integer lattice, this strategy has been successfully applied, first to reversible dynamics [Hol71, HS75] and later also to non-reversible systems [Sul77, Kü84, JK19, JK23, JK25].

1.3. Contributions of this work

Our goal is to provide a first investigation of the long-time behavior of birth-and-death dynamics in continuum space. In particular, our main contribution is two-fold. First, we show, in Theorem 2.3 a free-energy dissipation result which, to the best of our knowledge, so far existed only in the non-interacting case, e.g. [DSHS24, HS25]. Second, in Theorem 2.4, we apply this free-energy dissipation to establish the attractor property of the dynamics, in the sense that any of its long-time weak limit points is a Gibbs point process. In other words, we show that the $\omega$-limit set of the dynamics is precisely given by the set of Gibbs measures.

On the way, we furthermore prove various technical results for birth-and-death dynamics in continuum, which could be of independent interest. Most importantly, we show that there is a finite speed of propagation, i.e., with very high probability no information about the state of the process at a given position or its boundary condition can travel faster than linearly, see Lemma 4.5.

While we focus on a precise model, a birth-and-death process whose rates come from an area interaction, we believe the foundations laid here could serve as a stepping stone for future work, see Section 2.4.

1.4. Outline of the paper

The rest of the paper is structured as follows: in Section 2, after introducing the notation and setting, we present our main results in Section 2.3 and discuss possible future directions in Section 2.4. The proofs are then split up into various steps and provided in Section 3–6. We start by deriving a more convenient characterization of reversible Gibbs measures in Section 3. In Section 4 we approximate our global dynamic by a process that only acts locally and estimate the thereby introduced error. With this approximation at hand, we then show in Section 5 that the relative-entropy density is indeed non-increasing along trajectories and then provide a full characterization of all possible weak limit points in Section 6.

2. Setting and main result

We consider Markov processes on the space $\Omega$ of simple, locally-finite point configurations in $\mathbb{R}^{d}$, $d\geq 1$. An element $\eta\in\Omega$ is of the form

[TABLE]

We write $\eta_{\Lambda}:=\eta\cap\Lambda$, with $\Lambda\subset\mathbb{R}^{d}$, and correspondingly denote by $\Omega_{\Lambda}$ the space of point configurations in $\Lambda$. The space $\mathbb{R}^{d}$ is endowed with the Euclidean norm $\left\lvert\cdot\right\rvert$ and the associated Borel $\sigma$-algebra $\mathfrak{B}(\mathbb{R}^{d})$. We denote $\Lambda\Subset\mathbb{R}^{d}$, or $\Lambda\in\mathfrak{B}_{b}(\mathbb{R}^{d})$, if $\Lambda$ is a bounded Borel subset of $\mathbb{R}^{d}$, and set $\Lambda_{n}:=[-n/2,n/2)^{d}$. For $\Lambda\Subset\mathbb{R}^{d}$, let $\eta\mapsto N_{\Lambda}(\eta)=|\eta_{\Lambda}|$ denote the number of points of $\eta$ in $\Lambda$. On $\Omega$, we consider the usual $\sigma$-algebra $\mathscr{F}$ induced by the counting functions $\eta\mapsto N_{\Lambda}(\eta)$.

We call point process any probability measure on the set of point configurations; the set of all such measures is denoted by $\mathcal{P}(\Omega)$. The restriction of such a $\mu\in\mathcal{P}(\Omega)$ to $\Lambda\Subset\mathbb{R}^{d}$ is denoted by $\mu_{\Lambda}\in\mathcal{P}(\Omega_{\Lambda})$.

We will often use the short-hand notation $f\lesssim g$ if there exists a finite constant $c>0$, independent of $f$ and $g$, such that $f\leq c\cdot g$. We write $f\lesssim_{A}g$ if $c=c(A)$ depends on some parameter $A$, for example a subset $\Lambda\subset\mathbb{R}^{d}$.

2.1. Gibbs point processes

We are interested in translation-invariant Gibbs point processes based on an energy functional $H\colon\Omega\to\mathbb{R}\cup\{+\infty\}$ defined as follows. Let $\pi$ denote the homogeneous intensity-one Poisson point process on $\Omega$, and define the finite-volume Gibbs point process in $\Lambda\Subset\mathbb{R}^{d}$ with boundary condition $\omega\in\Omega$ as the following probability measure on $\Omega_{\Lambda}$:

[TABLE]

where the partition function $Z_{\Lambda}(\omega)$ is the usual normalization constant, and

[TABLE]

In order to avoid well-definedness issues and to keep the exposition of the main ideas as clear as possible, we restrict our attention to the area interaction, where

[TABLE]

and $B_{R}(x)$ denotes the ball with radius $R>0$ centered at $x\in\mathbb{R}^{d}$; see Section 2.4 for a discussion on this assumption. Note that the interaction range of the area interaction is $2R$, in the sense that

[TABLE]

Now, a (grand-canonical) infinite-volume Gibbs point process is any probability measure $\nu$ on $\Omega$ that satisfies the DLR equations (after Dobrushin, Lanford, and Ruelle), i.e.,

[TABLE]

It is a classical result that the area interaction model exhibits a phase transition. In our parametrization, this means that for all sufficiently large $R$, the set of translation-invariant infinite-volume Gibbs point processes $\mathscr{G}_{\theta}$ contains more than one element, see, e.g. [Rue71, GLM95, CCK95]. Moreover, $\mathscr{G}_{\theta}$ is never empty and it is one of the features of our main result that it works in both the uniqueness as well as the phase-transition regime.

2.2. Birth-and-death processes

Based on the equilibrium setting just described, we consider Markovian dynamics that are reversible with respect to elements of $\mathscr{G}_{\theta}$. For this, let

[TABLE]

be the conditional energy of a point $x\in\mathbb{R}^{d}$ in a configuration $\eta\in\Omega$, and define the birth rate, based on $h$, at $x$ given $\eta$ as

[TABLE]

Note that, in the terminology of the GNZ equations, see e.g. [Geo76, NZ79], the birth rate is the Papangelou intensity associated to the Hamiltonian $H$. Furthermore, due to the attractive nature of the area interaction, it is higher close to the points of $\eta$.

Now, we consider the birth-and-death process associated to the (formal) generator

[TABLE]

The following result from [GK06, Theorem 2.13] establishes the existence of the associated Markov process $(X^{\scriptscriptstyle{({\eta}})}_{t})_{t\geq 0}$, with starting configuration $X_{0}^{\scriptscriptstyle{({\eta}})}=\eta$, representing it as a thinning of a space-time Poisson random measure, see Figure 1 for an illustration.

Proposition 2.1** (Graphical representation).**

Let $\mathcal{N}$ be a Poisson random measure on $\mathbb{R}^{d}\times[0,\infty)^{3}$ with intensity measure $\mathrm{d}x\otimes\mathrm{d}u\otimes\mathrm{e}^{-r}\mathrm{d}r\otimes\mathrm{d}s$, $\eta=\sum_{i\geq 1}\delta_{x_{i}}\in\Omega$, and $\hat{\eta}=\sum_{i\geq 1}\delta_{(x_{i},\tau_{i})}$ be the point process on $\mathbb{R}^{d}\times[0,\infty)$ obtained by associating to each $x_{i}\in\eta$ an independent (of $\mathcal{N}$) unit-exponential random variable $\tau_{i}$. Suppose $\mathcal{F}=(\mathcal{F}_{t})_{t\geq 0}$ is a filtration such that $\mathcal{N}$ is $\mathcal{F}$-compatible, i.e., such that for each bounded Borel set $A\subset\mathbb{R}^{d}\times[0,\infty)^{2}$, $\mathcal{N}(A,\cdot)$ is $\mathcal{F}$-adapted, and $\mathcal{N}(A,t+s)-\mathcal{N}(A,t)$ is independent of $\mathcal{F}_{t}$, for any $s,t\geq 0$. Then, for any Borel measurable set $B\subset\mathbb{R}^{d}$, there exists a unique solution of

[TABLE]

The equation (2.1) has a unique solution that corresponds to the unique solution of the martingale problem for $\mathscr{L}$, see also [EK86, Theorem 4.4.2]. Note that such a process will take values in the space of càdlàg functions on counting measures on $\mathbb{R}^{d}$, embedded with the Skorokhod $J_{1}$-topology.

We denote the associated semigroup by

[TABLE]

for all $f$ for which the right-hand side is well-defined, and note that, as shown below in Lemma 4.2,

[TABLE]

where $\mathcal{L}$ denotes the set of all bounded, local, measurable functions $f\colon\Omega\to\mathbb{R}$. In particular, the domain of $\mathscr{L}$ contains $\mathcal{L}$, as well as all $T_{t}f$, when $f\in\mathcal{L}$. Moreover, let us mention that, due to the finite speed of propagation exhibited in Lemma 4.5, the existence of the infinite-volume dynamics can also be proven by other means.

2.3. Main results

In order to analyze the long-time behavior of the dynamics for arbitrary initial distributions, we rely on a Lyapunov-type approach based on the relative-entropy density. The advantage of this strategy is that it also works in the phase-transition regime and for initial distributions that are not absolutely continuous with respect to the reversible measure. However, we do require the existence of local densities in the following sense.

Definition 2.2** (Regular measures).**

A translation-invariant probability measure $\mu$ on $\Omega$ is said to be regular if there exists $z\in(0,\infty)$ such that, for any bounded $\Lambda\Subset\mathbb{R}^{d}$,

[TABLE]

Regular measures are also known as quasi-Gibbs measures, see [Osa13], as this notion is weaker than the requirement that $\mu$ is a canonical (fixed number of points per unit volume) Gibbs measure. However, not every grand-canonical Gibbs measure is regular: a Gibbs point process with energy functional $H$ is regular if $\left\lvert H_{\Lambda}(\omega_{\Lambda}\omega_{\Lambda^{c}})\right\rvert\leq C\left\lvert\omega_{\Lambda}\right\rvert$ is regular, cf. Penrose stability in [PZ21]. This is the case, for example, for any area-interaction Gibbs point process. Furthermore, note that the shifted lattice $\mathbb{Z}^{d}+U$, where $U$ is the uniform distribution on $[0,1]^{d}$, is translation invariant but not regular.

In order to state our main result, we introduce the local relative entropy for $\Lambda\Subset\mathbb{R}^{d}$ as

[TABLE]

if the density $\mathrm{d}\mu_{\Lambda}/\mathrm{d}\nu_{\Lambda}$ exists and $\infty$ otherwise. Similarly, we will write$I(\mu|\nu):=\int\mu(\mathrm{d}\eta)\log(\mathrm{d}\mu/\mathrm{d}\nu(\eta))$ with $\mathrm{d}\mu/\mathrm{d}\nu$ instead of $\mathrm{d}\mu_{\Lambda}/\mathrm{d}\nu_{\Lambda}$. The relative entropy density, also called specific entropy, is then defined as

[TABLE]

Note that the specific entropy even exists as a limit if $\mu$ is translation-invariant, see Proposition 5.4 below.

Next, if $\mu$ is a translation-invariant point process on $\Omega$ with finite intensity $\lambda>0$, there exists a unique measure $\mu_{o}^{!}\in\mathcal{P}(\Omega)$, its reduced Palm measure, such that

[TABLE]

where $\eta-x=\sum_{y\in\eta}\delta_{y-x}-\delta_{o}$ is the shift of the configuration together with the removal of the origin $o\in\mathbb{R}^{d}$, see for example [Kal83].

We can now present our first main result, which establishes the entropy dissipation, connecting entropy production as time progresses and the Fisher information, which we introduce next. Let $\nu$ be a point process on $\mathbb{R}^{d}$ and let $\mu_{\Lambda}$ be another point process on a bounded domain $\Lambda\Subset\mathbb{R}^{d}$ such that $\mu_{\Lambda}\ll\nu_{\Lambda}$. Then, the (rescaled) modified Fisher information, cf. [DSHS24, HS25], is defined as

[TABLE]

where

[TABLE]

is the add-one-cost operator and $b^{\nu}_{\Lambda}(x,\eta_{\Lambda}):=\int\nu(\mathrm{d}\zeta_{\Lambda^{c}}|\eta_{\Lambda})\ b(x,\eta_{\Lambda}\zeta_{\Lambda^{c}})$ is a localized birth rate.

Theorem 2.3**.**

Let $\nu\in\mathscr{G}_{\theta}$.

1. i.

For any translation-invariant starting measure $\mu$, the map $t\mapsto\mathscr{I}(\mu T_{t}|\nu)$ is non-increasing. 2. ii.

If $\mu$ is regular, then

[TABLE]

where $T^{\scriptscriptstyle{({\Lambda}})}$ is the semigroup associated to the localized dynamics with birth rate $b^{\nu}_{\Lambda}$, introduced in Definition 4.1 below, and

[TABLE] 3. iii.

For regular $\mu$ we have $\xi^{\mu}(s)=0$ if and only if $\mu T_{s}\in\mathscr{G}_{\theta}$.

Our main take away from Theorem 2.3 is the following attractor property, where the $\tau_{\mathcal{L}}$-topology is the smallest topology on $\mathcal{P}(\Omega)$ such that the mappings $\mu\mapsto\mu[f]$ are continuous for all $f\in\mathcal{L}$.

Theorem 2.4**.**

Let $\mu$ be a regular starting measure. Then, for any $\mu^{*}$ such that there exists an increasing sequence of times $(t_{k})_{k\geq 0}$, with $t_{k}\uparrow\infty$ and $\lim_{k\uparrow\infty}\mu T_{t_{k}}=\mu^{*}$ in the $\tau_{\mathcal{L}}$-topology, we have that $\mu^{*}\in\mathscr{G}_{\theta}$. In particular, $\mu^{*}$ is reversible.

Our main results confirm the physical intuition that the relative entropy (or free energy) is non-increasing in time and moreover governs the long-term behavior of large systems of interacting particles. The proof of our two main results is inspired by the method developed in [Sul77] for interacting particle systems on the lattice and split into various separate steps in Section 3–6.

2.4. Generalizations and outlook

Let us briefly discuss some possible generalizations and comment on future research directions.

2.4.1. Model assumptions

As we have restricted our study to the area interaction model, it is natural to ask under what general conditions the statements and proofs presented here still hold. In order to facilitate this quest, let us note that in the course of the proofs, we mainly make use of the following three properties of the area interaction: (1) its finite range; (2) its globally bounded birth rates (both needed, e.g., to prove finite speed of propagation, see Lemma 4.4); and (3) the factor property of Lemma 5.2. We believe our proofs, in particular Lemmas 4.4, 4.2 and 4.3 could be adapted to the case of a birth rate which is only locally bounded, that is, for every bounded $\Delta\Subset\mathbb{R}^{d}$, $\sup_{x\in\Delta,\eta}b(x,\eta)$ is finite, cf. [GK06, Condition 2.1]. However, the new interactions should still satisfy a (possibly weaker version of the) factor property as in Lemma 5.2.

2.4.2. Outlook

Is the second result in Theorem 2.3 optimal? Indeed, we conjecture that it holds with an equality and that the $\liminf$ is a true limit, thus recovering the classical de Bruijn identity. At the moment, the proof of such a statement following the line of argument as presented in this manuscript has two major hurdles. First, while one would intuitively believe that the converse upper bound in Lemma 5.5 should also hold, the structure of the Donsker–Varadhan variational formula makes it much harder to prove. While for the lower bound we only needed to use a single appropriately chosen test function, the upper bound requires bounding the functional from above for arbitrary test functions. Second, while for the Poisson case the density limit of the Fisher information is shown to exist with a subadditivity argument, see [HS25, Lemma 7.8] in the interacting case the situation is not as clear, as the measure is not exactly factorizable and one looses the precise superadditivity. In contrast to the proof of Lemma 5.3, the factor property of $\nu$ does not seem to yield an $o(\left\lvert\Lambda\right\rvert)$ error term.

Some other possible directions which are inspired by analogous results for interacting particle systems on the lattice are as follows. First, one could see how well our results can be extended to the case of non-reversible birth-and-death dynamics as for example in [Kü84, JK23].

Second, it would be interesting to see if one can also apply a similar strategy based on the change of relative entropy to analyze the long-time behavior without shift-invariance. For reversible lattice systems, this can be done in one and two dimensions, see [HS77] and [JK25], and it is not clear if these arguments can be extended to the continuum.

3. Characterizing reversible measures

Our main objective is to understand the birth-and-death process if we start it out of equilibrium from another translation-invariant measure. In order to identify all weak limit points of the dynamics as Gibbs measures, we start out by providing a more convenient characterization of reversible measures and Gibbs measures.

For this, let $\mathscr{R}_{\theta}$ denote the set of translation-invariant reversible measures for the above dynamics. The main objective of this section is to prove the equivalence of reversible measures and the infinite-volume Gibbs point processes described above.

Proposition 3.1**.**

We have that $\mathscr{G}_{\theta}=\mathscr{R}_{\theta}$.

The proof of this equivalence rests on a couple of elementary but technical lemmas. Before we start, let us introduce the reduced Campbell measure $C_{\mu}^{!}$ of $\mu\in\mathcal{P}(\Omega)$, which is a measure on $\mathbb{R}^{d}\times\Omega$ defined by

[TABLE]

Following [Geo76], the following characterization of Gibbs measures in terms of their reduced Campbell measures was derived in [Glö81].

Proposition 3.2** (Proposition, [Glö81]).**

For a translation-invariant $\mu\in\mathcal{P}(\Omega)$ we have $\mu\in\mathscr{G}_{\theta}$ if and only if

[TABLE]

for $\mathrm{d}x\otimes\mathrm{d}\mu$ almost all $(x,\eta)$.

In addition to this, we will often use the following technical helper to simplify calculations.

Lemma 3.3**.**

Let $\nu\in\mathcal{P}(\Omega)$, $B\in\mathfrak{B_{b}}(\mathbb{R}^{d})$, $n\in\mathbb{N}_{0}$ and $H,F\in\mathscr{F}$ with $H\subseteq\{N_{B}=n\}$ and $F\subseteq\{N_{B}=n+1\}$.

Then,

1. i.

[TABLE] 2. ii.

[TABLE]

Proof.

Ad i.: By definition of the generator $\mathscr{L}$ and because of the conditions on $H$ and $F$ we have

[TABLE]

Ad ii.: Note that the roles of $F,H$ on the left-hand side are now reversed. Hence, the assumptions on the sets $F,H$ imply that

[TABLE]

Now the claimed identity follows from using the definition of the reduced Campbell measure. ∎

The second ingredient is the following GNZ characterization of Gibbs measures using their reduced Palm measure. This is classical, but for convenience we nevertheless provide a proof.

Lemma 3.4**.**

Let $\mu\in\mathcal{P}(\Omega)$ be translation-invariant. Then, $\mu\in\mathscr{G}_{\theta}$ if and only if

[TABLE]

Proof.

By Proposition 3.2 it is equivalent to show that 3.1 holds if and only if

[TABLE]

For this, first assume that $b(o,\cdot)\mu=\mu_{o}^{!}$. Then, by definition,

[TABLE]

where $\theta_{-x}\eta:=\sum_{y\in\eta}\delta_{y-x}$ represents the shift operator. For the other direction, let $\varphi$ be some positive measurable function with $\int\mathrm{d}x\,\varphi(x)=1$. Then,

[TABLE]

as desired. ∎

We can now show Proposition 3.1, where we mainly follows the arguments in [Glö81, Theorem] reported here for the convenience of the reader.

Proof of Proposition 3.1.

The inclusion $\mathscr{G}_{\theta}\subseteq\mathscr{R}_{\theta}$ follows from a direct calculation that verifies the self-adjointness criterion $\nu[f\mathscr{L}g]=\nu[g\mathscr{L}f]$. Indeed, for any pair $f,g\in\mathcal{L}$, by the GNZ equations, see Lemma 3.4, we have that

[TABLE]

and therefore,

[TABLE]

For the other direction, i.e., $\mathscr{R}_{\theta}\subseteq\mathscr{G}_{\theta}$, we need to show that the reduced Campbell measure associated to any reversible measure $\nu$ is given by 3.1. Let be $B\in\mathfrak{B}_{b}(\mathbb{R}^{d})$ and for all $H\in\mathcal{F}$ and $n\in\mathbb{N}_{0}$ put $H_{n}:=H\cap\{N_{B}=n\}$ and $H_{n}^{+}:=\{\eta\colon\eta=\zeta+\delta_{x},\zeta\in H_{n},x\in B\}$. Then, for $\nu\in\mathscr{R}_{\theta}$ we have, using Lemma 3.3 twice and the reversibility of $\nu$,

[TABLE]

Now $\nu\in\mathscr{G}_{\theta}$ follows, since $\{H_{n}\times B\colon H\in\mathcal{F},n\in\mathbb{N}_{0},B\in\mathfrak{B}_{b}(\mathbb{R}^{d})\}$ is a generating system. ∎

4. Local and global dynamics

Since the projection of the dynamics to a finite volume $\Lambda$ is not Markovian due to interactions with $\Lambda^{c}$, the finite-volume relative entropy functionals $t\mapsto h_{\Lambda}(\mu T_{t}\lvert\nu)$ are generally not monotone. To remedy this, we define a modified dynamics $(T^{\scriptscriptstyle{({\Lambda}})}_{t})_{t\geq 0}$, in which the birth rate incorporates random boundary conditions sampled from a Gibbs measure $\nu\in\mathscr{G}_{\theta}$, and later compare the behavior of this dynamics to the original one.

Definition 4.1** (Local dynamics).**

Let $\nu\in\mathscr{G}_{\theta}$, $\Lambda\Subset\mathbb{R}^{d}$, and denote by \big{(}T^{\scriptscriptstyle{({\Lambda}})}_{t}\big{)}_{t\geq 0} the semigroup associated to the following (formal) generator

[TABLE]

with $b^{\nu}_{\Lambda}(x,\eta_{\Lambda}):=\int\nu(\mathrm{d}\zeta_{\Lambda^{c}}|\eta_{\Lambda})\,b(x,\eta_{\Lambda}\zeta_{\Lambda^{c}})$.

The associated Markov process can, as before, be constructed in the canonical coupling of all the related birth-and-death processes using one shared Poisson measure as driving noise. The birth rate in the corresponding stochastic integral equation will then of course also be $b^{\nu}_{\Lambda}$ as defined above, so that for any Borel measurable $B\subset\mathbb{R}^{d}$ the corresponding SDE with initial configuration $\eta\in\Omega$ reads

[TABLE]

and we denote the corresponding semigroup and formal generator by $T^{\scriptscriptstyle{({\Lambda}})}$ and $\mathscr{L}_{\Lambda}$, respectively. It can be seen, as in Proposition 3.1 above, that the restriction $\nu_{\Lambda}$ of $\nu$ is reversible with respect to $\mathscr{L}_{\Lambda}$.

4.1. Identification of the generators

Let us start by showing that the Markov process constructed via the graphical representation in terms of a driving Poisson point process as in Proposition 2.1 actually has the formal generator $\mathscr{L}$ (respectively $\mathscr{L}_{\Lambda}$). The technical properties we show in this section will then later be used in the proofs of the main results. For this, first recall that, for any starting configuration $\omega\in\Omega$, $(X_{t}^{\scriptscriptstyle{({\omega}})})_{t\geq 0}$, respectively $(X_{t}^{\scriptscriptstyle{({\Lambda,\omega}})})_{t\geq 0}$, denotes the Markov process solution of 2.1, respectively  4.1, starting from $\omega$.

4.1.1. Generator properties

We first show some basic properties related to the two generators.

Lemma 4.2**.**

We have that

[TABLE]

Proof.

We denote by $B_{\Lambda}(t)$, resp. $D_{\Lambda}(t)$, the set of times $s\leq t$ in which a birth, resp. death, occurs in $\Lambda$. Assume that $f\in\mathcal{L}$ with $f(\cdot)=f(\cdot_{\Lambda})$ for some $\Lambda\Subset\mathbb{R}^{d}$, and write

[TABLE]

The birth term is given by

[TABLE]

so that, by applying the Mecke formula,

[TABLE]

For the death term, we first see, by boundedness of $f$ and by boundedness of the birth rates, that

[TABLE]

Moreover, recalling that the death times are independent and distributed as an exponential random variable $\tau$ of parameter $1$, we have, by memorylessness,

[TABLE]

Therefore,

[TABLE]

Putting the two terms together, we obtain

[TABLE]

as desired. ∎

A similar result holds for the modified dynamics.

Lemma 4.3**.**

Let $f\colon\Omega_{\Lambda}\to\mathbb{R}$ be measurable with $|f|\leq z^{N_{\Lambda}}$. Then,

[TABLE]

Proof.

The proof goes along similar steps as the one of Lemma 4.2, considering $(X_{t}^{\scriptscriptstyle{({\Lambda,\omega}})})_{t\geq 0}$ in place of $(X_{t}^{\scriptscriptstyle{({\omega}})})_{t\geq 0}$. The birth term is handled as above, while for the death term, the assumption on $f$ guarantees not only well-definedness of all involved expressions but also that still

[TABLE]

as desired. ∎

4.2. Comparing local and global dynamics: finite speed of propagation

For any initial condition $\eta\in\Omega$, we have a canonical coupling between the infinite-volume Markov process $(X_{t}^{\scriptscriptstyle{({\eta}})})_{t\geq 0}$ and the Markov process $(X_{t}^{\scriptscriptstyle{({\Lambda,\eta}})})_{t\geq 0}$ in the finite volume $\Lambda$ with stochastic boundary conditions, using a common Poisson random measure $\mathcal{N}$, as in Proposition 2.1, as driving noise. We use this coupling to compare the corresponding evolutions, obtaining the following “finite-speed of propagation” result.

Lemma 4.4** (Finite speed of propagation).**

For any $\Lambda\Subset\mathbb{R}^{d}$, let $\widehat{\Lambda}$ be some superset of $\Lambda$, with $\ell:=\mathrm{dist}(\Lambda,\widehat{\Lambda}^{c})$. We have

[TABLE]

for some $\rho$ with $\rho(\ell)\xrightarrow[\ell\uparrow\infty]{}\infty$.

Proof.

Let $\|b\|_{\infty}:=\sup_{x,\eta}b(x,\eta)$, which is finite. First note that $X_{t,\Lambda}^{\scriptscriptstyle{({\widehat{\Lambda},\eta}})}\neq X_{t,\Lambda}^{(\eta)}$ for some $t\in[0,T]$ is possible only if the following event happens

[TABLE]

Tiling $\mathbb{R}^{d}$ into rectangles $Q_{i}$ of side length $4R$, we have that $E_{\Lambda}(T,\ell)\subseteq\widetilde{E}_{\Lambda}(T,\ell)$, where

[TABLE]

We can now bound the probability of $\widetilde{E}_{\Lambda}(T,\ell)$ via a union bound

[TABLE]

In particular, the right-hand side decays faster than exponential in $\ell$ for fixed $\Lambda,T$. ∎

The preceding upper bound on the probability that the coupling of the two processes fails can be converted into an estimate of the total variation distance of the laws $\mu T_{t}$ and $\mu T^{(\hat{\Lambda})}_{t}$.

Lemma 4.5**.**

For any $\Lambda\Subset\mathbb{R}^{d}$, let $\widehat{\Lambda}\supset\Lambda$, with $\ell:=\mathrm{dist}(\Lambda,\widehat{\Lambda}^{c})$. Further, let $\mu\in\mathcal{P}(\Omega)$ be translation-invariant and with finite first moment, i.e., for any $\Lambda\Subset\mathbb{R}^{d}$, $\mu[N_{\Lambda}]=:c_{\Lambda}<\infty$. Then, for any $f\in\mathcal{L}$ with $f(\cdot)=f(\cdot_{\Lambda})$, we have

[TABLE]

Moreover, consider a family $\{f_{\Lambda}\colon\Lambda\Subset\mathbb{R}^{d}\}$ of local functions such that $|f_{\Lambda}|\lesssim N_{\Lambda}$ (uniformly in $\Lambda$). If $\Lambda,\widehat{\Lambda}\uparrow\mathbb{R}^{d}$ in such a way that $\log(|\Lambda|)/\mathrm{dist}(\Lambda,\widehat{\Lambda}^{c})$ stays bounded from above, then, as $\Lambda\uparrow\mathbb{R}^{d}$,

[TABLE]

Proof.

Clearly,

[TABLE]

and therefore, by Lemma 4.4,

[TABLE]

For the second statement, note that

[TABLE]

Moreover, from Lemma 4.4,

[TABLE]

since $\log(|\Lambda|)\leq C\,\mathrm{dist}(\Lambda,\widehat{\Lambda}^{c})$, for some $C>0$, by assumption. Similarly,

[TABLE]

as well as

[TABLE]

We can then conclude that

[TABLE]

as desired. ∎

5. Monotonicity of the relative entropy density

To show that the relative entropy is non-increasing along the measure-valued trajectories of our original global dynamics, we introduced the local dynamics as an auxiliary process. With the technical helpers from the previous section at hand, we are now in a position to bound the error we made by this approximation. For this, we define the relative entropy density functional associated to the local dynamics via

[TABLE]

The main goal of this section is to prove the first part of our main result Theorem 2.3, which is implied by the following comparison result.

Proposition 5.1**.**

We have that $\mathscr{I}(\mu T_{t}|\nu)\leq\widetilde{\mathscr{I}}_{\mu}(t)$ and $t\mapsto\mathscr{I}(\mu T_{t}|\nu)$ is non-increasing.

The proof rests on two technical ingredients. First, we show that the finite-volume relative entropy is quasi-superadditive, i.e., superadditive up to sub-volume-order corrections. Second, we show that, as a consequence of the finite speed of propagation, the change in relative entropy under the global dynamics can be lower bounded by the change with respect to the local dynamics.

5.1. Quasi-superadditivity of the relative entropy

It will be useful to see that for large boxes $\Lambda$, any Gibbs measure $\nu$ is not too far from the measure $\nu_{\Lambda}\otimes\nu_{\Lambda^{c}}$, where the particles in $\Lambda$ and $\Lambda^{c}$ are independently distributed according to the correct marginal of $\nu$.

Lemma 5.2** (Factor property of Gibbs measures).**

For every $\epsilon>0$, there exists some $L=L(\epsilon)>0$ such that for every box $\Lambda$ with side length $\geq L$, any Gibbs measure $\nu$ is absolutely continuous with respect to $\nu_{\Lambda}\otimes\nu_{\Lambda^{c}}$ and we have the bound

[TABLE]

for the Radon–Nikodym derivative $g_{\Lambda}:=\mathrm{d}\nu/\mathrm{d}(\nu_{\Lambda}\otimes\nu_{\Lambda^{c}})$.

Proof.

An easy calculation using the DLR equations gives

[TABLE]

which means that actually $g_{\Lambda}\leq\exp(c|\partial\Lambda|)$. ∎

This weak dependence under $\nu$ between regions $\Lambda$ and $\Lambda^{c}$ translates to a “quasi-superadditivity” of the entropy in the following sense.

Lemma 5.3** (Quasi-superadditivity of the entropy w.r.t. Gibbs measures).**

Consider the situation of Lemma 5.2, and let $\mu\in\mathcal{P}(\Omega)$. Then, for every $\Delta\subseteq\Lambda^{c}$,

[TABLE]

Proof.

We report here the proof from [Sul77, Lemma 4.1]. Assume $\Delta=\Lambda^{c}$; the general statement requires the Radon–Nikodym derivative $\mathrm{d}\nu/(\mathrm{d}(\nu_{\Lambda}\otimes\nu_{\Delta})$, which is obtained from $g_{\Lambda}$ by conditional expectation.

Of course, if $I(\mu|\nu)=\infty$, the statement is trivial, so we can assume it to be finite, which means there exists a density $f$ such that $\mu=f\nu=fg_{\Lambda}\nu_{\Lambda}\otimes\nu_{\Lambda^{c}}$. In this case, since $0\leq g_{\Lambda}(\omega_{\Lambda},\omega_{\Lambda^{c}})\leq\exp(\epsilon|\Lambda|)$, we have

[TABLE]

which yields the desired estimate, since

[TABLE]

which finishes the proof. ∎

As a by-product, we obtain the existence (and lower semi-continuity) of the relative entropy with respect to $\nu$. Let us note that in continuum this existence usually requires quite some work, see e.g. [JKSZ24, Section 4.4], whereas the proof given here is very short and elementary.

Proposition 5.4**.**

For any translation invariant $\mu$ the relative entropy density $\mathscr{I}(\mu|\nu)$ with respect to $\nu$ exists as a limit and is lower semicontinuous.

Proof.

*i. Existence of the limit: * Let $\epsilon>0$ be arbitrary and $n\geq L(\epsilon)$ from Lemma 5.2. It suffices to show that there exists $M=M(\epsilon,n)>0$ such that for all $m\geq M$ we have

[TABLE]

Indeed, if $(m_{k})_{k\in\mathbb{N}}$, $(n_{k})_{k\in\mathbb{N}}$ are increasing sequences tending to $\infty$ such that

[TABLE]

then (5.2) implies that

[TABLE]

Since this holds for every $\epsilon>0$, the two sides of the inequality have to agree and the claim follows. To prove (5.2), take a box $\Lambda_{m}$ of side-length $m$ and first note that there is nothing to prove if the right-hand side is infinite. Otherwise, we can decompose $\Lambda_{m}$ into

[TABLE]

where $\Lambda^{i}$, $i=1,\dots,\ell$ are disjoint translates of $\Lambda_{n}$ and $\Delta\subset\Lambda_{m}$ is the possibly empty remainder. By choosing $M=M(n,\epsilon)$ sufficiently large, this can be arranged in a way such that

[TABLE]

To see that such a partition always exists one can for example rewrite $m=qn+r$ where $q>0$ and $r\in[0,n)$. Then, one can cover $\Lambda_{m}$ by disjoint translates of $\Lambda_{n}$ in a way such that $\left\lvert\Delta\right\rvert\lesssim_{d}n^{d}$. To such a partition we can now repeatedly apply the estimate from Lemma 5.3 and use the translation invariance of $\mu$ and $\nu$ to obtain

[TABLE]

Combining this with (5.4) and noting that $\left\lvert\Lambda_{m}\right\rvert=\ell\left\lvert\Lambda_{n}\right\rvert+\left\lvert\Delta\right\rvert$ yields

[TABLE]

as claimed.

*ii. Lower semicontinuity: * First note that for every finite volume $\Lambda\Subset\mathbb{R}^{d}$, the Donsker–Varadhan variational formula implies that the functional $\mu\mapsto I_{\Lambda}(\mu|\nu)$ is lower semicontinuous with respect to the $\tau_{\mathcal{L}}$-topology.

Now, let $(\mu_{k})_{k\in\mathbb{N}}$ be a sequence of measures that converges to $\mu$ in the $\tau_{\mathcal{L}}$-topology and let $\epsilon>0$. We can again take (5.2) as a starting point and see that for every sufficiently large $n\in\mathbb{N}$ we have

[TABLE]

The claimed lower semicontinuity now follows from taking the limit $n\uparrow\infty$ and noting that $\epsilon>0$ was arbitrary. ∎

5.2. Comparison of relative entropies under global and local dynamics

We now wish to compare finite-volume entropies in increasing volumes. Consider a set $\Lambda$, contained in a larger set $\Lambda^{\ast}$; the lemma below shows that the finite-volume entropy $I_{\Lambda}(\mu T_{t}|\nu)$ is at most only slightly larger than I_{\Lambda}\bigl{(}\mu T_{t}^{\scriptscriptstyle{({\Lambda^{\ast}}})}\,\big{|}\,\nu\bigr{)} if $\Lambda$ is sufficiently far from the boundary $\partial\Lambda^{\ast}$.

Lemma 5.5**.**

Let $\Lambda,\Lambda^{\ast}\Subset\mathbb{R}^{d}$ be bounded and measurable. For every $\epsilon>0$, there exists $\ell=\ell(t,\Lambda)>0$ such that

[TABLE]

for every translate $\widetilde{\Lambda}\subset\Lambda^{\ast}$ of $\Lambda$ such that $\mathrm{dist}(\widetilde{\Lambda},(\Lambda^{\ast})^{c})\geq\ell$.

Proof.

By translation invariance and the Donsker–Varadhan formula, cf. [DV76, Theorem 2.1] or [BLM13, Corollary 4.15],

[TABLE]

By dominated convergence, we can choose $F=F(\cdot_{\Lambda})$ to be bounded such that

[TABLE]

Now set $G=F\circ\theta_{x}$, where $x\in\mathbb{R}^{d}$ is such that $\Lambda=\theta_{x}\widetilde{\Lambda}$. By Lemma 4.5,

[TABLE]

We then have

[TABLE]

so that the claim follows by taking \ell=\log\bigl{(}2/(\epsilon c_{t,\Lambda}\|F\|_{\infty})\bigr{)}. ∎

5.3. Proof of monotonicity

With the two main technical ingredients at hand, we are now ready to provide the proof of Proposition 5.1.

Proof of Proposition 5.1.

For every finite volume $\Lambda$, the function t\mapsto I_{\Lambda}\bigl{(}\mu T^{\scriptscriptstyle{({\Lambda}})}_{t}\big{|}\nu\bigr{)} is decreasing by Jensen’s inequality. Hence, $\widetilde{\mathscr{I}}_{\mu}(t)$ is a decreasing function of $t$ as a limit of decreasing functions.

By the semigroup property, and since $\mathscr{I}(\mu T_{0}|\nu)=\mathscr{I}(\mu|\nu)=\widetilde{\mathscr{I}}_{\mu}(0)$, in order to see that $t\mapsto\mathscr{I}(\mu T_{t}|\nu)$ is decreasing, we only have to show that

[TABLE]

Fix a box $\Lambda\Subset\mathbb{R}^{d}$, $\epsilon>0$, and choose a sequence $(\Lambda^{\ast}_{k})_{k}$ of boxes such that

[TABLE]

By repeatedly using Lemma 5.3, we can choose $K$ large enough, such that

[TABLE]

for all $k\geq K$ and for any disjoint translates $\Lambda_{i}\subset\Lambda^{\ast}_{k}$ of $\Lambda$ such that $\Lambda^{\ast}_{k}$ is almost partitioned by $(\Lambda_{i})_{1\leq i\leq n_{k}}$. Now, fix some $r\in\mathbb{R}$ such that $I_{\Lambda}(\mu T_{t}|\nu)>r$. By Lemma 5.5, there exists a fixed $\ell>0$ such that $I_{\Lambda_{i}}(\mu T_{t}^{\scriptscriptstyle{({\Lambda^{\ast}_{k}}})}|\nu)>r$, if the distance between $\Lambda_{i}$ and $(\Lambda^{\ast}_{k})^{c}$ is greater than $\ell$.

It follows that

[TABLE]

As $\epsilon>0$ was arbitrary, and also $r$ can be made arbitrarily close to $I_{\Lambda}(\mu T_{t}|\nu)$, we have

[TABLE]

i.e., since the choice of $\Lambda$ was also arbitrary, we have obtained the desired estimate $\widetilde{\mathscr{I}}_{\mu}(t)\geq\mathscr{I}(\mu T_{t}|\nu)$. ∎

6. Fisher information and the attractor property

So far we have seen that for any initial distribution $\mu$ the map $t\mapsto\mathscr{I}(\mu_{t}|\nu)$ is non-increasing. In order to show the claimed lower bound for the change of relative entropy in terms of the Fisher information and the claim that all possible weak limit points for regular starting measures $\mu$ are necessarily Gibbs measures we will now derive a more explicit representation of the change of the relative entropy under the local dynamics. Our first goal for this section is to derive the rather explicit representation

[TABLE]

for regular starting measures $\mu$. This implies the second statement of Theorem 2.3 and is done in two steps.

6.1. The double-layer representation

First, we show that the left-hand side of the above representation can be written in terms of the (spatially averaged) relative entropy of some particular measures on a double layer system, i.e., with configuration space $\Omega_{\Lambda}\times\Omega_{\Lambda}$. These measures are related to the addition and removal of a point and are defined as follows.

Definition 6.1**.**

Let $(\mu^{x})_{x\in\Lambda}$ be (up to a rescaling factor) the local Palm distributions (cf. [DVJ08, Equations 13.1.4 and 13.1.5]) for $\mu_{\Lambda}$, i.e.,

[TABLE]

for suitable functions $f$ (non-negative or integrable with respect to the Campbell measure).

We introduce the measures $\mu\ast G^{\scriptscriptstyle{({\Lambda}})}_{x}$ and $\mu\circ G^{\scriptscriptstyle{({\Lambda}})}_{x}$ on $\Omega_{\Lambda}\times\Omega_{\Lambda}$ defined, respectively, via

[TABLE]

and

[TABLE]

where $b^{\nu}_{\Lambda}(x,\eta_{\Lambda})$ is the birth rate of the modified dynamics introduced in Definition 4.1.

We first make the following observation.

Lemma 6.2**.**

The measures $\mu\ast G^{\scriptscriptstyle{({\Lambda}})}_{x}$ and $\mu\circ G^{\scriptscriptstyle{({\Lambda}})}_{x}$ are supported on the set of pairs of configurations $(\eta_{\Lambda},\zeta_{\Lambda})\in\Omega_{\Lambda}\times\Omega_{\Lambda}$ differing only at $x$, that is, their support is

[TABLE]

Moreover, for such configurations, we have

[TABLE]

for $\mathrm{d}x$-a.e. $x$.

Proof.

Let $g$ be any bounded measurable function on $\Omega_{\Lambda}\times\Omega_{\Lambda}$, let $f_{0}:=\mathrm{d}\mu_{\Lambda}/\mathrm{d}\nu_{\Lambda}$, and compute

[TABLE]

Hence,

[TABLE]

for $\mathrm{d}x$-a.e. $x$. As we only have to check countably many $g$’s to decide whether two measures on $\Omega_{\Lambda}\times\Omega_{\Lambda}$ (with the canonical $\sigma$-algebra) coincide (cf. [Jan18, Proposition 2.8]), the statement of this lemma is shown. ∎

We can now formulate the first desired representation of the change of relative entropy under the local dynamics.

Proposition 6.3** (Double-layer representation).**

For regular measures $\mu$ we have

[TABLE]

Proof.

First note that the statement can be written as

[TABLE]

In order to see this, thanks to the semigroup property, it is enough to show that

[TABLE]

By reversibility,

[TABLE]

and since $\mu$ is a regular measure, we can apply Lemma 4.3 to find that

[TABLE]

It is easy to check that for $t\in[0,T]$ there exists $z^{\prime}\in(1,\infty)$ such that

[TABLE]

Therefore, denoting $F(t,\omega)=f_{t}(\omega)\log(f_{t}(\omega))$, we have that

[TABLE]

for some $z^{\prime\prime}\in(1,\infty)$. By dominated convergence ($(z^{\prime\prime})^{N_{\Lambda}}$ is integrable w.r.t. $\nu$) we find that

[TABLE]

We then have

[TABLE]

where $f_{0}=\mathrm{d}\mu_{\Lambda}/\mathrm{d}\nu_{\Lambda}$, and the last equality follows from Lemma 6.2.

The conclusion then easily follows from 6.1 and continuity of the derivative, which is seen in the form

[TABLE]

shown above. ∎

6.1.1. Fisher information

Now we show that the double-layer representation of the relative entropy change as in Proposition 6.3 can be expressed in terms of an appropriately rescaled modified Fisher information.

Proposition 6.4**.**

We have the representations

[TABLE]

Proof.

Recall that, by Lemma 6.2, the measures $\mu*G^{(\Lambda)}_{x}$ and $\mu\circ G^{\Lambda}_{x}$ are supported on the set of pairs of configurations $(\eta_{\Lambda},\zeta_{\Lambda})$ which only differ in $x$, i.e.,

[TABLE]

This allows us to rewrite

[TABLE]

By definition of $\mu*G^{(\Lambda)}_{x}$ the inner integral can be rewritten as

[TABLE]

Now note that, for $x\in\Lambda$, $d\mu_{\Lambda}^{x}\mathrm{d}\nu_{\Lambda}^{x}=d\mu_{\Lambda}/d\nu_{\Lambda}$. Indeed, by definition of the local Palm distributions, Definition 6.1, of $\nu$ and $\mu$, we have

[TABLE]

Since the restriction $\nu_{\Lambda}$ of $\nu$ is reversible for the process with birth rates $b^{\nu}_{\Lambda}$, we can use the characterization of $\nu$ from Lemma 3.4 to rewrite 6.4 as

[TABLE]

and putting it together with 6.3 yields

[TABLE]

as desired. ∎

6.2. Bounding the Fisher information from below

To conclude the proof of Theorem 2.3 it remains to show that we can lower bound the density limit of the Fisher information by the more explicit functional $\mu\mapsto\xi^{\mu}(s)$ that already appeared in the statement of our first main result.

Proposition 6.5**.**

Let $\mu$ be a regular starting measure. Then, for all $s\geq 0$ we have

[TABLE]

Moreover, we have $\xi^{\mu}(s)=0$ if and only if $\mu T_{s}\in\mathscr{G}_{\theta}$.

For the proof we make use of the double-layer representation derived in Proposition 6.3.

Proof.

First note that, by definition, we have

[TABLE]

for all $s\geq 0$ and $x\in\mathbb{R}^{d}$. Now fix $\epsilon>0$. Suppose that $\xi^{\mu}(s)<\infty$. Then, by the Donsker–Varadhan variational formula and approximation, there exists a bounded and $\Delta$-local measurable function $F$ such that

[TABLE]

with $F_{x}=F\circ\theta_{x}$ for all $x\in\mathbb{R}^{d}$. We will go on to show that

[TABLE]

If we had $\xi^{\mu}(s)=\infty$, we could use the same argument to find for every $N\in\mathbb{N}$ a bounded and $\Delta$-local measurable function $F$ such that

[TABLE]

and hence

[TABLE]

or ultimately

[TABLE]

Let $\Lambda^{\ominus}$ denote the set

[TABLE]

which is such that $|\Lambda^{\ominus}|/|\Lambda|\xrightarrow[\Lambda\uparrow\mathbb{R}^{d}]{}1$ but $\mathrm{dist}(\Lambda^{\ominus},\Lambda^{c})\xrightarrow[\Lambda\uparrow\mathbb{R}^{d}]{}\infty$. Then,

[TABLE]

By definition,

[TABLE]

and therefore

[TABLE]

We have to compare this with

[TABLE]

By Lemma 4.5, we have

[TABLE]

This concludes the proof, as the other terms can be dealt with completely analogously, and we can send $\epsilon\downarrow 0$ after taking the $\liminf_{\Lambda\uparrow\mathbb{R}^{d}}$.

To see that $\xi^{\mu}(s)$ vanishes if and only if $\mu T_{s}\in\mathscr{G}_{\theta}$ first note that $\xi^{\mu}(s)=0$ if and only if $b(o,\cdot)\mu=\mu_{o}^{!}$. But this is equivalent to $\mu T_{s}\in\mathscr{G}_{\theta}$ by Lemma 3.4. ∎

This concludes the proof of our first main result Theorem 2.3.

6.3. Identification of limit points as Gibbs measures

We are now finally in the position to identify all possible weak limit points as Gibbs measures.

Proof of Theorem 2.4.

Suppose $\rho$ is some limit point in the $\tau_{\mathcal{L}}$-topology of $(\mu T_{t})_{t\geq 0}$ along the sequence $(t_{k})_{k\geq 0}$, but $\rho\not\in\mathscr{G}_{\theta}$. Then, again by Lemma 3.4, and thus by the Donsker–Varadhan variational formula and approximation, there is a bounded, local function $F$ such that

[TABLE]

with $G(\eta):=\sum_{x\in\eta_{[0,1]^{d}}}\exp(F(\theta_{x}\eta\setminus\{o\}))$. Now, again by the Donsker–Varadhan variational formula, we have

[TABLE]

We also have the bound

[TABLE]

and the right-hand side goes to zero for $n\uparrow\infty$ uniformly in $t>0$, which can be seen by graphical representation, i.e., the construction of the process given in Proposition 2.1. Indeed, we can estimate

[TABLE]

where

[TABLE]

The first term on the right-hand side does not depend on $t$ and tends to [math] as $n$ goes to infinity by regularity of $\mu$. For the second term, note that for fixed initial configuration $\eta$, we can estimate the point counting variable $Y^{(\eta)}_{t}$ by

[TABLE]

Now it suffices to note that this is simply a Poisson random variable with mean $\left\lVert b\right\rVert_{\infty}(1-e^{-t})$. By combining the inequalities (6.5) and (6.6) with the definition of $\delta$ we obtain

[TABLE]

and, for $\widetilde{F}:=b(o,\cdot)F-G\mathds{1}_{N_{[0,1]^{d}}\leq n}$ and $n$ large enough, uniformly in $t>0$,

[TABLE]

Now, for any $\epsilon^{\prime}>0$, we see that

[TABLE]

where $q_{1}:=\mathrm{e}^{-\epsilon^{\prime}|\Lambda|\|b\|_{\infty}}$ and $q_{2}:=\mathrm{e}^{-\epsilon^{\prime}}$ for all $\epsilon^{\prime}>0$. But, for all $t>0$,

[TABLE]

Therefore, choosing $\epsilon>0$ small enough, we have, for all $\epsilon^{\prime}\in[0,\epsilon]$ and $k\in\mathbb{N}$,

[TABLE]

As $\mu T_{t_{k}}\xrightarrow[k\uparrow\infty]{}\rho$ in the $\tau_{\mathcal{L}}$-topology, for $k\in\mathbb{N}$ large enough and for all $\epsilon^{\prime}\in[0,\epsilon]$, we get

[TABLE]

Hence, for $K$ large enough,

[TABLE]

which is impossible. ∎

Acknowledgments

AZ thanks Lorenzo Dello Schiavo for the useful discussion. BJ and JK gratefully received support by the Leibniz Association within the Leibniz Junior Research Group on Probabilistic Methods for Dynamic Communication Networks as part of the Leibniz Competition (grant no. J105/2020). BJ gratefully received support from Deutsche Forschungsgemeinschaft through DFG Project no. P27 within the SPP 2265. AZ is also affiliated with the University of Potsdam.