# Exponential equilibration of genetic circuits using entropy methods

## Abstract

We analyse a continuum model for genetic circuits based on a partial integro-differential equation initially proposed in Friedman et al. (Phys Rev Lett 97(16):168302, 2006) as an approximation of a chemical master equation. We use entropy methods to show exponentially fast convergence to equilibrium for this model with explicit bounds. The asymptotic equilibration for the multidimensional case of more than one gene is also obtained under suitable assumptions on the equilibrium stationary states. The asymptotic equilibration property for networks involving one and more than one gene is investigated via numerical simulations.

## Mathematics Subject Classification

35B40 92Dxx 39B99 65M99## 1 Introduction

Translation of the information encoded in genes is responsible for all cellular functions. The decoding of DNA can be summarised, following the central dogma of molecular biology, in two steps: the transcription into messenger RNA and the translation into proteins. Cells produce responses to environmental signals, thanks to the regulation of DNA expression via certain feedback mechanism activating or inhibiting the genes. Typically, regulation is produced by the union of proteins to the DNA binding sites. Moreover, the number of species involved in gene regulatory networks (gene expression together with their regulation) is small, which makes its behaviour inherently stochastic (Elowitz et al. 2002; Gillespie 2007; Kaeet al. 2005; McAdams and Arkin 1997; Paulsson 2004). This underlying stochastic behaviour in gene regulatory networks is captured by using the chemical master equation (CME) (Kepler and Elston 2001; Mackey et al. 2011; Paulsson 2005; Sherman and Cohen 2014). However, the CME solution is unavailable in most cases, due to the large (even infinite) number of coupled equations.

There are two main ways to obtain the CME solution: via stochastic simulation or via approximations of the CME. One of the most extended methods to reproduce the CME dynamics using stochastic realisations is the stochastic simulation algorithm (SSA) (Gillespie 1976, 2007). This method has no restrictions in its applicability, even though it is computationally expensive. On the other hand, CME approximations which remain valid under certain conditions include the finite state projection (Munsky and Khammash 2006), moment methods (Engblom 2006; Hasenauer et al. 2015), linear noise approximations (Thomas et al. 2014; Kampen 2007; Wallace et al. 2012) or hybrid models (Jahnke 2011).

In addition to the above mentioned methods, assuming that protein production takes place in bursts one can obtain a partial integro-differential equation (PIDE) as a continuous approximation of the CME. This PIDE has a mathematical structure very similar to kinetic and transport equations in mathematical biology (Perthame 2007) and it admits an analytical solution for its steady state in the case of networks involving only one gene. In the next subsections, we describe both the one dimensional PIDE model (Friedman et al. 2006) for self-regulated gene networks and the generalised PIDE model (Pájaro et al. 2017) for arbitrary genetic circuits. We will discuss the main properties of the stationary states in one dimension to finally explain the main results of this work.

### 1.1 1-dimensional PIDE model

*X*is shown. Note that DNA transcribes into messenger RNA not only from the active state at rate (per unit time \(\tau \)) \(k_m\), but also from the inactive state with rate constant \(k_{\varepsilon }\) lower than \(k_m\), which is known as

*basal transcription level*or

*transcriptional leakage*(Friedman et al. 2006; Ochab-Marcinek and Tabaka 2015; Pájaro et al. 2015). The messenger RNA transcribes into protein

*X*following a first-order process with rate constant (per unit time) \(k_x\). The messenger RNA and protein are degraded at rate constants \(\gamma _m\) and \(\gamma _x\) respectively.

*x*, denoted by \(\rho :{{\mathbb {R}}}_{+} \rightarrow [0, \ 1]\) (see Ochab-Marcinek and Tabaka 2015; Pájaro et al. 2015):

*H*proteins bound to the DNA inhibiting their production (negative feedback) and negative if |

*H*| proteins bound to the DNA activating their production (positive feedback). Then, the rate \(R_T\) of messenger RNA production (transcription) can be written as function of the Hill expression (1.1), \(R_T=k_m c(x)\), with the input function \(c(x):= \left( 1 - \rho (x)\right) + \rho (x)\varepsilon \), where \(\varepsilon \) is the leakage constant defined as \(\varepsilon :=\frac{k_{\varepsilon }}{k_m}\). Note that the function \(R_T\) accounts for the messenger RNA production both from the DNA active state (with probability \(1 - \rho (x)\)) with rate constant \(k_m\) and from the inactive DNA (with probability \(\rho (x)\)) with lower rate constant \(k_{\varepsilon }\).

*mRNA*is much faster than the corresponding to protein, \({\gamma }_m / {\gamma }_x \gg 1\). Such condition is verified in many gene regulatory networks, both in prokaryotic and eukaryotic organisms (Shahrezaei and Swain 2008; Dar et al. 2012), and results in protein being produced in bursts. As suggested in Friedman et al. (2006) and Elgart et al. (2011), the burst size (denoted by \(b=\frac{k_x}{\gamma _m}\)) is typically modelled by an exponential distribution. The conditional probability for protein level to jump from a state

*y*to a state \(x > y\) after a burst is proportional to:

*stationary solution*of Eq. (1.3) (which we sometimes call

*equilibrium*) as \(P_{\infty }(x)\), which therefore verifies the following equation:

*mass*) is equal to 1. This equation has a unique solution with mass 1, which can be written out explicitly as (Ochab-Marcinek and Tabaka 2015; Pájaro et al. 2015):

*Z*being a normalising constant such that \(\int _{0}^{\infty }P_{\infty }(x)\, \mathrm {d}x=1\). Alternatively, stationary solutions may be studied by considering the zero-flux case; see for example Bokes and Singh (May 2017); Bokes et al. (Jul 2018).

### 1.2 Generalised *n*-dimensional PIDE model

Recently the 1D PIDE model has been extended to overcome more general gene regulatory networks than the self-regulation considered by Friedman et al. (2006). As a first step in this extension, Bokes and Singh (2015) propose the use of variable protein degradation rate, in order to accommodate gene networks with decoy binding sites (Lee and Maheshri 2012) to the PIDE model structure. Finally, including the previous models and considering genetic networks involving more than one gene Pájaro et al. (2017) proposed the generalised PIDE model for any number of genes.

In Pájaro et al. (2017) a general gene regulatory network comprising *n* genes, \({\varvec{G}}=\{DNA_1, \ldots , DNA_i, \ldots , DNA_n\}\), is proposed. These genes encoded by DNA-subchains are transcribed into *n* different messenger RNAs \({\varvec{M}}=\{mRNA_1,\)\( \ldots , mRNA_i, \ldots , mRNA_n\}\), which are translated into *n* proteins types \({\varvec{X}}=\{X_1, \ldots ,\)\( X_i, \ldots , X_n\}\). We show a schematic representation of the general network in Fig. 2, which is similar to the self-regulation circuit. The main differences are that: (i) each DNA type can be regulated by others different proteins than the one expressed by the considered gene (cross regulation), and (ii) the protein degradation rate can be a variable function of all proteins types considered.

The structure of this multidimensional network is equivalent to the previous self-regulation case. Each promoter can switch from the inactive states (\(DNAi_{\mathrm {off}}\)) to the active one (\(DNAi_{\mathrm {on}}\)) or vice versa with rate constants \(k_{\mathrm {on}}^{i}\) and \(k_{\mathrm {off}}^{i}\) respectively. The leakage (basal) messenger RNA production from the inactive promoter is conserved at lower rate constant (\(k_{\varepsilon }^i\)) than its production from the active state (\(k_m^i\)). Each *i* messenger RNA type is translated into the protein \(X_i\) at rate constant \(k_x^i\). Both messengers RNA and proteins are degraded with rates \(\gamma _m^i\) and \(\gamma _x^i({\mathbf {x}})\) respectively.

*mRNAi*rate constant from the total repressed \(DNA_i\) (the lowest rate of \(mRNA_i\) production).

*n*proteins \({\mathbf {X}}=\{X_1, \ldots , X_n\}\), we define the

*n*-vector \({\mathbf {x}}=(x_1, \ldots , x_n) \in {\mathbb {R}}_+^n\) as the amount of each protein type. The generalised (

*n*-dimensional) PIDE model, proposed in Pájaro et al. (2017), describes the temporal evolution of the joint density distribution function of

*n*proteins \(p:{{\mathbb {R}}}_{+}\times {{\mathbb {R}}}_{+}^{n} \rightarrow {{\mathbb {R}}}_{+}\):

*i*-th position changed to \(y_i\), (that is: \(({\mathbf {y}}_i)_j=x_j \ \text {if} \ j\ne i \) and \(({\mathbf {y}}_i)_j=y_i \ \text {if} \ j=i \)), and \(\gamma _x^i({\mathbf {x}})\) is the degradation rate function of each protein. The first term in the right-hand side of the equation accounts for protein degradation whereas the integral describes protein production by bursts. The burst size is assumed to follow an exponential distribution, what leads to the conditional probability for protein jumping from a state \(y_i\) to a state \(x_i\) after a burst be given by:

### 1.3 Main results

In this work we will apply entropy methods in order to analyse the asymptotic equilibration for the kinetic equations (1.3) and (1.8). These equations bear a similar structure to the self-similar fragmentation and the growth-fragmentation equations (Perthame and Ryzhik 2005; Laurençot and Perthame 2009; Doumic 2010; Cáceres et al. 2011; Balagué et al. 2013), used for instance in cell division modelling. In those cases, the transport term makes the cluster size of particles grow while the integral term breaks the particles into pieces of smaller size. In our present models, the transport term degrades the number density of proteins while the integral term makes the protein number density to grow.

In fact, the kinetic equations (1.3) and (1.8) have the structure of linear population models as in Michel et al. (2004, 2005) and Carrillo et al. (2011) for which the so-called general relative entropy applies. This fact already reported in Pájaro et al. (2016) implies the existence of infinitely many Lyapunov functionals for these models useful for different purposes among which to analyse their asymptotic behavior. We will make a summary of the main properties of Eq. (1.3) in Sect. 2 together with a quick treatment of the well-posedness theory for these models. They are easily generalisable to the multidimensional case (1.8).

In Sects. 3 and 4, we will improve over the direct application of the general relative entropy method in Pájaro et al. (2016). On one hand, we study in Sect. 3 the case of gene circuits involving one gene, Eq. (1.3), a direct functional inequality between the \(L^2\)-relative entropy and its production leading to exponential convergence. In order to fix our setting, we recall that \(\omega \) is given by (1.2) for some \(b > 0\), and \(c = c(x)\) is given by (1.4), for some constants \(K > 0\), \(H \in {\mathbb {Z}}{\setminus }\{0\}\) and \(0 < \epsilon \le 1\); and \(a > 0\) is a constant. For \(1 \le p < +\infty \) we denote by \(L^p(\Omega )\) the usual Lebesgue spaces of real functions *f* on \(\Omega \) such that \(|f|^p\) is integrable in the Lebesgue sense. We also write \(L^p(\Omega , w)\) to denote the corresponding spaces of functions *f* such that \(|f|^p\) is integrable with a weight *w*.

### Theorem 1.1

*p*be the mild solution to Eq. (1.3) with initial data \(p_0\) (see Definition 2.1). There exists a constant \(\lambda > 0\) depending only on the parameters of the equation (and not on \(p_0\)) such that

The value of \(\lambda \) can be estimated explicitly from the arguments in the proof, though we do not consider the specific value to be a good approximation of the optimal decay rate. The behaviour of the stationary solutions \(P_\infty (x)\) near the origin and infinity is crucial for direct functional inequalities involving the relative entropy and its production in the one dimensional case.

What we are showing is essentially a spectral gap in a weighted \(L^2\) norm, and some remarks are in order regarding the specific choice of space \(L^2((0,+\infty ), P_\infty ^{-1})\) that we have made. As will be seen later, this space is very natural for the technique we are going to use, since the evolution operator is contractive in this norm, and a similar observation is true for any Markov semigroup with an equilibrium. However, it is very likely that this operator also has a spectral gap in \(L^2\) norms with different weights, in weighted \(L^1\) norms, and in other metrics, as is often the case with Markov operators. In many examples (such as the Fokker-Planck equation) it is known that the spectral gap property breaks for weights which are slowly decaying, so that there may not be a spectral gap in \(L^1\), for example. In those cases there are well-known examples of initial data with slowly-decaying tails whose associated solution converges to equilibrium as slowly as one wishes. The same happens for example to the Boltzmann equation from kinetic theory; we refer to Gualdani et al. (2010) for details on the extension of spectral gaps to different weights. So the weight is not only a technical assumption: there may be norms and weights in which the convergence is *not* exponential. However, exponential weights as the ones we use are probably far from being the optimal ones where one can show a similar result.

Section 4 is devoted to the analysis of the multidimensional Eq. (1.8) corresponding to multiple genes involved in the gene transcription. In this case, solutions to the stationary problem (1.9) are not explicit and hence we are not able to control precisely the behaviour of the stationary solutions near the origin and infinity as before. For this reason, we are only able to show convergence towards a unique equilibrium solution assuming its existence with suitable behavior near the origin and infinity:

### Theorem 1.2

*nD*model) Given any mild solution

*p*with normalised nonnegative initial data \(p_0 \in L^1({\mathbb {R}}_+)\) to Eq. (1.8) and given a normalised stationary solution \(P_\infty ({\mathbf {x}})\) to (1.8) satisfying the technical Assumption 4.1 from Sect. 4, it holds that

The proof is based on a weaker variant of our one-dimensional inequality, in which the control between the relative entropy and its production is obtained except for an error term which happens to be small under the assumptions of the behavior of the stationary solution \(P_{\infty }({\mathbf {x}})\). Both results of equilibration are illustrated with numerical simulations in their corresponding sections.

## 2 Mathematical preliminaries and entropy methods

### 2.1 Properties of stationary solutions

*a*due to the presence of the function \(\rho (x)\) and its dependence on

*H*. It is as follows:

- 1.
If \(H>0\), then \(P_{\infty }(x) \simeq x^{a-1}\) as \(x\rightarrow 0^+\) and \(P_{\infty }(x) \simeq x^r e^{-x/b}\) as \(x\rightarrow +\infty \). Then the stationary state \(P_\infty (x)\) exhibits a singularity at zero for \(0<a<1\) and it is smooth otherwise having zero limit for \(a> 1\) and a positive limit for \(a=1\).

- 2.
If \(H< 0\), then \(P_{\infty }(x) \simeq x^r\) as \(x\rightarrow 0^+\) and \(P_{\infty }(x) \simeq x^{a-1} e^{-x/b}\) as \(x\rightarrow +\infty \). Then the stationary state \(P_\infty (x)\) exhibits a singularity at zero for \(a\varepsilon <1\) and it is smooth otherwise having zero limit for \(a\varepsilon >1\) and a positive limit for \(a\varepsilon =1\).

- 1.
If \(a<\dfrac{1}{\varepsilon }\), then \(\lim _{x \rightarrow 0} P_{\infty }(x) = \infty \). 1.1 Only one peak in \(x=0\) (Case 1 Fig 3). 1.2 Two peaks one in \(x=0\) and another in \(x>0\) (Case 2 Fig 3).

- 2.
If \(a > \dfrac{1}{\varepsilon }\), then \(\lim _{x \rightarrow 0} P_{\infty }(x) = 0\). If \(a \ge \dfrac{1}{\varepsilon }\), then \(\lim _{x \rightarrow 0} P_{\infty }(x) = M\) with \(M\ge 0\). 2.1 Only one peak in \(x > 0\) but close to \(x=0\) (Case 3 Fig 3). 2.2 Two different peaks at two points \(x_1, x_2 > 0\) (Case 4 Fig 3). 2.3 Only one peak in \(x \ge 0\) (Case 5 Fig 3).

### 2.2 Well-posedness

*classical solution*to Eq. (1.3) with initial data \(p_0 \in {\mathcal {C}}^1([0,+\infty ))\) is a function \(p \in {\mathcal {C}}^1([0,+\infty ) \times (0,+\infty ))\) which satisfies (1.3) for all \((t,x) \in [0,+\infty ) \times (0,+\infty )\), and such that \(p(0,x) = p_0(x)\) for all \(x \in (0,+\infty )\). It is not hard to show that, given an integrable initial condition \(p_0 \in {\mathcal {C}}^{1,\mathrm {b}}([0,+\infty ))\), there exists a unique mass-conserving classical solution. In order to give a brief sketch of the proof it is perhaps easier to work with

*mild solutions*, which we will introduce now. Given \(p(t)=p(t,\cdot ) \in L^1(0,+\infty )\), we denote by

*L*[

*p*(

*t*)] the right-hand side of (1.3) given by

*p*to (1.3) must satisfy

### Definition 2.1

Let \(p_0 \in L^1(0,+\infty )\). We say that \(p \in {\mathcal {C}}([0,\infty ); L^1(0,+\infty ))\) is a *mild solution* to Eq. (1.3) with initial data \(p_0\) if it satisfies (2.1) for all \(t \ge 0\), for almost all \(x > 0\).

### Theorem 2.2

### Proof

*T*]. Since the equation is linear (and our equation is invariant under time translations), this argument can be iterated to find solutions on \([0,+\infty )\). We refer to Engel and Nagel (2006) and Cañizo et al. (2013) for full details of this standard argument.

If the initial condition \(p_0\) is in \({\mathcal {C}}^{1, \mathrm {b}}(0,+\infty )\), one can see that the iteration above can also be done in the space \(Z := \{ p \in {\mathcal {C}}^{1,\mathrm {b}}([0,T] \times (0,+\infty )) \mid p(0,x) = p_0(x) \text { for x > 0} \}\). This gives the existence of a unique classical solution in this space. \(\square \)

The constructed solutions have basic properties: positivity preserving, \(L^1\)-contraction, and maximum principle.

### Lemma 2.3

*p*be the unique mild solution to Eq. (1.3) given by Theorem 2.2.

- 1.
Positivity is preserved: if \(p_0 \ge 0\) a.e. then \(p(t) \ge 0\) a.e., for all \(t \ge 0\).

- 2.The \(L^1\) norm is decreasingleading to \(L^1\)-contraction by linearity. If \(p_0 \ge 0\), the above inequality becomes an identity.$$\begin{aligned} \Vert p(t) \Vert _1 \le \Vert p_0\Vert _1 \qquad \text {for all}~ {t \ge 0,} \end{aligned}$$
- 3.Maximum principle:$$\begin{aligned} \mathop {{{\mathrm{ess\,inf}}}}\limits _{x> 0} \frac{p_0(x)}{P_\infty (x)} \le \frac{p(t,x)}{P_\infty (x)} \le \mathop {{{\mathrm{ess\,sup}}}}\limits _{x > 0} \frac{p_0(x)}{P_\infty (x)}. \end{aligned}$$

### Proof

*p*is nonnegative if \(p_0\) is nonnegative, since

*p*is a fixed point of the positivity-preserving operator \(\Psi \), which is also contractive in the \(L^\infty \) norm (for example) for

*t*small enough. Now, for a mild solution we obtain the same result by approximation from classical solutions, taking into account the \(L^1\)-stability (2.2).

*f*, so that \(f = f_+ - f_-\). The positivity and mass preservation imply that:

*M*is the supremum on the right hand side, the function \(q = M P_\infty - p\) is a mild solution with nonnegative initial data. Due to preservation of positivity we obtain the inequality on the right-hand side. The minimum principle is obtained analogously. \(\square \)

### 2.3 Entropy and *H*-theorem

*p*(

*t*) is a solution to (1.3), see Michel et al. (2004, 2005), Carrillo et al. (2011) and Pájaro et al. (2016).

### Proposition 2.4

*p*be a classical solution to (1.3) with integrable initial condition \(p_0 \in {\mathcal {C}}^{1,b}[0,+\infty )\) such that \(|p_0(x)| \le M P_\infty (x)\) for some \(M > 0\). Thus, the relative entropy satisfies

### Remark 2.5

Notice that the dependence on the time variable in (2.4) has been omitted for simplicity. Observe that the right-hand side in (2.4) is non-positive since the convexity of *H* implies \(H(u)-H(v)+H'(u)(v-u) \le 0\) for all \(u,v\in {\mathbb {R}}\).

Proposition 2.4 is very close to the results in Section 2 of Michel et al. (2005), but is strictly not contained there due to the form of the integral operator. It is worth giving a derivation of the result, so we include a proof here. We first obtain a technical lemma involving some classical computations in Michel et al. (2005):

### Lemma 2.6

### Proof

### Proof of Proposition 2.4

## 3 Exponential convergence for the 1D PIDE model

*H*is chosen as \(H(u)=(u-1)^2\), and

*p*(

*t*,

*x*) and \(P_\infty (x)\) are probability density functions. Now, by replacing the value of the considered convex function in Proposition 2.4, we obtain the following identity

### Lemma 3.1

### Proof

### 3.1 Entropy-entropy production inequality

We start by obtaining bounds for the steady state solution \(P_{\infty }\), of the Friedman Eq. (1.3).

### Lemma 3.2

*k*.

### Proof

*a*, (this term is increasing if \(a>1\), constant if \(a=1\) and decreasing if \(a<1\)). So that, we can bound \(P_{\infty }(x)\) in the interval \( I_{k,\delta }\) as follows:

*A*and

*B*being,

Note that inequality (3.5) can be directly checked for the simplest open loop case, whose stationary solution is given by (1.7).

### Lemma 3.3

### Proof

In order to prove the exponential convergence of the Friedman Eq. (1.3) we are going to split the proof of inequality (3.4) in the following two propositions.

### Proposition 3.4

### Proof

**Step 1:**\({\mathcal {H}}_{21}(u)\) bound.- We start working on the term \({\mathcal {H}}_{21}(u)(\tau )\), where \(0<\delta<y<x\). By swapping (

*x*,

*y*) in the domain of integration, we get

*i*,

*j*and call \(n:=j-i\ge 1\). We use \(n-1\) “intermediate reactions” to write the following: introduce \(n-1\) dummy integration variables \(z_{i+1}, \dots , z_{j-1}\) and denote averaged integrals with a stroke. Thus, we have:where the last step is just renaming \(x\equiv z_j\) and \(y\equiv z_i\). Observe that nothing has been done in the case \(j=i+1\). Using the Cauchy-Schwarz inequality and (3.7), we haveHence, we deduce that

**Step 2:**\({\mathcal {H}}_{22}(u)\) bound.- To prove that there exists \(\lambda _2>0\) such that

### Proposition 3.5

### Proof

*x*, then

*c*(

*x*) is bounded, \(\varepsilon \le c(x) \le 1 \) for all \(x \in {\mathbb {R}}^+\). So that:

### Proof of Theorem 1.1

Putting together (3.9) and (3.15) from Propositions 3.4 and 3.5, we deduce that the entropy-entropy production inequality (3.4) holds. Lemma 3.1 together with (3.4) finally implies (3.2). As consequence, we deduce the exponential convergence towards \(P_{\infty }\) for all mild solutions of (1.3). \(\square \)

### 3.2 Numerical illustration of exponential convergence

## 4 The *nD* PIDE model

We can generalise the entropy functional (2.3) defined for the one dimension PIDE model in order to study the convergence of the multidimensional model. A well-posedness theory of mild and classical solutions satisfying the positivity and mass preservation, the \(L^1\)-contraction principle, and the maximum principle can be analogously obtained from the one dimensional strategy in Sect. 2. Let us summarize these properties in the next proposition.

### Proposition 4.1

- (i)Mass conservation:$$\begin{aligned} \int _{{{\mathbb {R}}}_{+}^{n}}p(t,{\mathbf {x}})\mathrm {d} {\mathbf {x}}=\int _{{{\mathbb {R}}}_{+}^{n}}p_0({\mathbf {x}})\mathrm {d} {\mathbf {x}} =1 \end{aligned}$$
- (ii)
If \(p_0\) is nonnegative, then the solution

*p*(*t*) of Eq. (1.8) is nonnegative for all \(t\ge 0\). - (iii)\(L^1\)-contraction principle:$$\begin{aligned} \int _{{{\mathbb {R}}}_{+}^{n}}|p(t,{\mathbf {x}})|\mathrm {d} {\mathbf {x}} \le \int _{{{\mathbb {R}}}_{+}^{n}}|p_0({\mathbf {x}})|\mathrm {d} {\mathbf {x}}. \end{aligned}$$
- (iv)\(L^q\) bounds, \(1<q<\infty \):$$\begin{aligned}&\int _{{{\mathbb {R}}}_{+}^{n}}P_{\infty }({\mathbf {x}})|u(t,{\mathbf {x}})|^q\mathrm {d} {\mathbf {x}}\\&\qquad \le \int _{{{\mathbb {R}}}_{+}^{n}}P_{\infty }({\mathbf {x}})|u_0({\mathbf {x}})|^q\mathrm {d} {\mathbf {x}} ~~\text {with}~~ u(t,{\mathbf {x}}):=\frac{p(t,{\mathbf {x}})}{P_{\infty }({\mathbf {x}})}\quad \text{ and }\quad u_0({\mathbf {x}}):=\frac{p_0({\mathbf {x}})}{P_{\infty }({\mathbf {x}})}. \end{aligned}$$
- (v)Maximum principle:$$\begin{aligned} \mathop {{{\mathrm{ess\,inf}}}}\limits _{{\mathbf {x}} \in {{\mathbb {R}}}_{+}^{n}} u_0({\mathbf {x}})\le u(t,{\mathbf {x}}) \le \mathop {{{\mathrm{ess\,sup}}}}\limits _{{\mathbf {x}} \in {{\mathbb {R}}}_{+}^{n}} u_0({\mathbf {x}}). \end{aligned}$$

*H*(

*u*) any convex function of

*u*, we define the

*n*-dimensional general relative entropy functional as:

### Assumption 4.1

*H*(

*u*) and for all differentiable \(p \in L^1((0,+\infty )) \cap L^2((0,+\infty ), P_\infty ^{-1})\).

Similarly to the one dimensional case, we can obtain the following identity. The proof is totally analogous to the one of Lemma 2.6 and we skip it here for brevity.

### Lemma 4.2

*p*be a differentiable function on \({\mathbb {R}}^n_+\). For any \(i=1,\ldots ,n\) the following equality is verified:

With this identity, we can now derive the evolution of the relative entropy as in the one dimensional case. We will not make explicit the time dependency of the solutions again for simplicity.

### Proposition 4.3

*p*be a classical solution to the

*nD*PIDE model with initial data \(p_0 \in L^1({\mathbb {R}}^n_+) \cap {\mathcal {C}}^1({\mathbb {R}}^n_+)\). For any convex function \(H(u({\mathbf {x}}))\), the general entropy functional \({\mathcal {G}}_H^n(u)\) satisfies

### Proof of proposition 4.3

*H*(

*u*), we deduce that \(H(u)-H(v)+H'(u)(v-u) \le 0\) for all

*u*,

*v*leading to final claim. \(\square \)

### 4.1 Approach to equilibrium

Based on the Assumption 4.1 on stationary solutions, we are now able to control the entropy by the entropy production except for a small error term.

### Lemma 4.4

### Proof

*n*terms, each of which being a difference of values of

*u*at points which differ only by one coordinate

### Theorem 4.5

*p*with normalised nonnegative initial data \(p_0 \in L^1({\mathbb {R}}_+)\) to Eq. (1.8) and given a stationary solution \(P_\infty ({\mathbf {x}})\) to (1.8) satisfying Assumption 4.1, then

### Proof

**Step 1: Proof for “nice” initial data.**We first prove the result for initial data \(p_0 \in L^1({\mathbb {R}}^n_+) \cap {\mathcal {C}}^2({\mathbb {R}}^n_+)\) such that \(p_0 \le C_1 P_\infty \), for some constant \(C_1 > 0\). Observe that this implies in particular that \(p_0 \in L^2({\mathbb {R}}^n_+, P_\infty ({\mathbf {x}})^{-1} \,\mathrm {d}{\mathbf {x}})\). For such initial data we deduce that for all \(t \ge 0\)

*T*] in Eq. (4.10), the following equality holds for all \(T>0\):

*t*, this shows that \(\lim _{t \rightarrow +\infty } {\mathcal {G}}_2^n(u)(t) \le \epsilon \). Since \(\epsilon \) is arbitrary chosen, we deduce that:

**Step 2: Proof for all integrable initial data.**It is now classical to extend the result in step 1 to all initial data in \(L^1({\mathbb {R}}^n_+)\) by the \(L^1\)-contraction principle. In fact, any \(p_0 \in L^1({\mathbb {R}}^n_+)\) can be approximated in \(L^1({\mathbb {R}}^n_+)\) by a sequence \((p_0^s)_{s \ge 1}\) such that \(p_0^s \le s P_\infty \), for all \(s \ge 1\). Thus consider the solution \(p^s\) associated to initial data \(p_0^s\). By step 1, we get

### 4.2 Numerical exploration of the convergence rates

The entropy functional, \({\mathcal {G}}_2^n(u)(t)\), is represented in the plots B of Figs. 9, 10 and 11, which address three possible steady states (plots A of Figs. 9, 10 and 11) that have been obtained using the SELANSI toolboox (Pájaro et al. 2018). For all cases, these functions are represented in a semi-logarithm scale to numerically check if the convergence shown in the previous section is exponential in higher dimensions.

For each example described above, a multivariate Gaussian distribution with means 10 and standard deviations 1, \({\mathcal {N}}([10, \ 10], [1, \ 1])\), has been considered as initial condition.

## 5 Conclusions

Analytical results for the *nD* model show convergence to equilibrium via a very general method, but do not give a bound on the convergence rate. The numerical simulations we have carried out clearly support the idea that exponential convergence also holds in the multidimensional case, though we have not been able to prove this using the same entropy method as in the one-dimensional case. Approach to equilibrium seems to follow a steady exponential speed, being quickly dominated by the spectral gap expected from our analysis. There also seem to be initial regimes where the approach to equilibrium can occur much faster; our interpretation is that smaller (more negative) eigenvalues can dominate at initial stages of time evolution, but are overcome by the dominant eigenvalue as equilibrium is approached.

## Notes

### Acknowledgements

J. A. Cañizo and J. A. Carrillo were supported by Projects MTM2014-52056-P and MTM2017-85067-P, funded by the Spanish government and the European Regional Development Fund. J. A. Carrillo was partially supported by the EPSRC Grant Number EP/P031587/1. M. Pájaro acknowledges support from Spanish MINECO fellowships BES-2013-063112, EEBB-I-16-10540 and EEBB-I-17-12182.

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