Publications

We prove new results on the asymptotic behavior of the nonlinear integrate-and-fire neuron model. Among them, we give a criterion for the linearized stability or instability of equilibria, without restriction on the connectivity parameter, which provides a proof of stability or instability in some cases. In all cases, this criterion can be checked numerically, allowing us to give a full picture of the stable and unstable equilibria depending on the connectivity parameter and transmission delay. We also give further spectral results on the associated linear equation, and use them to give improved results on the nonlinear stability of equilibria for weak connectivity, and on the link between linearized and nonlinear stability.

There is a wide range of mathematical models that describe populations of large numbers of neurons. In this article, we focus on nonlinear noisy leaky integrate and fire (NNLIF) models that describe neuronal activity at the level of the membrane potential of neurons. We introduce a set of novel states, which we call “pseudo-equilibria”, and give evidence of their defining role in the behaviour of the NNLIF system when a significant synaptic delay is considered. The advantage is that these states are determined solely by the system’s parameters and are derived from a sequence of firing rates that result from solving a recurrence equation. We propose a new strategy to show convergence to an equilibrium for a weakly connected system with large transmission delay, based on following the sequence of pseudo-equilibria. Unlike with the direct entropy dissipation method, this technique allows us to see how a large delay favours convergence. We also present a detailed numerical study to support our results. This study explores the overall behaviour of the NNLIF system and helps us understand, among other phenomena, periodic solutions in strongly inhibitory networks.

We consider minimizers of the N-particle interaction potential energy and briefly review numerical methods used to calculate them. We consider simple pair potentials which are repulsive at short distances and attractive at long distances, focusing on examples which are sums of two powers. The range of powers we look at includes the well-known case of the Lennard-Jones potential, but we are also interested in less singular potentials which are relevant in collective behavior models. We report on results using the software GMIN developed by Wales and collaborators for problems in chemistry. For all cases, this algorithm gives good candidates for the minimizers for relatively low values of the particle number $N$. This is well-known for potentials similar to Lennard-Jones, but not for the range which is of interest in collective behavior. Standard minimization procedures have been used in the literature in this range, but they are likely to yield stationary states which are not minimizers. We illustrate numerically some properties of the minimizers in 2D, such as lattice structure, Wulff shapes, and the continuous large-$N$ limit for locally integrable (that is, less singular) potentials.

We prove uniqueness of self-similar profiles for the one-dimensional inelastic Boltzmann equation with moderately hard potentials, that is with collision kernel of the form $| \cdot |^\gamma$ for $\gamma > 0$ small enough (explicitly quantified). Our result provides the first uniqueness statement for self-similar profiles of inelastic Boltzmann models allowing for strong inelasticity besides the explicitly solvable case of Maxwell interactions (corresponding to $\gamma = 0$). Our approach relies on a perturbation argument from the corresponding Maxwell model through a careful study of the associated linearised operator. In particular, a part of the paper is devoted to the trend to equilibrium for the Maxwell model in suitable weighted Sobolev spaces, an extension of results which are known to hold in weaker topologies. Our results can be seen as a first step towards a full proof, in the one-dimensional setting, of a conjecture in Ernst & Brito (2002) regarding the determination of the long-time behaviour of solutions to inelastic Boltzmann equation.

We provide simple and constructive proofs of Harris-type theorems on the existence and uniqueness of an equilibrium and the speed of equilibration of discrete-time and continuous-time stochastic semigroups. Our results apply both to cases where the relaxation speed is exponential (also called geometric) and to those with no spectral gap, with non-exponential speeds (also called subgeometric). We give constructive estimates in the subgeometric case and discrete-time statements which seem both to be new. The method of proof also differs from previous works, based on semigroup and interpolation arguments, valid for both geometric and subgeometric cases with essentially the same ideas. In particular, we present very simple new proofs of the geometric case.

We consider Smoluchowski’s coagulation equation with a kernel of the form $K=2+\epsilon W$, where $W$ is a bounded kernel of homogeneity zero. For small $\epsilon$, we prove that solutions approach a universal, unique self-similar profile for large times, at almost the same speed as the constant kernel case (the speed is exponential when self-similar variables are considered). All the constants we use can be explicitly estimated. Our method is a constructive perturbation analysis of the equation, based on spectral results on the linearisation of the constant kernel case. To our knowledge, this is the first time the scaling hypothesis can be fully proved for a family of kernels which are not explicitly solvable.

We show that the Smoluchowski coagulation equation with the solvable kernels $K(x,y)$ equal to $2$, $x+y$ or $xy$ is contractive in suitable Laplace norms. In particular, this proves exponential convergence to a self-similar profile in these norms. These results are parallel to similar properties of Maxwell models for Boltzmann-type equations, and extend already existing results on exponential convergence to self-similarity for Smoluchowski’s coagulation equation.

We study the long-time behaviour of the growth-fragmentation equation, a nonlocal linear evolution equation describing a wide range of phenomena in structured population dynamics. We show the existence of a spectral gap under conditions that generalise those in the literature by using a method based on Harris’s theorem, a result coming from the study of equilibration of Markov processes. The difficulty posed by the non-conservativeness of the equation is overcome by performing an $h$-transform, after solving the dual Perron eigenvalue problem. The existence of the direct Perron eigenvector is then a consequence of our methods, which prove exponential contraction of the evolution equation. Moreover the rate of convergence is explicitly quantifiable in terms of the dual eigenfunction and the coefficients of the equation.

We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus $(x,v)∈ \mathbb{T}^d \times \R^d$ or on the whole space $(x,v) ∈\R^d \times \R^d$ with a confining potential. We present explicit convergence results in total variation or weighted total variation norms (alternatively $L^1$ or weighted $L^1$ norms). The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we give algebraic convergence rates when subquadratic potentials considered. We use a method from the theory of Markov processes known as Harris’s Theorem.

We show uniform-in-time propagation of algebraic and stretched exponential moments for the Becker-Döring equations. Our proof is based upon a suitable use of the maximum principle together with known rates of convergence to equilibrium.

We analyse a continuum model for genetic circuits based on a partial integro-differential equation initially proposed in Friedman, Cai & Xie (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.

We study two population models describing the dynamics of interacting neurons, initially proposed by Pakdaman, Perthame, and Salort (2010, 2014). In the first model, the structuring variable $s$ represents the time elapsed since its last discharge, while in the second one neurons exhibit a fatigue property and the structuring variable is a generic “state”. We prove existence of solutions and steady states in the space of finite, nonnegative measures. Furthermore, we show that solutions converge to the equilibrium exponentially in time in the case of weak nonlinearity (i.e., weak connectivity). The main innovation is the use of Doeblin’s theorem from probability in order to show the existence of a spectral gap property in the linear (no-connectivity) setting. Relaxation to the steady state for the nonlinear models is then proved by a constructive perturbation argument.

We prove an energy inequality for nonlocal diffusion operators of the following type, and some of its generalisations: $Lu (x) := \int_{\mathbb{R}^N} K(x,y) (u(y) - u(x)) \, \mathrm{d}y$, where $L$ acts on a real function u defined on $\mathbb{R}^N$, and we assume that $K(x,y)$ is uniformly strictly positive in a neighbourhood of $x=y$. The inequality is a nonlocal analogue of the Nash inequality, and plays a similar role in the study of the asymptotic decay of solutions to the nonlocal diffusion equation $\partial_t u=Lu$ as the Nash inequality does for the heat equation. The inequality allows us to give a precise decay rate of the $L^p$ norms of $u$ and its derivatives. As compared to existing decay results in the literature, our proof is perhaps simpler and gives new results in some cases (particularly, and surprisingly, in dimensions $N=1,2$).

Under suitable technical conditions we show that minimisers of the discrete interaction energy for attractive-repulsive potentials converge to minimisers of the corresponding continuum energy as the number of particles goes to infinity. We prove that the discrete interaction energy $\Gamma$-converges in the narrow topology to the continuum interaction energy. As an important part of the proof we study support and regularity properties of discrete minimisers: we show that continuum minimisers belong to suitable Morrey spaces and we introduce the set of empirical Morrey measures as a natural discrete analogue containing all the discrete minimisers.

In this work we present several quantitative results of convergence to equilibrium for the linear Boltzmann operator with soft potentials under Grad’s angular cutoff assumption. This is done by an adaptation of the famous entropy method and its variants, resulting in explicit algebraic, or even stretched exponential, rates of convergence to equilibrium under appropriate assumptions. The novelty in our approach is that it involves functional inequalities relating the entropy to its production rate, which have independent applications to equations with mixed linear and non-linear terms. We also briefly discuss some properties of the equation in the non-cutoff case and conjecture what we believe to be the right rate of convergence in that case.

In this work we investigate the rate of convergence to equilibrium for subcritical solutions to the Becker-Döring equations with physically relevant coagulation and fragmentation coefficients and mild assumptions on the given initial data. Using a discrete version of the log-Sobolev inequality with weights we show that in the case where the coagulation coefficient grows linearly and the detailed balance coefficients are of typical form, one can obtain a linear functional inequality for the dissipation of the relative free energy. This results in showing Cercignani’s conjecture for the Becker-Döring equations and consequently in an exponential rate of convergence to equilibrium. We also show that for all other typical cases one can obtain an ’almost’ Cercignani’s conjecture that results in an algebraic rate of convergence to equilibrium. Additionally, we show that if one assumes an exponential moment condition one can recover Jabin and Niethammer’s rate of decay to equilibrium, i.e. an exponential to some fractional power of $t$.

We study general quadratic reaction-diffusion systems with detailed balance, in space dimension $d \leq 4$. We show that close-to-equilibrium solutions (in an $L^2$ sense) are regular for all times, and that they relax to equilibrium exponentially in a strong sense. That is: all detailed balance equilibria are exponentially asymptotically stable in all $L^p$ norms, at least in dimension $d \leq 4$. The results are given in detail for the four-species reaction-diffusion system, where the involved constants can be estimated explicitly. The main novelty is the regularity result and exponential relaxation in Lp norms for p > 1, which up to our knowledge is new in dimensions 3 and 4.

We show that solutions of the 2D Fokker-Planck equation for bosons are defined globally in time and converge to equilibrium, and this convergence is shown to be exponential for radially symmetric solutions. The main observation is that a variant of the Hopf-Cole transformation relates the 2D equation in radial coordinates to the usual linear Fokker-Planck equation. Hence, radially symmetric solutions can be computed analytically, and our results for general (non radially symmetric) solutions follow from comparison and entropy arguments. In order to show convergence to equilibrium we also prove a version of the Csiszár-Kullback inequality for the Bose-Einstein-Fokker-Planck entropy functional.

We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres (with constant restitution coefficient $\alpha \in (0,1)$) under the thermalization induced by a host medium with a fixed Maxwellian distribution. We prove that the solution to the associated initial-value problem converges exponentially fast towards the unique equilibrium solution. The proof combines a careful spectral analysis of the linearised semigroup as well as entropy estimates. The trend towards equilibrium holds in the weakly inelastic regime in which $\alpha$ is close to $1$, and the rate of convergence is explicit and depends solely on the spectral gap of the elastic linearised collision operator.

We consider a collective behavior model in which individuals try to imitate each others’ velocity and have a preferred speed. We show that a phase change phenomenon takes place as diffusion decreases, bringing the system from a “disordered” to an “ordered” state. This effect is related to recently noticed phenomena for the diffusive Vicsek model. We also carry out numerical simulations of the system and give further details on the phase transition.

We prove a linear inequality between the entropy and entropy dissipation functionals for the linear Boltzmann operator (with a Maxwellian equilibrium background). This provides a positive answer to the analogue of Cercignani’s conjecture for this linear collision operator. Our result covers the physically relevant case of hard-spheres interactions as well as Maxwellian kernels, and we always work with a cut-off assumption. For Maxwellian kernels, the proof of the inequality is surprisingly simple and relies on a general estimate of the entropy of the gain part operator due to Villani (1998) and Matthes and Toscani (2012). For more general kernels, the proof relies on a comparison principle. Finally, we also show that in the grazing collision limit our results allow to recover known logarithmic Sobolev inequalities.

The existence of compactly supported global minimisers for continuum models of particles interacting through a potential is shown under almost optimal hypotheses. The main assumption on the potential is that it is catastrophic or not H-stable, which is the complementary assumption to that in classical results on thermodynamic limits in statistical mechanics. The proof is based on a uniform control on the local mass around each point of the support of a global minimiser, together with an estimate on the size of the ’holes’ that a minimiser may have. The class of potentials for which we prove existence of minimisers includes power-law potentials and, for some range of parameters, Morse potentials, widely used in applications. Finally, using Euler-Lagrange conditions on local minimisers we give a link to classical obstacle problems in the calculus of variations.

We present a refined duality estimate for parabolic equations. This estimate entails new results for systems of reaction-diffusion equations, including smoothness and exponential convergence towards equilibrium for equations with quadratic right-hand sides in two dimensions. For general systems in any space dimension, we obtain smooth solutions of reaction-diffusion systems coming out of reversible chemistry under an assumption that the diffusion coefficients are sufficiently close one to another.

We prove that any subcritical solution to the Becker-Döring equations converges exponentially fast to the unique steady state with same mass. Our convergence result is quantitative and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation, for which several bounds are provided. This improves the known convergence result by Jabin & Niethammer (see ref. [14]). Our approach is based on a careful spectral analysis of the linearized Becker-Döring equation (which is new to our knowledge) in both a Hilbert setting and in certain weighted $\ell^1$ spaces. This spectral analysis is then combined with uniform exponential moment bounds of solutions in order to obtain a convergence result for the nonlinear equation.

We study the creation and propagation of exponential moments of solutions to the spatially homogeneous $d$-dimensional Boltzmann equation. In particular, when the collision kernel is of the form $|v-v_*|^\beta b(\cos(\theta))$ for $\gamma \in (0, 2]$ with $\cos (\theta) = |v - v_*|^{-1}(v - v *) \cdot σ$ and $σ \in \mathbb{S}^{d-1}$, and assuming the classical cut-off condition $b(\cos (\theta))$ integrable in $\mathbb{S}^{d-1}$, we prove that there exists $a > 0$ such that moments with weight $\exp (a \min\{ t, 1\} |v|^\beta)$ are finite for $t > 0$, where $a$ only depends on the collision kernel and the initial mass and energy. We propose a novel method of proof based on a single differential inequality for the exponential moment with time-dependent coefficients.

We are concerned with the long-time behavior of the growth-fragmentation equation. We prove fine estimates on the principal eigenfunctions of the growth-fragmentation operator, giving their first-order behavior close to $0$ and $+\infty$. Using these estimates we prove a spectral gap result by following the technique in [Cáceres, Cañizo, Mischler 2011, JMPA], which implies that solutions decay to the equilibrium exponentially fast. The growth and fragmentation coefficients we consider are quite general, essentially only assumed to behave asymptotically like power laws.

We give a direct proof of well-posedness of solutions to general selection-mutation and structured population models with measures as initial data. This is motivated by the fact that some stationary states of these models are measures and not $L^1$ functions, so the measures are a more natural space to study their dynamics. Our techniques are based on distances between measures appearing in optimal transport and common arguments involving Picard iterations. These tools provide a simplification of previous approaches and are applicable or adaptable to a wide variety of models in population dynamics.

We analyse qualitative properties of the solutions to a mean-field equation for particles interacting through a pairwise potential while diffusing by Brownian motion. Interaction and diffusion compete with each other depending on the character of the potential. We provide sufficient conditions on the relation between the interaction potential and the initial data for diffusion to be the dominant term. We give decay rates of Sobolev norms showing that asymptotically for large times the behavior is then given by the heat equation. Moreover, we show an optimal rate of convergence in the L1-norm towards the fundamental solution of the heat equation.

We consider the continuous version of the Vicsek model with noise, proposed as a model for collective behaviour of individuals with a fixed speed. We rigorously derive the kinetic mean-field partial differential equation satisfied when the number $N$ of particles tends to infinity, quantifying the convergence of the law of one particle to the solution of the PDE. For this we adapt a classical coupling argument to the present case in which both the particle system and the PDE are defined on a surface rather than on the whole space $\mathbb{R}^d$. As part of the study we give existence and uniqueness results for both the particle system and the PDE.

We consider the spatially homogeneous Boltzmann equation for inelastic hard-spheres (with constant restitution coefficient $\alpha \in (0,1)$) under the thermalization induced by a host medium with a fixed Maxwellian distribution. We prove uniqueness of the stationary solution (with given mass) in the weakly inelastic regime, i.e., for any inelasticity parameter $\alpha \in (\alpha_0,1)$, with some constructive $\alpha_0 \in [0, 1)$. Our analysis is based on a perturbative argument which uses the knowledge of the stationary solution in the elastic limit and quantitative estimates of the convergence of stationary solutions as the inelasticity parameter goes to 1. In order to achieve this proof we give an accurate spectral analysis of the associated linearized collision operator in the elastic limit. Several qualitative properties of this unique steady state $F_\alpha$ are also derived; in particular, we prove that $F_\alpha$ is bounded from above and from below by two explicit universal (i.e., independent of $\alpha$) Maxwellian distributions.

We study the asymptotic behavior of linear evolution equations of the type $\partial_t g = Dg + Lg − \lambda g$, where $L$ is the fragmentation operator, $D$ is a differential operator, and $\lambda$ is the largest eigenvalue of the operator $Dg+Lg$. In the case $Dg=−x \partial g$, this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case $Dg=−x \partial(xg)$, it is known that $\lambda=1$ and the equation is the self-similar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation $\partial_t f=Lf$. By means of entropy-entropy dissipation inequalities, we give general conditions for $g$ to converge exponentially fast to the steady state $G$ of the linear evolution equation, suitably normalized. In other words, the linear operator has a spectral gap in the natural $L^2$ space associated to the steady state. We extend this spectral gap to larger spaces using a recent technique based on a decomposition of the operator in a dissipative part and a regularizing part.

We consider general stochastic systems of interacting particles with noise which are relevant as models for the collective behavior of animals, and rigorously prove that in the mean-field limit the system is close to the solution of a kinetic PDE. Our aim is to include models widely studied in the literature such as the Cucker–Smale model, adding noise to the behavior of individuals. The difficulty, as compared to the classical case of globally Lipschitz potentials, is that in several models the interaction potential between particles is only locally Lipschitz, the local Lipschitz constant growing to infinity with the size of the region considered. With this in mind, we present an extension of the classical theory for globally Lipschitz interactions, which works for only locally Lipschitz ones.

In a recent result by the authors it was proved that solutions of the self-similar fragmentation equation converge to equilibrium exponentially fast. This was done by showing a spectral gap in weighted $L^2$ spaces of the operator defining the time evolution. In the present work we prove that there is also a spectral gap in weighted $L^1$ spaces, thus extending exponential convergence to a larger set of initial conditions. The main tool is an extension result in ref. [4].

We present existence, uniqueness and continuous dependence results for some kinetic equations motivated by models for the collective behavior of large groups of individuals. Models of this kind have been recently proposed to study the behavior of large groups of animals, such as flocks of birds, swarms, or schools of fish. Our aim is to give a well-posedness theory for general models which possibly include a variety of effects: an interaction through a potential, such as a short-range repulsion and long-range attraction; a velocity-averaging effect where individuals try to adapt their own velocity to that of other individuals in their surroundings; and self-propulsion effects, which take into account effects on one individual that are independent of the others. We develop our theory in a space of measures, using mass transportation distances. As consequences of our theory we show also the convergence of particle systems to their corresponding kinetic equations, and the local-in-time convergence to the hydrodynamic limit for one of the models.

We consider Smoluchowski’s equation with a homogeneous kernel of the form $a(x,y) = x^\alpha y^\beta + y^\beta x^\alpha$, with $-1 < \alpha \leq \beta < 1$, and $-1 < \alpha + \beta < 1$. We first show that self-similar solutions of this equation are infinitely differentiable and prove sharp results on the behavior of self-similar profiles at $y = 0$ in the case $\alpha < 0$. We also give some partial uniqueness results for self-similar profiles: in the case $\alpha = 0$ we prove that two profiles with the same mass and moment of order $\alpha+\beta$ are necessarily equal, while in the case $\alpha < 0$ we prove that two profiles with the same moments of order $\alpha$ and $\beta$, and which are asymptotic at $y=0$, are equal. Our methods include a new representation of the coagulation operator, and estimates of its regularity using derivatives of fractional order.

We show in this work that gelation does not occur for a class of discrete coagulation-fragmentation models with size-dependent diffusion. With respect to a previous work [2], we do not assume here that the diffusion rates of clusters are bounded below. The proof uses a duality argument first devised for reaction-diffusion systems with a finite number of equations [13].

We show that solutions to Smoluchowski’s equation with a constant coagulation kernel and an initial datum with some regularity and exponentially decaying tail converge exponentially fast to a self-similar profile. This convergence holds in a weighted Sobolev norm which implies the $L^2$ convergence of derivatives up to a certain order $k$ depending on the regularity of the initial condition. We prove these results through the study of the linearized coagulation equation in self-similar variables, for which we show a spectral gap in a scale of weighted Sobolev spaces. We also take advantage of the fact that the Laplace or Fourier transforms of this equation can be explicitly solved in this case.

We present a new a priori estimate for discrete coagulation–fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a priori estimate provides a global $L^2$ bound on the mass density and was previously used, for instance, in the context of reaction–diffusion equations. In this paper we demonstrate two lines of applications for such an estimate: On the one hand, it enables to simplify parts of the known existence theory and allows to show existence of solutions for generalised models involving collision-induced, quadratic fragmentation terms for which the previous existence theory seems difficult to apply. On the other hand and most prominently, it proves mass conservation (and thus the absence of gelation) for almost all the coagulation coefficients for which mass conservation is known to hold true in the space homogeneous case.

Abstract Under the condition of detailed balance and some additional restrictions on the size of the coefficients, we identify the equilibrium distribution to which solutions of the discrete coagulation-fragmentation system of equations converge for large times, thus showing that there is a critical mass which marks a change in the behavior of the solutions. This was previously known only for particular cases as the generalized Becker–Döring equations. Our proof is based on an inequality between the entropy and the entropy production which also gives some information on the rate of convergence to equilibrium for solutions under the critical mass.

This work treats mainly the problem of existence of solutions for two different equations: the continuous coagulation-fragmentation equations and the Wigner-Poisson-Fokker-Planck equation. In addition, some aspects of the qualitative behavior of the coagulation-fragmentation equations are studied. The thesis is organized as follows: in this introduction we briefly present the context of both equations and the main results obtained. In chapters 2–4 we give some preliminar results and background which is needed for the later treatment of the continuous coagulation-fragmentation system of equations; in chapter 5 we state and prove our existence results, and on the way we rederive some of the already known results on the topic, as the techniques involved are similar. In particular, section 5.7.3 contains results which show the interplay between singular coagulation and fragmentation coefficients. Chapter 6 contains a new result on the asymptotic behavior of the generalized Becker-Döring system of equations (which is a particular case of the discrete coagulation-fragmentation equations, as explained below), and chapter 7 shows an explicit approximation to the behavior of solutions of the Becker-Döring equations in a particular case, together with numerical solutions that back up the validity of the approximation. Finally, chapter 8 contains our result on the Wigner-Poisson-Fokker-Planck equation, which essentially consists of an existence theory in $L^1$. Some appendices are given which contain a summary of known results which are necessary in the development of the rest of this work.

We prove the following asymptotic behavior for solutions to the generalized Becker-Döring system for general initial data: under a detailed balance assumption and in situations where density is conserved in time, there is a critical density rho_s such that solutions with an initial density rho_0 leq rho_s converge strongly to the equilibrium with density rho_0, and solutions with initial density rho_0 > rho_s converge (in a weak sense) to the equilibrium with density rho_s. This extends the previous knowledge that this behavior happens under more restrictive conditions on the initial data. The main tool is a new estimate on the tail of solutions with density below the critical density.

A global existence, uniqueness and regularity theorem is proved for the simplest Markovian Wigner–Poisson–Fokker–Planck model incorporating friction and dissipation mechanisms. The proof relies on Green function and energy estimates under mild formulation, making essential use of the Husimi function and the elliptic regularization of the Fokker-Planck operator.

Micellization is the precipitation of lipids from aqueous solution into aggregates with a broad distribution of aggregation number. Three eras of micellization are characterized in a simple kinetic model of Becker-Döring type. The model asigns the same constant energy to the $(k-1)$ monomer-monomer bonds in a linear chain of $k$ particles. The number of monomers decreases sharply and many clusters of small size are produced during the first era. During the second era, nuclei are increasing steadily in size until their distribution becomes a self-similar solution of the diffusion equation. Lastly, when the average size of the nuclei becomes comparable to its equilibrium value, a simple mean-field Fokker-Planck equation describes the final era until the equilibrium distribution is reached.