Interacting particle systems (IPSs) are a very important class of dynamical systems, arising in different domains like biology, physics, sociology and engineering. In many applications, these systems can be very large, making their simulation and control, as well as related numerical tasks, very challenging. Kernel methods, a powerful tool in machine learning, offer promising approaches for analyzing and managing IPS. This paper provides a comprehensive study of applying kernel methods to IPS, including the development of numerical schemes and the exploration of mean-field limits. We present novel applications and numerical experiments demonstrating the effectiveness of kernel methods for surrogate modelling and state-dependent feature learning in IPS. Our findings highlight the potential of these methods for advancing the study and control of large-scale IPS.

A system of interacting multi-class finite-state jump processes is analyzed. The model under consideration consists of a block-structured network with dynamically changing multi-color nodes. The interactions are local and described through local empirical measures. Two levels of heterogeneity are considered: between and within the blocks where the nodes are labeled into two types. The central nodes are those connected only to nodes from the same block, whereas the peripheral nodes are connected to both nodes from the same block and nodes from other blocks. Limits of such systems as the number of nodes tends to infinity are investigated. In particular, under specific regularity conditions, propagation of chaos and the law of large numbers are established in a multi-population setting. Moreover, it is shown that, as the number of nodes goes to infinity, the behavior of the system can be represented by the solution of a McKean–Vlasov system. Then, we prove large deviations principles for the vectors of empirical measures and the empirical processes, which extends the classical results of Dawson and Gärtner (Stochastics 20, 1987) and Léonard (Ann. Inst. H. Poincaré Prob. Statist. 31, 1995).

We analyze average-based distributed algorithms relying on simple and pairwise random interactions among a large and unknown number of anonymous agents. This allows the characterization of global properties emerging from these local interactions. Agents start with an initial integer value, and at each interaction keep the average integer part of both values as their new value. The convergence occurs when, with high probability, all the agents possess the same value, which means that they all know a property of the global system. Using a well-chosen stochastic coupling, we improve upon existing results by providing explicit and tight bounds on the convergence time. We apply these general results to both the proportion problem and the system size problem.

Dielectric breakdown in a thin oxide is presented in terms of an interacting particle system on a two-dimensional lattice. All edges in the system are initially assumed to be closed. An edge between two adjacent vertices will open according to an exponentially distributed random variable. Breakdown occurs at the time an open path connects the top layer of the lattice to the bottom layer. Using the extreme value theory, we show that the time until breakdown is asymptotically Weibull distributed.

In this paper we are concerned with a model in econophysics, the subfield of statistical physics that applies concepts from traditional physics to economics. Our model is an example of an interacting particle system with disorder, meaning that some of the transition rates are not identical but rather drawn from a fixed distribution. Economical agents are represented by the vertices of a connected graph and are characterized by the number of coins they possess. Agents independently spend one coin at rate one for their basic need, earn one coin at a rate chosen independently from a distribution ϕ, and exchange money at rate µ with one of their nearest neighbors, with the richest neighbor giving one coin to the other neighbor. If an agent needs to spend one coin when his/her fortune is at 0, he/she dies, i.e. the corresponding vertex is removed from the graph. Our first results focus on the two extreme cases of lack of cooperation µ=0 and perfect cooperation µ = ∞ for finite connected graphs. These results suggest that, when overall the agents earn more than they spend, cooperation is beneficial for the survival of the population, whereas when overall the agents earn less than they spend, cooperation becomes detrimental. We also study the infinite one-dimensional system. In this case, when the agents earn less than they spend on average, the density of agents that die eventually is bounded from below by a positive constant that does not depend on the initial number of coins per agent or the level of cooperation.

In the field of Life Sciences, it is very common to deal with extremely complex systems, from both analytical and computational points of view, due to the unavoidable coupling of different interacting structures. As an example, angiogenesis has revealed to be an highly complex, and extremely interesting biomedical problem, due to the strong coupling between the kinetic parameters of the relevant branching – growth – anastomosis stochastic processes of the capillary network, at the microscale, and the family of interacting underlying biochemical fields, at the macroscale. In this paper, an original revisited conceptual stochastic model of tumour-driven angiogenesis has been proposed, for which it has been shown that it is possible to reduce complexity by taking advantage of the intrinsic multiscale structure of the system; one may keep the stochasticity of the dynamics of the vessel tips at their natural microscale, whereas the dynamics of the underlying fields is given by a deterministic mean field approximation obtained by an averaging at a suitable mesoscale. While in previous papers, only an heuristic justification of this approach had been offered; in this paper, a rigorous proof is given of the so called ‘propagation of chaos’, which leads to a mean field approximation of the stochastic relevant measures associated with the vessel dynamics, and consequently of the underlying tumour angiogenic factor (TAF) field. As a side, though important result, the non-extinction of the random process of tips has been proven during any finite time interval.

We consider a class of multi-type particle systems whose structure is similar to that of a contact process and show that additivity is equivalent to the existence of a dual process, extending a result of Harris. We prove a necessary and sufficient condition for the model to preserve positive correlations. We then show that complete convergence on Zd holds for a large subclass of models including the two-stage contact process and a household model, and give examples.

In this paper, we continue the work started by Steve Krone on the two-stage contact process. We give a simplified proof of the duality relation and answer most of the open questions posed in Krone (1999). We also fill in the details of an incomplete proof.

We study a free energy computation procedure, introduced in[Darve and Pohorille, J. Chem. Phys.115 (2001) 9169–9183; Hénin and Chipot, J. Chem. Phys.121 (2004) 2904–2914], which relies on the long-timebehavior of a nonlinear stochasticdifferential equation. This nonlinearity comes from a conditionalexpectation computed with respect to one coordinate of the solution. The long-time convergence of the solutions tothis equation has been provedin [Lelièvre et al., Nonlinearity21 (2008) 1155–1181], under some existence and regularity assumptions.In this paper, we prove existence and uniqueness under suitable conditions for the nonlinear equation, andwe study a particle approximation technique based on a Nadaraya-Watson estimator ofthe conditional expectation. The particle system converges to the solutionof the nonlinear equation if the number of particles goes to infinityand then the kernel used in the Nadaraya-Watson approximation tends to aDirac mass. We derive a rate for this convergence, and illustrate it by numericalexamples on a toy model.

With the rapid adoption of transgenic crops, gene flow from transgenic crops to wild relatives through pollen dispersal is of significant concern and warrants both empirical and theoretical studies to assess the risk of introduction of transgenes into wild populations. We propose to use the (biased) voter model in a heterogeneous environment to investigate the effects of recurrent gene flow from transgenic crop to wild relatives. The model is defined on the d-dimensional integer lattice that is divided into two parts, Δ and Zd \ Δ. Individuals carrying the transgene and individuals carrying the wild type gene compete according to the evolution rules of a (biased) voter model on Zd \ Δ, while the process is conditioned to have only individuals carrying the transgene on Δ. Our main findings suggest that unless transgenes confer increased fitness in wild relatives, introgression of transgenes into populations of wild plants is slow and may even be reversible without intervention. Our study also addresses the effects of different spatial planting patterns of transgenic crops on the rate of introgression.


The Diffusion Monte Carlo method is devoted to the computation ofelectronic ground-state energies of molecules. In this paper, we focus onimplementations of this method which consist in exploring theconfiguration space with a fixed number of random walkers evolvingaccording to a stochastic differential equation discretized in time. Weallow stochastic reconfigurations of the walkers to reduce thediscrepancy between the weights that they carry. On a simpleone-dimensional example, we prove the convergence of the method for afixed number of reconfigurations when the number of walkers tends to+∞ while the timestep tends to 0. We confirm our theoreticalrates of convergence by numerical experiments. Various resamplingalgorithms are investigated, both theoretically and numerically.


This paper is helpful for understanding interactive-agent economies. It presents a user-friendly framework for exploring the implications of economic interaction, and demonstrates its wide applicability. Agents repeatedly face binary choices and receive i.i.d. shocks; payoffs depend upon one's own action and shocks, as well as upon the actions of a subset of other agents. Simulations explore how properties change as the number of agents, the number of neighbors, and the degree of interaction change. Key results in this framework include the following: The law of large numbers can readily be postponed or defeated, stationarity and ergodicity are trivial to deduce, any equilibrium distribution may be attained in “globally interactive” economies, continuous- and discrete-time analogues generally behave very similarly, dynamics can be sensitive to details of the interaction structure (implying that popular mean-field approximations are not always appropriate), and substantial persistence is typical.

We present an interacting particle system methodology for the numerical solving of the Lyapunov exponent of Feynman–Kac semigroups and for estimating the principal eigenvalue ofSchrödinger generators. The continuous or discrete time models studied in this work consists of N interacting particles evolving in an environment with soft obstacles related to a potential function V. These models are related to genetic algorithms and Moran type particle schemes. Their choice is not unique. We will examine a class of modelsextending the hard obstacle modelof K. Burdzy, R. Holyst and P. March and including the Moran type scheme presented by the authors in a previous work. We provide precise uniform estimates with respect to the time parameter and we analyze the fluctuations of continuous time particle models.

We consider an interacting particle system in which each site of the d-dimensional integer lattice can be in state 0, 1, or 2. Our aim is to model the spread of disease in plant populations, so think of 0 = vacant, 1 = healthy plant, 2 = infected plant. A vacant site becomes occupied by a plant at a rate which increases linearly with the number of plants within range R, up to some saturation level, F1, above which the rate is constant. Similarly, a plant becomes infected at a rate which increases linearly with the number of infected plants within range M, up to some saturation level, F2. An infected plant dies (and the site becomes vacant) at constant rate δ. We discuss coexistence results in one and two dimensions. These results depend on the relative dispersal ranges for plants and disease.

The estimation of critical values is one of the most interesting problems in the study of interacting particle systems. The bounds obtained analytically are not usually very tight and, therefore, computer simulation has been proved to be very useful in the estimation of these values. In this paper we present a new method for the estimation of critical values in any interacting particle system with an absorbing state. The method, based on the asymptotic behaviour of the absorption time of the process, is very easy to implement and provides good estimates. It can also be applied to processes different from particle systems.

In this paper we consider the continuous-time filtering problem and we estimate the order of convergence of an interacting particle system scheme presented by the authors in previous works. We will discuss how the discrete time approximating model of the Kushner-Stratonovitch equation and the genetic type interacting particle system approximation combine. We present quenched error bounds as well as mean order convergence results.

Two species (designated by 0's and 1's) compete for territory on a lattice according to the rules of a voter model, except that the 0's jump d0 spaces and the 1's jump d1 spaces. When d0 = d1 = 1 the model is the usual voter model. It is shown that in one dimension, if d1 > d0 and d0 = 1,2 and initially there are infinitely many blocks of 1's of length ≥ d1, then the 1's eliminate the 0's. It is believed this may be true whenever d1 > d0. In the biased annihilating branching process particles place offspring on empty neighbouring sites at rate λ and neighbouring pairs of particles coalesce at rate 1. In one dimension it is known to converge to the product measure density λ/(1+λ) when λ ≥ 1/3, and the initial configuration is non-zero and finite. This result is extended to λ ≥ 0.0347. Bounds on the edge-speed are given.

We introduce a new class of interacting particle systems on a graph G. Suppose initially there are Ni(0) particles at each vertex i of G, and that the particles interact to form a Markov chain: at each instant two particles are chosen at random, and if these are at adjacent vertices of G, one particle jumps to the other particle's vertex, each with probability 1/2. The process N enters a death state after a finite time when all the particles are in some independent subset of the vertices of G, i.e. a set of vertices with no edges between any two of them. The problem is to find the distribution of the death state, ηi = Ni(∞), as a function of Ni(0).

We are able to obtain, for some special graphs, the limiting distribution of Ni if the total number of particles N → ∞ in such a way that the fraction, Ni(0)/S = ξi, at each vertex is held fixed as N → ∞. In particular we can obtain the limit law for the graph S2, the two-leaf star which has three vertices and two edges.

We use an interacting particle system on ℤ to model an epidemic. Each site of ℤ can be in either one of three states: empty, healthy or infected. An empty site x gets occupied by a healthy individual at a rate βn1(x) where n1(x) is the number of healthy nearest neighbors of x. A healthy individual at x gets infected at rate αn2(x) where n2(x) is the number of infected nearest neighbors of x. An infected individual dies at rate δ independently of everything else. We show that for all α, β and δ> 0 and all initial configurations, all the sites of a fixed finite set remain either all empty or all healthy after an almost surely finite time. Moreover, if the initial configuration has infinitely many healthy individuals then the process converges almost surely (in the sense described above) to the all healthy state. We also consider a model introduced by Durrett and Neuhauser where healthy individuals appear spontaneously at rate β > 0 and for which coexistence of 1's and 2's was proved in dimension 2 for some values of α and β. We prove that coexistence may occur in any dimension.

One-dimensional, periodic and annihilating systems of Brownian motions and random walks are defined and interpreted in terms of sizeless particles which vanish on contact. The generating function and moments of the number pairs of particles which have vanished, given an arbitrary initial arrangement, are derived in terms of known two-particle survival probabilities. Three important special cases are considered: Brownian motion with the particles initially (i) uniformly distributed and (ii) equally spaced on a circle and (iii) random walk on a lattice with initially each site occupied. Results are also given for the infinite annihilating particle systems obtained in the limit as the number of particles and the size of the circle or lattice increase. Application of the results to the theory of diffusion-limited reactions is discussed.