Stochastic Particle Systems Research Articles

Double-loop importance sampling for McKean–Vlasov stochastic differential equation

This paper investigates Monte Carlo (MC) methods to estimate probabilities of rare events associated with the solution to the d-dimensional McKean–Vlasov stochastic differential equation (MV-SDE). MV-SDEs are usually approximated using a stochastic interacting P-particle system, which is a set of P coupled d-dimensional stochastic differential equations (SDEs). Importance sampling (IS) is a common technique for reducing high relative variance of MC estimators of rare-event probabilities. We first derive a zero-variance IS change of measure for the quantity of interest by using stochastic optimal control theory. However, when this change of measure is applied to stochastic particle systems, it yields a P×d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P \ imes d$$\\end{document}-dimensional partial differential control equation (PDE), which is computationally expensive to solve. To address this issue, we use the decoupling approach introduced in (dos Reis et al. 2023), generating a d-dimensional control PDE for a zero-variance estimator of the decoupled SDE. Based on this approach, we develop a computationally efficient double loop MC (DLMC) estimator. We conduct a comprehensive numerical error and work analysis of the DLMC estimator. As a result, we show optimal complexity of OTOLr-4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}\\left( \ extrm{TOL}_{\ extrm{r}}^{-4}\\right) $$\\end{document} with a significantly reduced constant to achieve a prescribed relative error tolerance TOLr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{TOL}_{\ extrm{r}}$$\\end{document}. Subsequently, we propose an adaptive DLMC method combined with IS to numerically estimate rare-event probabilities, substantially reducing relative variance and computational runtimes required to achieve a given TOLr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{TOL}_{\ extrm{r}}$$\\end{document} compared with standard MC estimators in the absence of IS. Numerical experiments are performed on the Kuramoto model from statistical physics.

Well-posedness and stability of a stochastic neural field in the form of a partial differential equation

A system of partial differential equations representing stochastic neural fields was recently proposed with the aim of modelling the activity of noisy grid cells when a mammal travels through physical space. The system was rigorously derived from a stochastic particle system and its noise-driven pattern-forming bifurcations have been characterised. However, due to its nonlinear and non-local nature, standard well-posedness theory for smooth time-dependent solutions of parabolic equations does not apply. In this article, we transform the problem through a suitable change of variables into a nonlinear Stefan-like free boundary problem and use its representation formulae to construct local-in-time smooth solutions under mild hypotheses. Then, we prove that fast-decaying initial conditions and globally Lipschitz modulation functions imply that solutions are global-in-time. Last, we improve previous linear stability results by showing nonlinear asymptotic stability of stationary solutions under suitable assumptions.

Variational Structures Beyond Gradient Flows: a Macroscopic Fluctuation-Theory Perspective

Macroscopic equations arising out of stochastic particle systems in detailed balance (called dissipative systems or gradient flows) have a natural variational structure, which can be derived from the large-deviation rate functional for the density of the particle system. While large deviations can be studied in considerable generality, these variational structures are often restricted to systems in detailed balance. Using insights from macroscopic fluctuation theory, in this work we aim to generalise this variational connection beyond dissipative systems by augmenting densities with fluxes, which encode non-dissipative effects. Our main contribution is an abstract theory, which for a given flux-density cost and a quasipotential, provides a decomposition into dissipative and non-dissipative components and a generalised orthogonality relation between them. We then apply this abstract theory to various stochastic particle systems—independent copies of jump processes, zero-range processes, chemical-reaction networks in complex balance and lattice-gas models—without assuming detailed balance. For macroscopic equations arising out of these particle systems, we derive new variational formulations that generalise the classical gradient-flow formulation.

A hypothesis test for the domain of attraction of a random variable

In this work, we address the problem of detecting whether a sampled probability distribution of a random variable V has infinite first moment. This issue is notably important when the sample results from complex numerical simulation methods. For example, such a situation occurs when one simulates stochastic particle systems with complex and singular McKean–Vlasov interaction kernels. As stated, the detection problem is ill-posed. We thus propose and analyze an asymptotic hypothesis test for independent copies of a given random variable which is supposed to belong to an unknown domain of attraction of a stable law. The null hypothesis H0 is: ‘√V is in the domain of attraction of the Normal law’ and the alternative hypothesis is H1: ‘X is in the domain of attraction of a stable law with index smaller than 2’. Our key observation is that X cannot have a finite second moment when H0 is rejected (and therefore H1 is accepted). Surprisingly, we find it useful to derive our test from the statistics of random processes. More precisely, our hypothesis test is based on a statistic which is inspired by methodologies to determine whether a semimartingale has jumps from the observation of one single path at discrete times. We justify our test by proving asymptotic properties of discrete time functionals of Brownian bridges.

Rigorous derivation of the degenerate parabolic-elliptic Keller-Segel system from a moderately interacting stochastic particle system. Part I Partial differential equation

The aim of this paper is to provide the analysis result for the partial differential equations arising from the rigorous derivation of the degenerate parabolic-elliptic Keller-Segel system from a moderately interacting stochastic particle system. The rigorous derivation is divided into two articles. In this paper, we establish the solution theory of the degenerate parabolic-elliptic Keller-Segel problem and its non-local version, which will be used in the second paper for the discussion of the mean-field limit. A parabolic regularized system is introduced to bridge the stochastic particle model and the degenerate Keller-Segel system. We derive the existence of the solution to this regularized system by constructing approximate solutions, giving uniform estimates and taking the limits, where a crucial step is to obtain the L∞ Bernstein type estimate for the gradient of the approximate solution. Based on this, we obtain the well-posedness of the corresponding non-local equation through perturbation method. Finally, the weak solution of the degenerate Keller-Segel system is obtained by using a nonlinear version of Aubin-Lions lemma.

Data-driven stochastic particle scheme for collisional plasma simulations

We present a novel framework, enabled by machine learning (ML) trainable models, for collisional plasma flow simulations governed by the Rosenbluth-Fokker-Planck (RFP) equation. The maximum entropy objective is applied to close the particle velocity distribution. The closure offers a flexible tool to reconstruct a large variety of distributions, using a few low-order statistical moments. The methodology is extended by introducing an artificial neural network (ANN) to couple the moments and the transport coefficients. In order to ensure the model's training and input data quality, we generate diverse training data sets using the high-fidelity Direct Simulation Monte-Carlo (DSMC) method. Moreover, energy constraints are integrated in the model training, thereby improving prediction accuracy. As a final step, the ANN is combined with a stochastic particle system, where the accuracy of the whole machinery for non-equilibrium thermal relaxation scenarios is evaluated. Simulation results indicate that our approach reproduces the correct relaxation behavior at lower computational cost compared to standard methods.

Derivation of a macroscopic model for Brownian hard needles

We study the role of anisotropic steric interactions in a system of hard Brownian needles in two dimensions. Despite having no volume, non-overlapping needles exclude a volume in configuration space that influences the macroscopic evolution of the system. Starting from the stochastic particle system, we use the method of matched asymptotic expansions and conformal mapping to systematically derive a nonlinear non-local partial differential equation for the evolution of the population density in position and orientation. We consider the regime of high rotational diffusion, resulting in an equation for the spatial density that allows us to compare the effective excluded volume of a hard-needle system with that of a hard-sphere system. We further consider spatially homogeneous solutions and find an isotropic to nematic transition as density increases, consistent with Onsager’s theory.

Hyperbolic–parabolic normal form and local classical solutions for cross-diffusion systems with incomplete diffusion

We investigate degenerate cross-diffusion equations, with a rank-deficient diffusion-matrix, modelling multispecies population dynamics driven by partial pressure gradients. These equations have recently been found to arise in a mean-field limit of interacting stochastic particle systems. To date, their analysis in multiple space dimensions has been confined to the purely convective case with equal mobility coefficients. In this article, we introduce a normal form for an entropic class of such equations which reveals their structure of a symmetric hyperbolic–parabolic system. Due to the state-dependence of the range and kernel of the singular diffusive matrix, our way of rewriting the equations is different from that classically used for symmetric second-order systems with a nullspace invariance property. By means of this change of variables, we solve the Cauchy problem for short times and positive initial data in for

Finite Dimensional Approximations of Hamilton–Jacobi–Bellman Equations for Stochastic Particle Systems with Common Noise

Finite Dimensional Approximations of Hamilton–Jacobi–Bellman Equations for Stochastic Particle Systems with Common Noise

Velocity jump process with volume exclusions in a narrow channel.

This paper analyzes the impact of collisions in a system of N identical hard-core particles driven according to a velocity jump process. The physical space is essentially a channel in R with a probability of occupants being able to pass each other. The system mimics what nature does, where individuals pass one another in a narrow channel while making incidental contact with those moving in the opposite direction. The passing probability may depend on the particles' size and the channel's width. Starting from the particle level model, we systematically derive a nonlinear transport equationbased on an asymptotic expansion. Under low-occupied fractions, numerical solutions of both the kinetic model and the stochastic particle system are compared well during biased and unbiased random velocity changes. Analysis of the subpopulation motility within a large population exhibits the consequences of volume exclusions and channel confinements on the traveling speeds.

Concentration inequalities for the hydrodynamic limit of a two-species stochastic particle system

We study a stochastic particle system which is motivated from grain boundary coarsening in two-dimensional networks. Each particles lives on the positive real line and is labeled as belonging to either Species 1 or Species 2. Species 1 particles drift at unit speed toward the origin, while Species 2 particles do not move. When a particle in Species 1 hits the origin, it is removed, and a randomly selected particle mutates from Species 2 to Species 1. The process described is an example of a high-dimensional piecewise deterministic Markov process (PDMP), in which deterministic flow is punctuated with stochastic jumps. Our main result is a proof of exponential concentration inequalities of the Kolmogorov-Smirnoff distance between empirical measures of the particle system and solutions of limiting nonlinear kinetic equations. Our method of proof involves a time and space discretization of the kinetic equations, which we compare with the particle system to derive recurrence inequalities for comparing total numbers in small intervals. To show these recurrences occur with high probability, we appeal to a state dependent Hoeffding type inequality at each time increment.

A stochastic particle system approximating the BGK equation

<p style='text-indent:20px;'>We consider a stochastic <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula>-particle system on a torus in which each particle moving freely can instantaneously thermalize according to the particle configuration at that instant. Following [<xref ref-type="bibr" rid="b2">2</xref>], we show that the propagation of chaos does hold and that the one-particle distribution converges to the solution of the BGK equation. The improvement with respect to [<xref ref-type="bibr" rid="b2">2</xref>] consists in the fact that here, as suggested by physical considerations, the thermalizing transition is driven only by the restriction of the particle configuration in a small neighborhood of the jumping particle. In other words, the Maxwellian distribution of the outgoing particle is computed via the empirical hydrodynamical fields associated to the fraction of particles sufficiently close to the test particle and not, as in [<xref ref-type="bibr" rid="b2">2</xref>], via the whole particle configuration.</p>

Explicit numerical approximations for McKean-Vlasov neutral stochastic differential delay equations

This paper studies the numerical methods to approximate the solutions for a sort of McKean-Vlasov neutral stochastic differential delay equations (MV-NSDDEs) that the growth of the drift coefficients is super-linear. First, we obtain that the solution of MV-NSDDE exists and is unique. Then, we use a stochastic particle method, which is on the basis of the results about the propagation of chaos for a stochastic particle system, to deal with the approximation of the law. Furthermore, we construct the tamed Euler-Maruyama numerical scheme with respect to the corresponding particle system and obtain the rate of convergence. Combining propagation of chaos and the convergence rate of the numerical solution to the particle system, we get the convergence error between the numerical solution and exact solution of the original MV-NSDDE in the stepsize and number of particles.

Recent progress on limit theorems for large stochastic particle systems

This article presents a selection of recent results in the mathematical study of physical systems described by a large number of particles, with various types of interactions (mean-field, moderate, nearest-neighbor). Limit theorems are obtained concerning either the large-scale or the long-time behavior of these systems. These results rely on the use of a large range of mathematical tools, arising from both probability theory and the analysis of partial differential equations, and thereby illustrate fruitful interactions between these two disciplines.

Propagation of chaos for stochastic particle systems with singular mean-field interaction of Lq−Lp type

Propagation of chaos for stochastic particle systems with singular mean-field interaction of Lq−Lp type

Well-posedness and numerical schemes for one-dimensional McKean–Vlasov equations and interacting particle systems with discontinuous drift

In this paper, we first establish well-posedness results for one-dimensional McKean–Vlasov stochastic differential equations (SDEs) and related particle systems with a measure-dependent drift coefficient that is discontinuous in the spatial component, and a diffusion coefficient which is a Lipschitz function of the state only. We only require a fairly mild condition on the diffusion coefficient, namely to be non-zero in a point of discontinuity of the drift, while we need to impose certain structural assumptions on the measure-dependence of the drift. Second, we study Euler–Maruyama type schemes for the particle system to approximate the solution of the one-dimensional McKean–Vlasov SDE. Here, we will prove strong convergence results in terms of the number of time-steps and number of particles. Due to the discontinuity of the drift, the convergence analysis is non-standard and the usual strong convergence order 1/2 known for the Lipschitz case cannot be recovered for all presented schemes.

Controlled pattern formation of stochastic Cucker–Smale systems with network structures

We present a new stochastic particle system on networks which exhibits flocking behavior and pattern formation. More precisely, we consider Cucker–Smale particles with decentralized formation control and multiplicative noises on symmetric connected networks. Under suitable assumptions on the initial configurations and the network structure, we establish time-asymptotic stochastic flocking behavior and pattern formation of solutions for the proposed stochastic particle system. Our approach is based on the Lyapunov functional energy estimates and does not require any spectral information of the graph associated with the network structure.

Spin q-Whittaker Polynomials and Deformed Quantum Toda

Spin q-Whittaker symmetric polynomials labeled by partitions \(\lambda \) were recently introduced by Borodin and Wheeler (Spin q-Whittaker Polynomials, 2017. arXiv preprint arXiv:1701.06292 [math.CO]) in the context of integrable \(\mathfrak {sl}_2\) vertex models. They are a one-parameter deformation of the \(t=0\) Macdonald polynomials. We present a new more convenient modification of spin q-Whittaker polynomials and find two Macdonald type q-difference operators acting diagonally in these polynomials with eigenvalues, respectively, \(q^{-\lambda _1}\) and \(q^{\lambda _N}\) (where \(\lambda \) is the polynomial’s label). We study probability measures on interlacing arrays based on spin q-Whittaker polynomials, and match their observables with known stochastic particle systems such as the q-Hahn TASEP. In a scaling limit as \(q\nearrow 1\), spin q-Whittaker polynomials turn into a new one-parameter deformation of the \(\mathfrak {gl}_n\) Whittaker functions. The rescaled Pieri type rule gives rise to a one-parameter deformation of the quantum Toda Hamiltonian. The deformed Hamiltonian acts diagonally on our new spin Whittaker functions. On the stochastic side, as \(q\nearrow 1\) we discover a multilevel extension of the beta polymer model of Barraquand and Corwin (Probab Theory Relat Fields 167(3–4):1057–1116, 2016. arXiv:1503.04117 [math.PR]), and relate it to spin Whittaker functions.

Propagation of chaos: A review of models, methods and applications. I. Models and methods

<p style='text-indent:20px;'>The notion of propagation of chaos for large systems of interacting particles originates in statistical physics and has recently become a central notion in many areas of applied mathematics. The present review describes old and new methods as well as several important results in the field. The models considered include the McKean-Vlasov diffusion, the mean-field jump models and the Boltzmann models. The first part of this review is an introduction to modelling aspects of stochastic particle systems and to the notion of propagation of chaos. The second part presents concrete applications and a more detailed study of some of the important models in the field.</p>

Poisson-Dirichlet asymptotics in condensing particle systems

We study measures on random partitions, arising from condensing stochastic particle systems with stationary product distributions. We provide fairly general conditions on the stationary weights, which lead to Poisson-Dirichlet statistics of the condensed phase in the thermodynamic limit. The Poisson-Dirichlet distribution is known to be the unique reversible measure of split-merge dynamics for random partitions, which we use to characterize the limit law. We also establish concentration results for the macroscopic phase, using size-biased sampling techniques and the equivalence of ensembles to characterize the bulk distribution of the system.