In this paper, we use Stein’s method to obtain optimal bounds, both in Kolmogorov and in Wasserstein distance, in the normal approximation for the empirical distribution of the ground state of a many-interacting-worlds harmonic oscillator proposed by Hall, Deckert and Wiseman (Phys. Rev. X 4 (2014) 041013). Our bounds on the Wasserstein distance solve a conjecture of McKeague and Levin (Ann. Appl. Probab. 26 (2016) 2540–2555).

We study the limiting behavior of interacting particle systems indexed by large sparse graphs, which evolve either according to a discrete time Markov chain or a diffusion, in which particles interact directly only with their nearest neighbors in the graph. To encode sparsity we work in the framework of local weak convergence of marked (random) graphs. We show that the joint law of the particle system varies continuously with respect to local weak convergence of the underlying graph marked with the initial conditions. In addition, we show that the global empirical measure converges to a nonrandom limit for a large class of graph sequences including sparse Erdős–Rényi graphs and configuration models, whereas the empirical measure of the connected component of a uniformly random vertex converges to a random limit. Along the way, we develop some related results on the time-propagation of ergodicity and empirical field convergence, as well as some general results on local weak convergence of Gibbs measures in the uniqueness regime which appear to be new. The results obtained here are also useful for obtaining autonomous descriptions of marginal dynamics of interacting diffusions and Markov chains on sparse graphs. While limits of interacting particle systems on dense graphs have been extensively studied, there are relatively few works that have studied the sparse regime in generality.

In this paper, we study a class of zero-sum two-player stochastic differential games with the controlled stochastic differential equations and the payoff/cost functionals of recursive type. As opposed to the pioneering work by Fleming and Souganidis [Indiana Univ. Math. J. 38 (1989) 293–314] and the seminal work by Buckdahn and Li [SIAM J. Control Optim. 47 (2008) 444–475], the involved coefficients may be random, going beyond the Markovian framework and leading to the random upper and lower value functions. We first prove the dynamic programming principle for the game, and then under the standard Lipschitz continuity assumptions on the coefficients, the upper and lower value functions are shown to be the viscosity solutions of the upper and the lower fully nonlinear stochastic Hamilton–Jacobi–Bellman–Isaacs (HJBI) equations, respectively. A stability property of viscosity solutions is also proved. Under certain additional regularity assumptions on the diffusion coefficient, the uniqueness of the viscosity solution is addressed as well.

To each edge $(\mathit{i},\mathit{j})$, $\mathit{i}<\mathit{j}$, of the complete directed graph on the integers we assign unit weight with probability p or weight x with probability $1-\mathit{p}$, independently from edge to edge, and give to each path weight equal to the sum of its edge weights. If ${\mathit{W}}_{0,\mathit{n}}^{\mathit{x}}$ is the maximum weight of all paths from 0 to n then ${\mathit{W}}_{0,\mathit{n}}^{\mathit{x}}/\mathit{n}\to {\mathit{C}}_{\mathit{p}}(\mathit{x})$, as $\mathit{n}\to \infty$, almost surely, where ${\mathit{C}}_{\mathit{p}}(\mathit{x})$ is positive and deterministic. We study ${\mathit{C}}_{\mathit{p}}(\mathit{x})$ as a function of x, for fixed $0<\mathit{p}<1$, and show that it is a strictly increasing convex function that is not differentiable if and only if x is a nonpositive rational or a positive integer except 1 or the reciprocal of it. We allow x to be any real number, even negative, or, possibly, $-\infty$. The case $\mathit{x}=-\infty$ corresponds to the well-studied directed version of the Erdős–Rényi random graph (known as Barak–Erdős graph) for which ${\mathit{C}}_{\mathit{p}}(-\infty )={lim}_{\mathit{x}\to -\infty }{\mathit{C}}_{\mathit{p}}(\mathit{x})$ has been studied as a function of p in a number of papers.

We consider the mean-field limit of systems of particles with singular interactions of the type $-log|\mathit{x}|$ or $|\mathit{x}{|}^{-\mathit{s}}$, with $0<\mathit{s}<\mathit{d}-2$, and with an additive noise in dimensions $\mathit{d}\ge 3$. We use a modulated-energy approach to prove a quantitative convergence rate to the solution of the corresponding limiting PDE. When $\mathit{s}>0$, the convergence is global in time, and it is the first such result valid for both conservative and gradient flows in a singular setting on ${\mathbb{R}}^{\mathit{d}}$. The proof relies on an adaptation of an argument of Carlen–Loss (Duke Math. J. 81 (1995) 135–157) to show a decay rate of the solution to the limiting equation, and on an improvement of the modulated-energy method developed in (SIAM J. Math. Anal. 48 (2016) 2269–2300; Duke Math. J. 169 (2020) 2887–2935; Nguyen, Rosenzweig and Serfaty (2021)), making it so that all prefactors in the time derivative of the modulated energy are controlled by a decaying bound on the limiting solution.

This paper makes three contributions to estimating the number of perfect matching in bipartite graphs. First, we prove that the popular sequential importance sampling algorithm works in polynomial time for dense bipartite graphs. More carefully, our algorithm gives a $(1\pm \mathit{ϵ})$-approximation for the number of perfect matchings of a λ-dense bipartite graph, using $\mathit{O}({\mathit{n}}^{\frac{1-2\mathit{\lambda }}{\mathit{\lambda }}{\mathit{ϵ}}^{-2}})$ samples. With size n on each side and for $\frac{1}{2}>\mathit{\lambda }>0$, a λ-dense bipartite graph has all degrees greater than $(\mathit{\lambda }+\frac{1}{2})\mathit{n}$.

Second, practical applications of the algorithm requires many calls to matching algorithms. A novel preprocessing step is provided which makes significant improvements.

Third, three applications are provided. The first is for counting Latin squares, the second is a practical way of computing the greedy algorithm for a card guessing game with feedback and the third is for stochastic block models. In all three examples, sequential importance sampling allows treating practical problems of reasonably large sizes.

Mean field games are concerned with the limit of large-population stochastic differential games where the agents interact through their empirical distribution. In the classical setting, the number of players is large but fixed throughout the game. However, in various applications, such as population dynamics or economic growth, the number of players can vary across time and this may lead to different Nash equilibria. In order to account for this evolution, we introduce a branching mechanism in the population of agents and obtain a variant of the original mean field game problem. As a first step, we study a simple model using a PDE approach to illustrate the main differences with the classical setting. We prove existence of a solution and show that it provides an approximate Nash-equilibrium for large population games. We also present a numerical example for a linear–quadratic model. Then we study the problem in a general setting by a probabilistic approach. It is based upon the relaxed formulation of stochastic control problems which allows us to obtain a general existence result.

Models of parking in which cars are placed randomly and then move according to a deterministic rule have been studied since the work of Konheim and Weiss in the 1960s. Recently, Damron, Gravner, Junge, Lyu, and Sivakoff ((2019) Ann. Appl. Probab. 29 2089–2113) introduced a model in which cars are both placed and move at random. Independently at each point of a Cayley graph G, we place a car with probability p, and otherwise an empty parking space. Each car independently executes a random walk until it finds an empty space in which to park. In this paper we introduce three new techniques for studying the model, namely the space-based parking model, and the strategies for parking and for car removal. These allow us to study the original model by coupling it with models where parking behaviour is easier to control. Applying our methods to the one-dimensional parking problem in $\mathbb{Z}$, we improve on previous work, showing that for $\mathit{p}<1/2$ the expected journey length of a car is finite, and for $\mathit{p}=1/2$ the expected journey length by time t grows like ${\mathit{t}}^{3/4}$ up to a polylogarithmic factor.

In the past decade, an intensive study of strong approximation of stochastic differential equations (SDEs) with a drift coefficient that has discontinuities in space has begun. In the majority of these results it is assumed that the drift coefficient satisfies piecewise regularity conditions and that the diffusion coefficient is globally Lipschitz continuous and nondegenerate at the discontinuities of the drift coefficient. Under this type of assumptions the best ${\mathit{L}}_{\mathit{p}}$-error rate obtained so far for approximation of scalar SDEs at the final time is $3/4$ in terms of the number of evaluations of the driving Brownian motion. In the present article, we prove for the first time in the literature sharp lower error bounds for such SDEs. We show that for a huge class of additive noise driven SDEs of this type the ${\mathit{L}}_{\mathit{p}}$-error rate $3/4$ can not be improved.

For the proof of this result we employ a novel technique by studying equations with coupled noise: we reduce the analysis of the ${\mathit{L}}_{\mathit{p}}$-error of an arbitrary approximation based on evaluation of the driving Brownian motion at finitely many times to the analysis of the ${\mathit{L}}_{\mathit{p}}$-distance of two solutions of the same equation that are driven by Brownian motions that are coupled at the given time-points and independent, conditioned on their values at these points. To obtain lower bounds for the latter quantity, we prove a new quantitative version of positive association for bivariate normal random variables $(\mathit{Y},\mathit{Z})$ by providing explicit lower bounds for the covariance $Cov(\mathit{f}(\mathit{Y}),\mathit{g}(\mathit{Z}))$ in case of piecewise Lipschitz continuous functions f and g. In addition it turns out that our proof technique also leads to lower error bounds for estimating occupation time functionals ${\int }_0^1\mathit{f}({\mathit{W}}_{\mathit{t}})\mathit{d}\mathit{t}$ of a Brownian motion W, which substantially extends known results for the case of f being an indicator function.

We analyze the ${\mathit{L}}^1$-mixing of a generalization of the averaging process introduced by Aldous (2011). The process takes place on a growing sequence of graphs which we assume to be finite-dimensional, in the sense that the random walk on those geometries satisfies a family of Nash inequalities. As a byproduct of our analysis, we provide a complete picture of the total variation mixing of a discrete dual of the averaging process, which we call binomial splitting process. A single particle of this process is essentially the random walk on the underlying graph. When several particles evolve together, they interact by synchronizing their jumps when placed on neighboring sites. We show that, given k the number of particles and n the (growing) size of the underlying graph, the system exhibits cutoff in total variation if $\mathit{k}\to \infty$ and $\mathit{k}=\mathit{O}({\mathit{n}}^2)$. Finally, we exploit the duality between the two processes to show that the binomial splitting process satisfies a version of Aldous’ spectral gap identity, namely, the relaxation time of the process is independent of the number of particles.

We study mixing times of the symmetric and asymmetric simple exclusion process on the segment where particles are allowed to enter and exit at the endpoints. We consider different regimes depending on the entering and exiting rates as well as on the rates in the bulk, and show that the process exhibits pre-cutoff and in some cases cutoff. Our main contribution is to study mixing times for the asymmetric simple exclusion process with open boundaries. We show that the order of the mixing time can be linear or exponential in the size of the segment depending on the choice of the boundary parameters, proving a strikingly different (and richer) behavior for the simple exclusion process with open boundaries than for the process on the closed segment. Our arguments combine coupling, second class particle and censoring techniques with current estimates. A novel idea is the use of multi-species particle arguments, where the particles only obey a partial ordering.

Consider a multitype branching process in a random environment, whose reproduction law of generation n depends on the random environment at time n, unlike a constant distribution assumed in the Galton–Watson process. The famous Kesten–Stigum theorem for a supercritical multitype Galton–Watson process gives a precise description of the exponential increasing rate of the population size via a criterion for the nondegeneracy of the fundamental martingale. Finding the corresponding result in the random environment case is a longstanding problem. For the single-type case the problem has been solved by Athreya and Karlin for the sufficiency (Ann. Math. Stat. 42 (1971) 1499–1520) and Tanny for the necessity (Stochastic Process. Appl. 28 (1988) 123–139), but for the multitype case it has been open for 50 years. Here we solve this problem in the typical case, by constructing a suitable martingale which reduces to the fundamental one in the constat environment case, and by establishing a criterion for the nondegeneracy of its limit. The convergence in law of the direction of the branching process is also considered. Our results open ways in establishing other limit theorems, such as law of large numbers, central limit theorems, Berry–Essen bound, and large deviation results.

We confirm a long-standing conjecture concerning shear-induced chaos in stochastically perturbed systems exhibiting a Hopf bifurcation. The method of showing the main chaotic property, a positive Lyapunov exponent, is a computer-assisted proof. Using the recently developed theory of conditioned Lyapunov exponents on bounded domains and the modified Furstenberg–Khasminskii formula, the problem boils down to the rigorous computation of eigenfunctions of the Kolmogorov operators describing distributions of the underlying stochastic process.

We give a new algorithm for the estimation of the cross-covariance matrix $\mathbb{E}\mathit{X}{\mathit{Y}}^{\prime }$ of two large-dimensional signals $\mathit{X}\in {\mathbb{R}}^{\mathit{n}}$, $\mathit{Y}\in {\mathbb{R}}^{\mathit{p}}$ in the context where the number T of observations of the pair $(\mathit{X},\mathit{Y})$ is large but $\mathit{n}/\mathit{T}$ and $\mathit{p}/\mathit{T}$ are not supposed to be small. In the asymptotic regime where n, p, T are large, with high probability, this algorithm is optimal for the Frobenius norm among rotationally invariant estimators, that is, estimators derived from the empirical estimator by cleaning the singular values, while letting singular vectors unchanged.

Despite the widespread usage of discrete generation ensemble Kalman particle filtering methodology to solve nonlinear and high-dimensional filtering and inverse problems, little is known about their mathematical foundations. As genetic-type particle filters (a.k.a. sequential Monte Carlo), this ensemble-type methodology can also be interpreted as mean-field particle approximations of the Kalman–Bucy filtering equation. In contrast with conventional mean-field type interacting particle methods equipped with a globally Lipschitz interacting drift-type function, Ensemble Kalman filters depend on a nonlinear and quadratic-type interaction function defined in terms of the sample covariance of the particles.

Most of the literature in applied mathematics and computer science on these sophisticated interacting particle methods amounts to designing different classes of useable observer-type particle methods. These methods are based on a variety of inconsistent but judicious ensemble auxiliary transformations or include additional inflation/localisation-type algorithmic innovations, in order to avoid the inherent time-degeneracy of an insufficient particle ensemble size when solving a filtering problem with an unstable signal.

To the best of our knowledge, the first and the only rigorous mathematical analysis of these sophisticated discrete generation particle filters is developed in the pioneering articles by Le Gland–Monbet–Tran and by Mandel–Cobb–Beezley, which were published in the early 2010s. Nevertheless, besides the fact that these studies prove the asymptotic consistency of the ensemble Kalman filter, they provide exceedingly pessimistic mean-error estimates that grow exponentially fast with respect to the time horizon, even for linear Gaussian filtering problems with stable one-dimensional signals.

In the present article we develop a novel self-contained and complete stochastic perturbation analysis of the fluctuations, the stability, and the long-time performance of these discrete generation ensemble Kalman particle filters, including time-uniform and nonasymptotic mean-error estimates that apply to possibly unstable signals. To the best of our knowledge, these are the first results of this type in the literature on discrete generation particle filters, including the class of genetic-type particle filters and discrete generation ensemble Kalman filters. The stochastic Riccati difference equations considered in this work are also of interest in their own right, as a prototype of a new class of stochastic rational difference equation.

We study random surfaces with a uniformly convex gradient interaction in the presence of quenched disorder taking the form of a random independent external field. Previous work on the model has focused on proving existence and uniqueness of infinite-volume gradient Gibbs measures with a given tilt and on studying the fluctuations of the surface and its discrete gradient.

In this work, we focus on the convergence of the thermodynamic limit, establishing convergence of the finite-volume distributions with Dirichlet boundary conditions to translation-covariant (gradient) Gibbs measures. Specifically, it is shown that, when the law of the random field has finite second moment and is symmetric, the distribution of the gradient of the surface converges in dimensions $\mathit{d}\ge 4$ while the distribution of the surface itself converges in dimensions $\mathit{d}\ge 5$. Moreover, a power-law upper bound on the rate of convergence in Wasserstein distance is obtained. The results partially answer a question discussed by Cotar and Külske (Ann. Appl. Probab. 22 (2012) 1650–1692).

We develop a theory for continuous-time non-Markovian stochastic control problems which are inherently time-inconsistent. Their distinguishing feature is that the classical Bellman optimality principle no longer holds. Our formulation is cast within the framework of a controlled non-Markovian forward stochastic differential equation, and a general objective functional setting. We adopt a game-theoretic approach to study such problems, meaning that we seek for subgame perfect Nash equilibrium points. As a first novelty of this work, we introduce and motivate a refinement of the definition of equilibrium that allows us to establish a direct and rigorous proof of an extended dynamic programming principle, in the same spirit as in the classical theory. This in turn allows us to introduce a system consisting of an infinite family of backward stochastic differential equations analogous to the classical HJB equation. We prove that this system is fundamental, in the sense that its well-posedness is both necessary and sufficient to characterise the value function and equilibria. As a final step, we provide an existence and uniqueness result. Some examples and extensions of our results are also presented.

This paper considers how to obtain MCMC quantitative convergence bounds which can be translated into tight complexity bounds in high-dimensional settings. We propose a modified drift-and-minorization approach, which establishes generalized drift conditions defined in subsets of the state space. The subsets are called the “large sets”, and are chosen to rule out some “bad” states which have poor drift property when the dimension of the state space gets large. Using the “large sets” together with a “fitted family of drift functions”, a quantitative bound can be obtained which can be translated into a tight complexity bound. As a demonstration, we analyze several Gibbs samplers and obtain complexity upper bounds for the mixing time. In particular, for one example of Gibbs sampler which is related to the James–Stein estimator, we show that the number of iterations required for the Gibbs sampler to converge is constant under certain conditions on the observed data and the initial state. It is our hope that this modified drift-and-minorization approach can be employed in many other specific examples to obtain complexity bounds for high-dimensional Markov chains.

We establish universal modified log-Sobolev inequalities for reversible Markov chains on the boolean lattice ${\{0,1\}}^{\mathit{n}}$, when the invariant law π satisfies a form of negative dependence known as the stochastic covering property. This condition is strictly weaker than the strong Rayleigh property, and is satisfied in particular by all determinantal measures, as well as the uniform distribution over the set of bases of any balanced matroid. In the special case where π is k-homogeneous, our results imply the celebrated concentration inequality for Lipschitz functions due to Pemantle and Peres (Combin. Probab. Comput. 23 (2014) 140–160). As another application, we deduce that the natural Monte-Carlo Markov chain used to sample from π has mixing time at most $\mathit{k}\mathit{n}loglog\frac{1}{\mathit{\pi }(\mathit{x})}$ when initialized in state x. To the best of our knowledge, this is the first work relating negative dependence and modified log-Sobolev inequalities.

We show that in branching Brownian motion (BBM) in ${\mathbb{R}}^{\mathit{d}}$, $\mathit{d}\ge 2$, the law of ${\mathit{R}}_{\mathit{t}}^{\ast }$, the maximum distance of a particle from the origin at time t, converges as $\mathit{t}\to \infty$ to the law of a randomly shifted Gumbel random variable.

One of the fundamental assumptions in stochastic control of continuous time processes is that the dynamics of the underlying (diffusion) process is known. This is, however, usually obviously not fulfilled in practice. On the other hand, over the last decades, a rich theory for nonparametric estimation of the drift (and volatility) for continuous time processes has been developed. The aim of this paper is bringing together techniques from stochastic control with methods from statistics for stochastic processes to find a way to both learn the dynamics of the underlying process and control in a reasonable way at the same time. More precisely, we study a long-term average impulse control problem, a stochastic version of the classical Faustmann timber harvesting problem. One of the problems that immediately arises is an exploration-exploitation dilemma as is well known for problems in machine learning. We propose a way to deal with this issue by combining exploration and exploitation periods in a suitable way. Our main finding is that this construction can be based on the rates of convergence of estimators for the invariant density. Using this, we obtain that the average cumulated regret is of uniform order $\mathit{O}({\mathit{T}}^{-1/3})$.

Supercooled Stefan problems describe the evolution of the boundary between the solid and liquid phases of a substance, where the liquid is assumed to be cooled below its freezing point. Following the methodology of Delarue, Nadtochiy and Shkolnikov, we construct solutions to the one-phase one-dimensional supercooled Stefan problem through a certain McKean–Vlasov equation, which allows to define global solutions even in the presence of blow-ups. Solutions to the McKean–Vlasov equation arise as mean-field limits of particle systems interacting through hitting times, which is important for systemic risk modeling. Our main contributions are: (i) A general tightness theorem for the Skorokhod ${\mathit{M}}_1$-topology which applies to processes that can be decomposed into a continuous and a monotone part. (ii) A propagation of chaos result for a perturbed version of the particle system for general initial conditions. (iii) The proof of a conjecture of Delarue, Nadtochiy and Shkolnikov, relating the solution concepts of so-called minimal and physical solutions, showing that minimal solutions of the McKean–Vlasov equation are physical whenever the initial condition is integrable.

We view the classical Lindeberg principle in a Markov process setting to establish a probability approximation framework by the associated Itô’s formula and Markov operator. As applications, we study the error bounds of the following three approximations: approximating a family of online stochastic gradient descents (SGDs) by a stochastic differential equation (SDE) driven by multiplicative Brownian motion, Euler–Maruyama (EM) discretization for multi-dimensional Ornstein–Uhlenbeck stable process and multivariate normal approximation. All these error bounds are in Wasserstein-1 distance.