There are a number of situations in which rescaled interacting particle systems have been shown to converge to a reaction diffusion equation (RDE) with a bistable reaction term. These RDEs have traveling wave solutions. When the speed of the wave is nonzero, block constructions have been used to prove the existence or nonexistence of nontrivial stationary distributions. Here, we follow the approach in a paper by Etheridge, Freeman, and Pennington to show that in a wide variety of examples when the RDE limit has a bistable reaction term and traveling waves have speed 0, one can run time faster and further rescale space to obtain convergence to motion by mean curvature. This opens up the possibility of proving that the sexual reproduction model with fast stirring has a discontinuous phase transition, and that in Region 2 of the phase diagram for the nonlinear voter model studied by Molofsky et al there were two nontrivial stationary distributions.
We establish a general perturbative method to prove entropic Ricci curvature bounds for interacting stochastic particle systems. We apply this method to obtain curvature bounds in several examples, namely: Glauber dynamics for a class of spin systems including the Ising and Curie--Weiss models, a class of hard-core models and random walks on groups induced by a conjugacy invariant set of generators.
We obtain a lower bound for the coarse Ricci curvature of continuous time pure jump Markov processes, with an emphasis on interacting particle systems. Applications to several models are provided, with a detailed study of the herd behavior of a simple model of interacting agents.
We establish the large deviation principle (LDP) for stochastic flows of interacting Brownian motions. In particular, we consider smoothly correlated flows, coalescing flows and Brownian motion stopped at a hitting moment.
We provide a probabilistic proof of a well known connection between a special case of the Allen-Cahn equation and mean curvature flow. We then prove a corresponding result for scaling limits of the spatial $Λ$-Fleming-Viot process with selection, in which the selection mechanism is chosen to model what are known in population genetics as hybrid zones. Our proofs will exploit a duality with a system of branching (and coalescing) random walkers which is of some interest in its own right.
We consider the model space of constant curvature in dimension n and characterize all co-adapted couplings of Brownian motions on this space for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of $ρ$ satisfying the above hypotheses.
We prove the Large Deviation Principle for the empirical process in a system of locally interacting Brownian motions in the nonequilibrium dynamic. Such a phenomenon has been proven only for two lattice systems: the symmetric simple exclusion process and zero-range process. Therefore, we have achieved the third result in this context and moreover the first result for the diffusion-type interacting particle system.
Given a submersion $ϕ: M \to N$, where $M$ is Riemannian, we construct a stochastic process $X$ on $M$ such that the image $Y:=ϕ(X)$ is a (reversed, scaled) mean curvature flow of the fibers of the submersion. The model example is the mapping $π: GL(n) \to GL(n)/O(n)$, whose image is equivalent to the space of $n$-by-$n$ positive definite matrices, $\pdef$, and the said MCF has deterministic image. We are able to compute explicitly the mean curvature (and hence the drift term) of the fibers w.r.t. this map, (i) under diagonalization and (ii) in matrix entries, writing mean curvature as the gra…
The Dirichlet forms related to various infinite systems of interacting Brownian motions are studied. For a given random point field $ μ$, there exist two natural infinite-volume Dirichlet forms $ (\mathcal{E}^{\mathsf{upr}},\mathcal{D}^{\mathsf{upr}})$ and $(\mathcal{E}^{\mathsf{lwr}},\mathcal{D}^{\mathsf{lwr}})$ on $ L^2(\mathsf{S} ,μ) $ describing interacting Brownian motions each with unlabeled equilibrium state $ μ$. The former is a decreasing limit of a scheme of such finite-volume Dirichlet forms, and the latter is an increasing limit of another scheme of such finite-volume Dirichlet for…
Consider a negatively drifted one dimensional Brownian motion starting at positive initial position, its first hitting time to 0 has the inverse Gaussian law. Moreover, conditionally on this hitting time, the Brownian motion up to that time has the law of a 3- dimensional Bessel bridge. In this paper, we give a generalization of this result to a family of Brownian motions with interacting drifts, indexed by the vertices of a conductance network. The hitting times are equal in law to the inverse of a random potential that appears in the analysis of a self-interacting process called the Vertex R…
We study the hydrodynamic scaling limit for the Glauber-Kawasaki dynamics. It is known that, if the Kawasaki part is speeded up in a diffusive space-time scaling, one can derive the Allen-Cahn equation which is a kind of the reaction-diffusion equation in the limit. This paper concerns the scaling that the Glauber part, which governs the creation and annihilation of particles, is also speeded up but slower than the Kawasaki part. Under such scaling, we derive directly from the particle system the motion by mean curvature for the interfaces separating sparse and dense regions of particles as a …
We prove that self-diffusion constants of interacting Brownian particles in $ \mathbb{R}$ always vanish if the particles do not collide with each other. We represent self-diffusion constants by additive functionals of reversible Markov processes as obtained in O. 1998.
We develop a new and systematic method for proving entropic Ricci curvature lower bounds for Markov chains on discrete sets. Using different methods, such bounds have recently been obtained in several examples (e.g., 1-dimensional birth and death chains, product chains, Bernoulli-Laplace models, and random transposition models). However, a general method to obtain discrete Ricci bounds had been lacking. Our method covers all of the examples above. In addition, we obtain new Ricci curvature bounds for zero-range processes on the complete graph. The method is inspired by recent work of Caputo, D…
We show that the pointed measured Gromov convergence of the underlying spaces implies (or under some condition, is equivalent to) the weak convergence of Brownian motions under Riemannian Curvature-Dimension (RCD) conditions.
We consider a system of interacting Brownian particles in R^d with a pairwise potential, which is radially symmetric, of finite range and attains a unique minimum when the distance of two particles becomes a>0. The asymptotic behavior of the system is studied under the zero temperature limit from both microscopic and macroscopic aspects. If the system is rigidly crystallized, namely if the particles are rigidly arranged in an equal distance a, the crystallization is kept under the evolution in macroscopic time scale. Then, assuming that the crystal has a definite limit shape under a macroscopi…
We derive a continuum mean-curvature flow as a certain hydrodynamic scaling limit of Glauber-Kawasaki dynamics with speed change. The Kawasaki part describes the movement of particles through particle interactions. It is speeded up in a diffusive space-time scaling. The Glauber part governs the creation and annihilation of particles. The Glauber part is set to favor two levels of particle density. It is also speeded up in time, but at a lesser rate than the Kawasaki part. Under this scaling, a mean-curvature interface flow emerges, with a homogenized `surface tension-mobility' parameter reflec…
An embedding of the group $\Diff(S^{1})$ of orientation preserving diffeomorphims of the unit circle $S^1$ into an infinite-dimensional symplectic group, $\Sp(\infty)$, is studied. The authors prove that this embedding is not surjective. A Brownian motion is constructed on $\Sp(\infty)$. This study is motivated by recent work of H. Airault, S. Fang and P. Malliavin. The Ricci curvature of the infinite-dimensional symplectic group is computed. The result shows that in almost all directions, the Ricci curvature is negative infinity.
We prove the central limit theorem of random variables induced by distances to Brownian paths and Green functions on the universal cover of Riemannian manifolds of finite volume with pinched negative curvature. We further provide some ergodic properties of Brownian motions and an application of the central limit theorem to the dynamics of geodesic flows in pinched negative curvature.
We review probabilistic approaches to the Gross-Pitaevskii theory describing interacting dilute systems of particles. The main achievement are large deviations principles for the mean occupation measure of a large system of interacting Brownian motions in a trapping potential. The corresponding rate functions are given as variational problems whose solution provide effective descriptions of the infinite system.
We characterize the minimal time horizon over which any equity market with $d \geq 2$ stocks and sufficient intrinsic volatility admits relative arbitrage with respect to the market portfolio. If $d \in \{2,3\}$, the minimal time horizon can be computed explicitly, its value being zero if $d=2$ and $\sqrt{3}/(2π)$ if $d=3$. If $d \geq 4$, the minimal time horizon can be characterized via the arrival time function of a geometric flow of the unit simplex in $\mathbb R^d$ that we call the minimum curvature flow.
We study the joint asymptotic behavior of spacings between particles at the edge of multilevel Dyson Brownian motions, when the number of levels tends to infinity. Despite the global interactions between particles in multilevel Dyson Brownian motions, we observe a decoupling phenomenon in the limit: the global interactions become negligible and only the local interactions remain. The resulting limiting objects are interacting particle systems which can be described as Brownian versions of certain totally asymmetric exclusion processes. This is the first appearance of a particle system with loc…
We consider a system of infinitely many interacting Brownian motions that models the height of a one-dimensional interface between two bulk phases. We prove that the large scale fluctuations of the system are well approximated by the solution to the KPZ equation provided the microscopic interaction is weakly asymmetric. The proof is based on the martingale solutions of Goncalves and Jara and the corresponding uniqueness result of Gubinelli and Perkowski.
This paper presents our study of the asymptotic behavior of a two-component system of Brownian motions undergoing certain singular interactions. In particular, the system is a combination of two different types of particles and the mechanical properties and interaction parameters depend on the corresponding type of particles. We prove that the hydrodynamic limit of the empirical densities of two types is the solution of a certain quasi-linear parabolic partial differential equation known as the Maxwell-Stefan equation.
In this note, we give a new proof of Liggett's theorem on the invariant measures of independent particle systems from [Lig78] in the particular case of independent drifted Brownian motions. This particular case has received a lot of attention recently due to its applications for the analysis of the local extrema of discrete Gaussian free field. The novelty of our proof is that it identifies directly the expected Poisson Point Process with exponential intensity without relying on the Choquet-Deny convolution equation $μ* P=μ$ ([ChoquetDeny60,Deny60]).
We define the Ricci curvature of Markov chains on metric spaces as a local contraction coefficient of the random walk acting on the space of probability measures equipped with a Wasserstein transportation distance. For Brownian motion on a Riemannian manifold this gives back the value of Ricci curvature of a tangent vector. Examples of positively curved spaces for this definition include the discrete cube and discrete versions of the Ornstein--Uhlenbeck process. Moreover this generalization is consistent with the Bakry--Émery Ricci curvature for Brownian motion with a drift on a Riemannian man…
We investigate the construction of diffusions consisting of infinitely numerous Brownian particles moving in $\mathbb{R}^d$ and interacting via logarithmic functions (two-dimensional Coulomb potentials). These potentials are very strong and act over a long range in nature. The associated equilibrium states are no longer Gibbs measures. We present general results for the construction of such diffusions and, as applications thereof, construct two typical interacting Brownian motions with logarithmic interaction potentials, namely the Dyson model in infinite dimensions and Ginibre interacting Bro…
