Schedule for: 18w5134 - Interacting Particle Systems and Parabolic PDEs

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Chemotaxis plays a crucial role in a variety of processes in biology and ecology. Quite often it acts to improve efficiency of biological reactions. One example is reproduction, where eggs release chemicals that attract sperm. Another example are infected tissues secreting chemokines, attracting monocytes to fight invading bacteria. A macro scale example is flower scent appealing to pollinator insects. I will start from a simplified single equation model inspired by coral spawning. In the framework of this model, chemotaxis can be shown to have dramatic effect on the efficiency of the reaction. Next I will talk about a more realistic two equation system set in two dimensions. The problem is linked with the analysis of Fokker-Planck operators with logarithmic potential, and in particular the rate of convergence to ground state. There is no spectral gap in this case, and new weighted Poincare inequalities will be needed to derive sufficiently sharp estimates. The talks are based on works joint with Lenya Ryzhik, Fedja Nazarov and Yao Yao.

We are interested in a class of parabolic integro-differential equations coming from evolutionary biology. Such equations describe the dynamics of a phenotypically structured population under the effect of selection and mutations. The solutions to such equations concentrate, in the limit of small diffusion (mutation) and in long time, as a sum of Dirac masses. In the first part of this talk, we present the basic ingredients of an approach based on Hamilton-Jacobi equations with constraint to study such problems. In the second part, we consider a more restricted framework with more regularity, where we are able to obtain the well-posedness of the corresponding Hamilton-Jacobi equation with constraint, and where we can obtain a rather precise approximation of the phenotypic distribution of the population. Finally, we show how this approach can lead to quantitative results in biological applications, in particular, in the case of a time periodic environment

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The distribution function of the rightmost particle in a branching Brownian motion satisfies the Fisher-KPP equation: \[ u_t=u_{xx}+u(1-u) \] Such an equation appears also in biology, chemistry or theoretical physics to describe a moving interface, or a front, between a stable and an unstable medium. Thirty years ago, Bramson gave rigorous sharp estimates on the position of the front, and, fifteen years ago, Ebert and van Saarloos heuristically identified universal vanishing corrections. In this presentation, I will present a novel way to study the position of such a front, which allows to recover all the known terms and find some new ones. We start by studying a front equation where the non-linearity is replaced by a condition at a free boundary, and we show how to extend our results to the actual Fisher-KPP.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

The maximum of independent random walks is governed by extreme-value statistics (e.g. Gumbel). What happens if the jump rates for the random walks are chosen randomly too (independent and identically distributed in space and time)? In this talk we demonstrate how a new class of extreme value statistics and scalings arise from this change. We show this for a particular class of "exactly solvable" models with Beta distributed jump probabilities. The statistics and scalings that arise relate to the Kardar-Parisi-Zhang universality class and limiting distributions from random matrix theory.

The Box-Ball System (BBS) is a cellular automaton introduced by Takahashi and Satsuma (TS) as a discrete counterpart of the Korteweg \& de Vries (KdV) differential equation. Both systems exhibit solitons, solitary waves that conserve shape and speed even after collision with other solitons. The BBS has state space $\{0,1\}^{Z}$, where each integer represents a box which may contain one ball or be empty. One iteration of the automaton consists of a carrier visiting successively all boxes from left to right, picking balls from occupied boxes and deposing one ball, if carried, at each visited empty box. Building on the TS identification of solitons, we provide a soliton decomposition of the ball configurations, show that the dynamics reduces to a hierarchical translation of the components and prove that shift-stationary measures with independent soliton components are invariant for the dynamics. We also prove that the asymptotic speed of a tagged soliton of size $k$ converges to a positive real number $v_k$ and exhibit the equations satisfied by the speeds $(v_k)_{k\ge 1}$. A detailed analysis shows that among many others, product measures and Ising measures have independent soliton components. Joint work with Chi Nguyen, Minmin Wang, Leonardo Rolla and Davide Gabrielli.

We investigate the large time behavior of solutions of reaction-diffusion equations with general reaction terms in periodic media, with a particular emphasis on the effect of the geometry of the domain. We first derive some conditions which guarantee that solutions with compactly supported initial data invade the domain. In particular, we relate such solutions with front-like solutions such as pulsating traveling fronts. Next, we focus on the homogeneous bistable equation set in a domain with periodic obstacles, and specifically on the cases where fronts are not known to exist. We show how the geometry of the domain can block or allow invasion, and we exhibit a periodic domain on which the propagation takes place in an asymmetric fashion. Finally, we focus on the speed of invasion. This talk is based on a paper in collaboration with Luca Rossi.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We will discuss how fluctuations of some interacting particle systems converge to stochastic PDEs (in our cases related to variants of the KPZ equation). The models considered include the six vertex model and two variants of the asymmetric simple exclusion process (ASEP) -- ASEP with inhomogeneous jump rates and ASEP on a bounded interval or half-line. Time permitting, we will highlight some of the key ideas behind these results -- namely Markov duality, and various types of heat kernel estimates.

The stochastic Burgers equation is one of the basic evolutionary SPDEs related to fluid dynamics and KPZ, among other things. The ergodic properties of the system in the compact space case were understood in 2000's. With my coauthors, Eric Cator, Kostya Khanin, Liying Li, I have been studying the noncompact case. The one force - one solution principle has been proved for positive and zero viscosity. The analysis is based on long-term properties of action minimizers and polymer measures. The latest addition to the program is the convergence of infinite volume polymer measures to Lagrangian one-sided minimizers in the limit of vanishing viscosity (or, temperature) which results in the convergence of the associated global solutions and invariant measures.

We establish a Large Deviation Principle (LDP) for the lower-tail of the Kardar-Parisi-Zhang (KPZ) equation with narrow wedge initial condition. Our analysis exploits an exact connection between the KPZ one-point distribution and the Airy Point Process (PP). The latter is a semi-infinite particle system arises at the spectral edge the Gaussian Unitary Ensemble. We develop the LDP for the Airy PP, with an explicitly rate function written in terms of electrostatic energy. From this rate function, we solve a variational problem to obtain the exact lower-tail LD rate function of the KPZ equation.

We prove the convergence, in the small mass limit, of statistically invariant states for a class of semi-linear damped wave equations, perturbed by an additive Gaussian noise, both with Lipschitz-continuous and with polynomial nonlinearities. In particular, we prove that the first marginals of any sequence of invariant measures for the stochastic wave equation converge in a suitable Wasserstein metric to the unique invariant measure of the limiting stochastic semi-linear parabolic equation obtained in the Smoluchowski-Kramers approximation.

The Landis conjecture, proposed in the 80s, states that if a solution of the equation $\Delta u +V(x)u=0$ decays faster than a suitable exponential then it must be identically equal to zero. It can be viewed as a unique continuation property at infinity (UCI). The conjecture has been disproved by Meshkov in the case of complex-valued functions, but it remains open in the real case. In this talk, I will recall some partial results obtained by Kenig and collaborators. Next, I will present the proof of the UCI and of the Landis conjecture for radial operators. Finally, I will discuss the validity of the UCI under the restriction on the sign either of the solution or of the generalized principal eigenvalue.

Many interesting Markov processes (e.g. SPDEs) are not strong Feller and have mutually singular transition kernels. Their ergodic properties cannot be analyzed by the standard classical methods and require a special treatment. We develop a generalized coupling method (inspired by ideas of [M. Hairer, 2002]) and establish verifiable sufficient conditions for exponential or subexponential ergodicity. We apply the obtained results to show exponential ergodicity of a variety of nonlinear stochastic parabolic PDEs with additive forcing, including 2D stochastic Navier-Stokes equations, stochastic Boussinesq equations and other models. This extends recent results of [N. Glatt-Holtz, J. Mattingly, G. Richards, 2017] [1] O. Butkovsky, A.Kulik, M.Scheutzow (2018). Generalized couplings and ergodic rates for SPDEs and other Markov models. arXiv:1806.00395

The Boltzmann equation describes the evolution of particle densities, in terms of space and velocity, for gases and plasma. We will discuss the regularization effect of this equation in the in-homogeneous, non-cutoff case. We prove that the solution remains bounded and Holder continuous for as long as its associated hydrodynamic quantities are bounded and away from vacuum. Our analysis is based on techniques that originate in the study of parabolic integro-differential equations.

I will describe an interacting particle system motivated by Stein variational gradient descent [Q. Liu and D. Wang, NIPS 2016], a deterministic algorithm for sampling from a given probability density with unknown normalization. We prove that in the large particle limit the empirical measure converges to a solution of a non-local and nonlinear PDE. We also prove global well-posedness and uniqueness of the solution to the limiting PDE. Finally, we prove that the solution to the PDE converges to the unique invariant solution in large time limit. This is joint work with Jianfeng Lu and Yulong Lu.

The long-time behavior of reaction-advection-diffusion equations, which, for example, model temperature change in a fluid undergoing reaction, is quite well understood in situations with rigid structure (e.g. constant or periodic coefficients). Less well-understood, however, are models where the advection is coupled to the system, despite this arising naturally in physical settings. In this talk, I will discuss some such situations. I will introduce the main tools to study these equations and then obtain various bounds on the propagation speeds depending on the parameters in the model. This is a joint work with François Hamel.

We consider a bistable reaction-diffusion equation on ${\bf R}^N$ in the presence of an obstacle $K$, which is a wall of infinite span with periodically arrayed holes. More precisely, $K$ is a closed subset of ${\bf R}^N$ with smooth surface such that its projection onto the $x_1$-axis is bounded, while it is periodic in the rest of variables $(x_2,\ldots, x_N)$. We assume that ${\bf R}^N \setminus K$ is connected. Our goal is to study what happens when a planar traveling front coming from $x_1 = +\infty$ meets the wall $K$. We first show that there is clear dichotomy between {\it propagation} and {\it blocking}. In other words, the traveling front either completely penetrates through the wall or is totally blocked, and that there is no intermediate behavior. This dichotomy result will be proved by what we call a De Giorgi type lemma for the elliptic equation $\Delta v + f(v) = 0$ on ${\bf R}^N$. Then we will discuss sufficient conditions for blocking, and those for propagation. This is joint work with Henri Berestycki and Fran\c{c}ois Hamel. If time allows, I will also talk about the non-KPP monostable equation with the nonlinearity $u^p(1-u)$, $p>1$, and discuss briefly whether or not blocking can occur for the front solution.

We outline a probabilistic proof of a well known connection between a special case of the Allen-Cahn equation and mean curvature flow. This proof is sufficiently flexible that it can be extended to capture a similar result for a model which captures the motion of what are known in population genetics as hybrid zones. As time permits we'll outline some extensions of this work. Joint work with Nic Freeman and Sarah Penington.

We add a competitive interaction between nearby particles in a branching Brownian motion (BBM). Each particle has a mass, which decays at rate proportional to the mass density in a window centred at the location of the particle. The total mass of the system increases through branching events. In standard BBM, we may define the front location at time t as the location of the rightmost particle. For BBM with decay of mass, it makes sense to instead define the front location as the location at which the local mass density drops to o(1). We can show that in a weak sense this front is ~$t^(1/3)$ behind the front for standard BBM. We can also show that at large times, over a bounded time interval, the local mass density for BBM with decay of mass is well approximated by the solution of a PDE known as the non-local Fisher-KPP equation. This relationship between the particle system and the PDE allows us to control the behaviour of the local mass density behind the front at large times, and also to use intuition from the particle system setting to prove results about the PDE. Several interesting questions about this model remain open. Joint work with Louigi Addario-Berry and Julien Berestycki.

The (d+1)-dimensional KPZ equation \[ \partial_t h = \nu \Delta h + \frac{\lambda}{2}|\nabla h|^2 + \sqrt{D}\dot{W}, \] in which \dot{W} is a space--time white noise, is a natural model for the growth of d-dimensional random surfaces. These surfaces are extremely rough due to the white noise forcing, which leads to difficulties in interpreting the nonlinear term in the equation. In particular, it is necessary to renormalize the mollified equations to achieve a limit as the mollification is turned off. The d = 1 case has been understood very deeply in recent years, and progress has been made in d ≥ 3, but little is known in d = 2. I will describe recent joint work with Sourav Chatterjee showing the tightness of a family of Cole--Hopf solutions to (2+1)-dimensional mollified and renormalized KPZ equations. This implies that subsequential limits exist, which we furthermore can show do not coincide with solutions to the linearized equation, despite the fact that our renormalization scheme involves a logarithmic attenuation of the nonlinearity as the mollification scale is taken to zero.

We will discuss the transition from homogenization to the Wong-Zakai approximation of stochastic heat equation in d=1. Joint work with Li-Cheng Tsai.

Consider the heat equation $\partial_t u = (1/2) \Delta u + \lambda V (t, x)u$, where $V(t,x)$ is a mean zero Gaussian field with correlation of compact support in space and time. I will describe weak limits of (rescaled) forms of $u$, in the form of the stochastic heat equation with additive white noise. The proofs use probabilistic techniques. Based on arXiv:1710.0034 (with Gu and Ryzhik) and arXiv:1601.01652 (with Mukherjee and Shamov), and recent developments.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.