Tuesday, June 3rd
- Asymptotic Behavior of SPDEs and PDEs:
Vincent Martinez (CUNY Hunter College)
math.hunter.cuny.edu/vmartine/
Title: On the infinite-rotation limit of the stochastic primitive equations
Abstract: This talk will discuss a recent result for the stochastic primitive equation in the limit of infinite-rotation , namely, that the laws of the solutions to the stochastic primitive equation are asymptotically attracted to the unique invariant probability measure corresponding to its limit resonant system. This result is obtained by establishing an averaging principle that allows one to properly identify the covariance structure of the noise in the infinite-rotation regime and an asymptotic coupling technique to establish the existence of a spectral gap for the limit resonant system. This is joint work with Quyuan Lin (Clemson) and Rongchang Liu (U Arizona).
Article: 2503.00991
Peter Rudzis (University of North Carolian at Chapel Hill)
Title: Fluctuations of the Atlas model from inhomogeneous stationary profiles
Abstract: The Atlas model is a prototypical example in the class of rank-based diffusions, with applications in stochastic portfolio theory, queuing theory, and other areas of active research. In this model, a collection of Brownian particles moves freely on the real line, and the leftmost particle is given a drift \gamma \geq 0 for all time (thus, the drift of any tagged particle depends on the particle order). In the infinite-dimensional setting, the process of inter-particle gaps is known to have an infinite, one-parameter family of stationary distributions. Our main results establish fluctuation-type scaling limits for the occupation measure and order statistics of the infinite particle system when started from any stationary regime in which the particle density grows at an exponential rate. We describe the fluctuations in terms of a stochastic partial differential equation (SPDE) in which the initial condition is Brownian motion, the linear operator is the infinitesimal generator of geometric Brownian motion, and the equation is driven by an additive space-time noise that is white in time and colored in space. A consequence of this characterization is that the asymptotic fluctuations of a tagged particle started from the bulk of the initial data can be described by an explicit Gaussian process which has the same Holder regularity as fractional Brownian motion with Hurst parameter 1/4. Unlike the case of homogeneous initial data (Dembo and Tsai, 2017), in our setting the variance of the Gaussian limit diverges as one approaches the lower edge, and one can show that the lowest particle, started from 0 and translated linearly in time, converges in distribution to an explicit non-Gaussian limit, which reduces to Gumbel distribution for an appropriate choice of parameters. This work is joint with Sayan Banerjee and Amarjit Budhiraja.
Article: 2310.04545
Adina Oprisan (New Mexico State University)
Home | New Mexico State University – BE BOLD. Shape the Future.
Title: The hitting time for a Brownian motion with drift
Abstract: We consider a Brownian motion with power law drifts, when the order of the drift at 0 and infinity is different. We find that the exit time from the half-line is finite almost surely, and study the asymptotic behavior of the process at the time of hitting. We derive subexponential estimates for the hitting time, which show that the behavior of the process near the origin does not influence the time to hit 0 over long time periods. Our results extend to the case in which the drifts are regularly varying functions at 0 respectively infinity. This talk is based on a joint work with Dante DeBlassie and Robert Smits.
Article: 2403.06043
Shivam Dhama (Boston University)
Title: Uniform-in-time bounds for a stochastic hybrid system with fast periodic sampling and small white noise.
Abstract: We study the asymptotic behavior, uniform-in-time, of a nonlinear dynamical system under the combined effects of fast sampling with period delta and small white noise of size epsilon. The dynamics depend on both the current and recent measurements of the state, and as such, it is not Markovian. Our results can be interpreted as Law of Large Numbers (LLN) and Central Limit Theorem (CLT) type results. LLN type result shows that the resulting stochastic process is close to an ordinary differential equation uniformly in time as small parameters tend to zero. Further, in regards to CLT, we provide quantitative and uniform-in-time control of the fluctuations process. The interaction of the small parameters provides an additional drift term in the limiting fluctuations, which captures both the sampling and noise effects. As a consequence, we obtain a first-order perturbation expansion of the stochastic process along with time-independent estimates on the reminder. The zeroth- and first-order terms in the expansion are given by an ODE and SDE, respectively. Simulation studies that illustrate and supplement the theoretical results are also provided. This is a joint work with K. Spiliopoulos.
Article: 2404.18242
- Studies of Populations and Interacting Particles:
Grzegorz Rempala (Ohio State University)
Grzegorz A. Rempala, PhD | College of Public Health | The Ohio State University
Title: Stochastic Dynamics in Transcription Models
abstract: The formalism of stochastic reaction networks (SRNs) provides building blocks for number of models in mathematical biology both at molecular and population levels (e.g., gene transcription or epidemic outbreak). In particular the SRNs allow to naturally incorporate both delay and multi-scale phenomena. In the first case the resulting models may be often expressed in the language of queuing theory, in the second case they lead to stochastic diffusions and ODE/PDE approximations. In this talk I will provide a brief overview of the applications of SRNs to modeling molecular biological systems emphasizing the recent work on multi-scaling for simple gene transcription model.
David Sivakoff (Ohio State University)
Title: Competing one- and two-dimensional growth dynamics in two dimensions.
Abstract: We introduce a competing growth model between two colored states, blue and red, that spread among empty sites of Z2 starting from small initial densities p and q. Blue grows only horizontally at unit speed, while red grows either only vertically or in both directions at a constant speed. Once a site is colored either blue or red, it remains that color forever. When each color spreads in only one direction, we show that three distinct phases are characterized by power-law relationships between p and q as p to 0: when q << p3/2 most sites are eventually blue; when q >> p2/3 most sites are eventually red; and when p3/2 << q << p2/3 most sites remain empty forever. When red spreads in all directions, every site is eventually either red or blue, and we precisely locate the power-law scaling of the phase transition using a multiscale argument. Based on joint work with Janko Gravner.
Article: 2405.14723
Jimmy He (Ohio State University)
Title: Cutoff profile for the ASEP with one open boundary
Abstract: Consider the ASEP on a finite interval with one open boundary. This system will eventually reach equilibrium, but how long does this take? This question was studied by Gantert, Nestoridi, and Schmid, who established the cutoff phenomenon for this Markov chain which says that the first order behavior is a sharp transition to equilibrium. In joint work with Dominik Schmid, we refine their result, studying the shape of the convergence. The proof uses ideas from integrable probability.
Articles: Boundary current fluctuations for the half‐space ASEP and six‐vertex model and 2307.14941
Dilshad Imon (University of North Carolina at Chapel Hill)
Graduate Students | UNC Statistics & Operations Research
Title: Flocking under Fast and Large Jumps: Stability, Chaos, and Traveling Waves
Abstract: We study a pure-jump n-particle system with attractive mean field interactions under which each particle jumps forward by a random amount, independently sampled from a given distribution \theta, at exponentially distributed times with rate given by a function w of its signed distance from the system center of mass. The function w is taken to be non-increasing which leads to a `flocking’ behavior: the particles below the center of mass jump forward at a higher rate than those above it. This model was introduced in Modeling flocks and prices: Jumping particles with an attractive interaction and some of its properties were studied for the case when w is bounded. In the current work we are interested in the setting where w is unbounded, and this feature, together with the mild integrability we impose on the jump sizes, results in a stochastic dynamical system for interacting particles with fast and large jumps for which little is available in the literature. We identify natural conditions under which the system is well-posed and study the large n limit (the so-called `fluid limit’) of the empirical measure process associated with the system. By establishing well-posedness of the associated McKean-Vlasov equation we characterize the fluid limit of the particle system and prove a propagation of chaos result. Next, for the centered n-particle system, by constructing suitable Lyapunov functions, we establish existence and uniqueness of stationary distributions and study their tail properties. In the special case where w is an exponential function and \theta is an exponential distribution, by establishing that all stationary solutions of the McKean-Vlasov equation must be the unique fixed point of the equation, we prove a propagation of chaos result at t= infinity and establish convergence of the particle system, starting from stationarity, in the large n limit, to a traveling wave solution of the McKean-Vlasov equation. The proof of this result may be of interest for other interacting particle systems where convexity properties or functional inequalities generally used for establishing such a result are not available. Our work answers several open problems posed in Modeling flocks and prices: Jumping particles with an attractive interaction.
Article: 2404.13117
Wednesday, June 4th
- Well-Posedness of Solutions:
Le Chen (Auburn University)
Le Chen – Faculty – Math – Departments – Auburn University College of Sciences and Mathematics
Title: Interpolating Stochastic Heat and Wave Equations Through Fractional SPDEs
Abstract: This talk explores Stochastic Partial Differential Equations (SPDEs) with fractional differential operators, which give a parametric family that bridges the stochastic heat equation (SHE) and the stochastic wave equation (SWE). We will showcase recent developments and findings pertinent to this interpolation.
Article: 2108.11473
Hongyi Chen (University of Illinois Chicago)
hchen238 | Dept of Math, Stat, & Comp Sci | University of Illinois Chicago
Title: Global Geometry and the Parabolic Anderson Model on Compact Manifolds
Abstract: We introduce a family of intrinsic Gaussian noises on compact manifolds that we call “colored noise” on manifolds. With this noise, we study the parabolic Anderson model (PAM) on compact manifolds. Under some curvature conditions, we show the well-posedness of the PAM and provide some preliminary (but sharp) bounds on the second moment of the solution. It is interesting to see that global geometry plays a role in obtaining the well-posedness of the equation.
Article: 2502.04572
Kazuo Yamazaki (University of Nebraska-Lincoln)
Kazuo Yamazaki | Department of Mathematics | Nebraska
Title: Recent developments on the uniqueness and non-uniqueness of stochastic PDEs
Abstract: I will discuss recent developments on the uniqueness and non-uniqueness issue of stochastic PDEs, with special interest on the singular stochastic PDEs such as those forced by space-time white noise. The non-uniqueness results refer to those obtained via the convex integration technique, while uniqueness results particularly refer to the recent new approach of Hairer and Rosati that utilizes results from Anderson Hamiltonian and harmonic analysis tools to handle (deterministic) logarithmically supercritical PDEs.
Articles: 2503.00343 and 2008.04760 and 2312.15558 and 2410.02196
Alexander Dunlap (Duke University)
Title: Viscous shock solutions in KPZ
Abstract: I will present some joint work with Evan Sorensen on the long-term behavior of “V-shaped” solutions for the KPZ equation, and describe how this understanding can be used to complete the classification of time-invariant measures for this equation.
Article: Viscous shock fluctuations in KPZ
- Numerical Approximations of SPDEs:
Cecilia Mondaini (Drexel University)
Cecilia F. Mondaini, PhD | CoAS | Drexel University
Title: Estimating the long-time bias in numerical approximations of SPDEs
Abstract: I will present a general result that yields uniform-in-time error estimates for numerical approximations of SPDEs. As a consequence, this allows for estimating the long-time numerical error, namely the distance between the corresponding invariant measures. The result relies on the following two crucial assumptions on the numerical approximation: the existence of finite-time error estimates, and a discretization-uniform Wasserstein contraction for the corresponding semigroup. This will be illustrated with an application to the 2D stochastic Navier-Stokes equations, with a space-time numerical approximation given by a spectral Galerkin and semi-implicit Euler schemes. Here, the proof of finite-time error estimates required technical improvements from the related literature. This is based on joint work with Nathan Glatt-Holtz (Indiana U).
Article: 2302.01461
Yi-Ming Chen (Ohio State University)
Yi-Ming Chen | Department of Mathematics
Title: Analysis of Fully Discrete Discontinuous Galerkin Method for Nonlinear Stochastic Convection-Diffusion Equations
Abstract: Recently there is a growing interest for designing efficient high order numerical algorithms to approximate solutions of stochastic partial differential equations (SPDEs). In this talk, we develop fully-discrete stability analysis and high order error estimates for nonlinear stochastic convection-diffusion equations in one and higher dimensions, using local discontinuous Galerkin (LDG) method in space and implicit-explicit Euler-Maruyama scheme in time. Existing work on DG methods for SPDEs mostly focus on linear equations and semi-discrete analysis. This is because solutions of SPDEs are generally not time differentiable and are not bounded in the path variable. In this work, we prove stability analysis and error estimates for fully nonlinear case on subsets whose probability converge to one. Furthermore, we establish high moment stability and error estimation and derive pathwise error estimates. Numerical experiments are provided to verify the strong order of accuracy of our numerical schemes. Future work includes theoretical analysis of DG methods for SPDEs with higher order spatial derivatives.
Juliane Dalben (Drexel University)
Teaching and Research Assistants | CoAS | Drexel University
Title: A general form of Harris’ theorem and its applications in stochastic fluid dynamics
Abstract: In this talk, we focus on the study of two systems of stochastic partial differential equations: the hydrostatic Navier-Stokes equations and the Boussinesq approximation for Rayleigh-Bénard convection, both subject to additive stochastic noise in a two-dimensional domain. For each of these systems, given suitably regular initial data, we can define a Markov semigroup associated to each dynamic. Then, we verify that both Markov semigroups satisfy all the assumptions of a weak form of the Harris’ theorem proved by Glatt-Holtz and Mondaini in 2024 and inspired by Hairer et al in 2011. This theorem yields that each of these Markov semigroups is a contraction in a suitable Wasserstein distance, which immediately implies existence and uniqueness of a corresponding invariant measure and, most importantly, provides exponential rates of convergence towards this measure, i.e. exponential mixing rates. By showing the Wasserstein contraction of these Markov semigroups, we also obtain some advantages, such as long-term accuracy estimates with respect to suitable approximations of such systems and numerical schemes, which we intend to pursue in future work. This is a joint work with Cecilia Mondaini (Drexel University).
Related Article: 2302.01461 and s00440-009-0250-6.pdf
- Maximization Problem:
Jie Xiao (Memorial Univesity of Newfoundland)
Jie Xiao | Mathematics and Statistics | Memorial University of Newfoundland
Title: The p-maximization problem for the convex stress functions
Abstract:
Thursday, June 5th
- Asymptotic Behavior of SPDEs:
Carl Mueller (University of Rochester)
Carl Mueller : Faculty : Department of Mathematics : University of Rochester
Title: The almost linear stochastic heat equation
Abstract: This is joint work with Davar Khoshnevisan and Kuwoo Kim. The parabolic Anderson model with nonnegative drift is a widely studied linear SPDE which is related to the KPZ equation and many other topics. We consider this equation on the torus [−1,1] with end points identified. It was previously shown that for certain choices of parameters and for positive initial data, solutions (u(t,·)) tend to 0 in L∞ as t →∞, and this convergence occurs at an exponential rate. We study certain perturbed versions of this equation. Under appropriate conditions, we show that solutions (w(t,·)) also approach 0 at an exponential rate, and furthermore w(t,·) closely tracks u(t,·) on a logarithmic scale as t → ∞. Thus one can read off properties of w, a solution to a nonlinear equation, from the solutions u of a linear equation.
Article: 2211.02795 and aihp199.dvi
Konstantinos Spiliopoulos (Boston University)
Konstantinos Spiliopoulos – homepage
Title: Metastability and exit problems for systems of stochastic reaction-diffusion equations
Abstract: We develop a metastability theory for a class of stochastic reaction-diffusion equations exposed to small multiplicative noise. We consider the case where the unperturbed reaction-diffusion equation features multiple asymptotically stable equilibria. When the system is exposed to small stochastic perturbations, it is likely to stay near one equilibrium for a long period of time, but will eventually transition to the neighborhood of another equilibrium. We are interested in studying the exit time from the full domain of attraction (in a function space) surrounding an equilibrium and therefore do not assume that the domain of attraction features uniform attraction to the equilibrium. This means that the boundary of the domain of attraction is allowed to contain saddles and limit cycles. Our method of proof is purely infinite dimensional, i.e., we do not go through finite dimensional approximations. In addition, we address the multiplicative noise case and we do not impose gradient type of assumptions on the nonlinearity. We prove large deviations logarithmic asymptotics for the exit time and for the exit shape, also characterizing the most probable set of shapes of solutions at the time of exit from the domain of attraction.
Article: 1903.06038 and 2502.01783
Cole Graham (University of Wisconsin-Madison)
Title: Flowing across scales in the 2D stochastic heat equation
Abstract: The stochastic heat equation is critical in spatial dimension two: noise at many different scales influences the solution. In this talk, I will explore this multi-scale structure through a renormalization group analysis. This approach flexibly handles nonlinear noise and offers insights into pointwise statistics and macroscopic fluctuations. This represents joint work with Alex Dunlap.
Articles: The 2D nonlinear stochastic heat equation:pointwise statistics and the decoupling function and 2405.09520
Benjamin Fehrman (Louisiana State University)
Title: Stochastic dynamics of conservative stochastic PDE
Abstract: In this talk, we will motivate the study of stochastic PDE with conservative noise through their application to the non-equilibrium statistical mechanics theories of fluctuating hydrodynamics and macroscopic fluctuation theory. We will discuss some of the difficulties that arise in the analysis of such equations, and explain an approach to their well-posedness based on the equation’s kinetic form. The well-posedness theory will then be used to study the stochastic dynamics, including through the construction a random dynamical system and invariant measure, and the dynamical fluctuations and large deviations.
Articles: 2206.14789 and 2410.00254
Parisa Fatheddin (Ohio State University)
Parisa Fatheddin | Ohio State University
Title: Large Deviation Principle for Stochastic Navier-Stokes and Schrodinger equations
Abstract: The asymptotic behavior of stochastic Navier-Stokes and stochastic Schrodinger equations will be presented. Theorems such as large and moderate deviations as well as law of iterated logarithm and central limit theorem will be discussed. For large and moderate deviations results employ two methods: the classical Azencott method that relies on time discretization and the more modern technique weak convergence approach that was introduced in 2008. These are from joint work with Zhaoyang Qiu and Hannelore Lisei.
Articles: 2003.09082 and 2407.09300 and 1911.00064
Wai Tong Fan (University of North Carolina at Chapel Hill)
Wai Tong (Louis) Fan | School of Data Science and Society
Title: Stochastic waves on metric graphs and their genealogies
Abstract: Stochastic reaction-diffusion equations are important models in mathematics and in applied sciences such as spatial population genetics and ecology. Some of them arise as the scaling limit of discrete systems such as interacting particle models and are robust against model perturbation. In this talk, I will discuss methods to compute extinction probability, quasi-stationary distribution, asymptotic speed and other long-time behaviors for stochastic reaction-diffusion equations of Fisher-KPP type. Importantly, we consider these equations on general metric graphs that flexibly parametrize the underlying space. This enables us to not only bypass the ill-posedness issue of these equations in higher dimensions, but also assess the impact of space and stochasticity on the coexistence and the genealogies of interacting populations. Based on joint work with Adrian Gonzalez-Casanova, Wenqing Hu, Zhenyao Sun, Greg Terlov, Oliver Tough and Yifan (Johnny) Yang.
Related Article: Stochastic PDEs on graphs as scaling limits of discrete interacting systems and 2309.10998
Johnny Yang (Indiana University Bloomington):
X Yang: Graduate Students: About: Department of Mathematics: Indiana University Bloomington
Title: SPDEs on metric measure spaces
Abstract: This talk explores parabolic stochastic partial differential equations (SPDEs) on metric spaces with fractional dimensions, a classical example being the Sierpinski gasket. I will talk about some qualitative properties of solutions to a general class of SPDEs on those spaces focusing on various comparison principles. Under mild conditions, we showed that the solution to the parabolic Anderson models is strictly positive everywhere for positive time given non-trivial non-negative initial condition. In addition, the solution to the super-Brownian density SPDE has compact support, given non-negative compactly supported initial conditions.
Kenneth Ng (Ohio State University)
Biography – Kenneth Ng’s Homepage
Title: Mean Field Analysis of Mutual Insurance
Abstract: A mutual insurance company (MIC) is a form of consumer co-operative owned by their policyholders, who are simultaneously the owners and customers of the company. It offers a unique surplus-sharing mechanism in the form of dividends to policyholders who insured from an MIC, and thus their proceeds will depend on the insurance purchases and claim experience of the other policyholders within the same MIC. This surplus-sharing mechanism creates an interactive environment among policyholders, which influences the insurance strategy of each individual. In this work, we formulate the problem of optimal insurance strategies of heterogeneous policyholders within an MIC under a mean field game framework. The complete solution is characterized by a mean field forward-backward stochastic differential equation (MF-FBSDE), whose global existence and uniqueness is established. To address generic scenarios where closed-form solutions are not available, we introduce a deep neural network approach to numerically solve the associated MF-FBSDE and thus the optimal insurance strategies.
Article: Mean field analysis of two-party governance: Competition versus cooperation among leaders – ScienceDirect and Optimal investment-withdrawal strategy for variable annuities under a performance fee structure – ScienceDirect