348 research outputs found
The Master Equation for Large Population Equilibriums
We use a simple N-player stochastic game with idiosyncratic and common noises
to introduce the concept of Master Equation originally proposed by Lions in his
lectures at the Coll\`ege de France. Controlling the limit N tends to the
infinity of the explicit solution of the N-player game, we highlight the
stochastic nature of the limit distributions of the states of the players due
to the fact that the random environment does not average out in the limit, and
we recast the Mean Field Game (MFG) paradigm in a set of coupled Stochastic
Partial Differential Equations (SPDEs). The first one is a forward stochastic
Kolmogorov equation giving the evolution of the conditional distributions of
the states of the players given the common noise. The second is a form of
stochastic Hamilton Jacobi Bellman (HJB) equation providing the solution of the
optimization problem when the flow of conditional distributions is given. Being
highly coupled, the system reads as an infinite dimensional Forward Backward
Stochastic Differential Equation (FBSDE). Uniqueness of a solution and its
Markov property lead to the representation of the solution of the backward
equation (i.e. the value function of the stochastic HJB equation) as a
deterministic function of the solution of the forward Kolmogorov equation,
function which is usually called the decoupling field of the FBSDE. The
(infinite dimensional) PDE satisfied by this decoupling field is identified
with the \textit{master equation}. We also show that this equation can be
derived for other large populations equilibriums like those given by the
optimal control of McKean-Vlasov stochastic differential equations. The paper
is written more in the style of a review than a technical paper, and we spend
more time and energy motivating and explaining the probabilistic interpretation
of the Master Equation, than identifying the most general set of assumptions
under which our claims are true
Smooth Approximation of Lipschitz functions on Riemannian manifolds
We show that for every Lipschitz function defined on a separable
Riemannian manifold (possibly of infinite dimension), for every continuous
, and for every positive number , there exists
a smooth Lipschitz function such that
for every and
. Consequently, every separable
Riemannian manifold is uniformly bumpable. We also present some applications of
this result, such as a general version for separable Riemannian manifolds of
Deville-Godefroy-Zizler's smooth variational principle.Comment: 10 page
Exploring Peer Instruction: Should Cohort Clicker Responses Appear During or After Polling?
During problem-based learning sessions, undergraduate students were tasked with answering chemistry-related questions using
clicker-handset technology in which the last response made by each handset would override any previous vote. The benefits, if any, of showing cohort responses from clicker questions during versus after polling were explored. Preliminary work suggested that cohort responses shown live during polling created greater unprompted peer instruction, which was inferred from a noticeably louder level of classroom debate. To test if subtle polling changes can promote greater peer instruction, this study monitored cohort performance, clicker response times, and voting-behavior patterns throughout the polling process. Profiling individual and team-based clicker activity in this manner highlighted contrasting performance data. No significant differences were seen when clicker handsets were used individually by students; however, certain trends were seen in the team-based model, which depended on how cohort responses were displayed and were also influenced by question type, with multiple-choice questions (MCQs) performing differently from true−false style questions. The results highlight improved performance in the team-based clicker model with peer instruction taking place during MCQ polls in which cohort responses were displayed live during voting. These findings highlight a clicker strategy embedded with peer instruction that bypasses the need for the standard three-phase process of polling, discussing, and then repolling. Displaying polling responses live enables multiple polling and discussion opportunities to occur in a single interchangeable phase, thus providing a time-efficient voting and peer-instruction method that may attract more instructors to adopt clicker technology within their teaching
A simple mean field model for social interactions: dynamics, fluctuations, criticality
We study the dynamics of a spin-flip model with a mean field interaction. The
system is non reversible, spacially inhomogeneous, and it is designed to model
social interactions. We obtain the limiting behavior of the empirical averages
in the limit of infinitely many interacting individuals, and show that phase
transition occurs. Then, after having obtained the dynamics of normal
fluctuations around this limit, we analize long time fluctuations for critical
values of the parameters. We show that random inhomogeneities produce critical
fluctuations at a shorter time scale compared to the homogeneous system.Comment: 37 pages, 2 figure
Hamilton Jacobi Bellman equations in infinite dimensions with quadratic and superquadratic Hamiltonian
We consider Hamilton Jacobi Bellman equations in an inifinite dimensional
Hilbert space, with quadratic (respectively superquadratic) hamiltonian and
with continuous (respectively lipschitz continuous) final conditions. This
allows to study stochastic optimal control problems for suitable controlled
Ornstein Uhlenbeck process with unbounded control processes
A Semi-Lagrangian scheme for a modified version of the Hughes model for pedestrian flow
In this paper we present a Semi-Lagrangian scheme for a regularized version
of the Hughes model for pedestrian flow. Hughes originally proposed a coupled
nonlinear PDE system describing the evolution of a large pedestrian group
trying to exit a domain as fast as possible. The original model corresponds to
a system of a conservation law for the pedestrian density and an Eikonal
equation to determine the weighted distance to the exit. We consider this model
in presence of small diffusion and discuss the numerical analysis of the
proposed Semi-Lagrangian scheme. Furthermore we illustrate the effect of small
diffusion on the exit time with various numerical experiments
Quantum Smoluchowski equation: Escape from a metastable state
We develop a quantum Smoluchowski equation in terms of a true probability
distribution function to describe quantum Brownian motion in configuration
space in large friction limit at arbitrary temperature and derive the rate of
barrier crossing and tunneling within an unified scheme. The present treatment
is independent of path integral formalism and is based on canonical
quantization procedure.Comment: 10 pages, To appear in the Proceedings of Statphys - Kolkata I
Allocating HIV Prevention Funds in the United States: Recommendations from an Optimization Model
The Centers for Disease Control and Prevention (CDC) had an annual budget of approximately $327 million to fund health departments and community-based organizations for core HIV testing and prevention programs domestically between 2001 and 2006. Annual HIV incidence has been relatively stable since the year 2000 [1] and was estimated at 48,600 cases in 2006 and 48,100 in 2009 [2]. Using estimates on HIV incidence, prevalence, prevention program costs and benefits, and current spending, we created an HIV resource allocation model that can generate a mathematically optimal allocation of the Division of HIV/AIDS Prevention’s extramural budget for HIV testing, and counseling and education programs. The model’s data inputs and methods were reviewed by subject matter experts internal and external to the CDC via an extensive validation process. The model projects the HIV epidemic for the United States under different allocation strategies under a fixed budget. Our objective is to support national HIV prevention planning efforts and inform the decision-making process for HIV resource allocation. Model results can be summarized into three main recommendations. First, more funds should be allocated to testing and these should further target men who have sex with men and injecting drug users. Second, counseling and education interventions ought to provide a greater focus on HIV positive persons who are aware of their status. And lastly, interventions should target those at high risk for transmitting or acquiring HIV, rather than lower-risk members of the general population. The main conclusions of the HIV resource allocation model have played a role in the introduction of new programs and provide valuable guidance to target resources and improve the impact of HIV prevention efforts in the United States
Consensus via multi-population robust mean-field games
In less prescriptive environments where individuals are told ‘what to do’
but not ‘how to do’, synchronization can be a byproduct of strategic thinking,
prediction, and local interactions. We prove this in the context of multipopulation
robust mean-field games. The model sheds light on a multi-scale
phenomenon involving fast synchronization within the same population and
slow inter-cluster oscillation between different populations
- …