Ergodic Constant Research Articles

Generalized convergence of solutions for nonlinear Hamilton–Jacobi equations with state-constraint

For a continuous Hamiltonian H:(x,p,u)∈T⁎Rn×R→R, we consider the asymptotic behavior of associated Hamilton–Jacobi equations with state-constraint{H(x,Du,λu)≤Cλ,x∈Ωλ⊂Rn,H(x,Du,λu)≥Cλ,x∈Ω‾λ⊂Rn, as λ→0+. When H satisfies certain convex, coercive and monotone conditions, the domain Ωλ:=(1+r(λ))Ω keeps bounded, star-shaped for all λ>0 with limλ→0+⁡r(λ)=0, and limλ→0+⁡Cλ=c(H) equals the ergodic constant of H(⋅,⋅,0), we prove the convergence of solutions uλ to a specific solution of the critical equation{H(x,Du,0)≤c(H),x∈Ω⊂Rn,H(x,Du,0)≥c(H),x∈Ω‾⊂Rn. We also discuss the generalization of such a convergence for equations with more general Cλ and Ωλ.

Weak KAM Approach to First-Order Mean Field Games with State Constraints

We study the asymptotic behavior of solutions to the constrained MFG system as the time horizon T goes to infinity. For this purpose, we analyze first Hamilton–Jacobi equations with state constraints from the viewpoint of weak KAM theory, constructing a Mather measure for the associated variational problem. Using these results, we show that a solution to the constrained ergodic mean field games system exists and the ergodic constant is unique. Finally, we prove that any solution of the first-order constrained MFG problem on [0, T] converges to the solution of the ergodic system as T goes to infinity.

Ergodic behavior of control and mean field games problems depending on acceleration

The goal of this paper is to study the long time behavior of solutions of the first-order mean field game (MFG) systems with a control on the acceleration. The main issue for this is the lack of small time controllability of the problem, which prevents to define the associated ergodic mean field game problem in the standard way. To overcome this issue, we first study the long-time average of optimal control problems with control on the acceleration: we prove that the time average of the value function converges to an ergodic constant and represent this ergodic constant as a minimum of a Lagrangian over a suitable class of closed probability measure. This characterization leads us to define the ergodic MFG problem as a fixed-point problem on the set of closed probability measures. Then we also show that this MFG ergodic problem has at least one solution, that the associated ergodic constant is unique under the standard monotonicity assumption and that the time-average of the value function of the time-dependent MFG problem with control of acceleration converges to this ergodic constant.

Weak KAM theory for potential MFG

We develop the counterpart of weak KAM theory for potential mean field games. This allows to describe the long time behavior of time-dependent potential mean field game systems. Our main result is the existence of a limit, as time tends to infinity, of the value function of an optimal control problem stated in the space of measures. In addition, we show a mean field limit for the ergodic constant associated with the corresponding Hamilton-Jacobi equation.

A convergent hierarchy of non-linear eigenproblems to compute the joint spectral radius of nonnegative matrices

We show that the joint spectral radius of a finite collection of nonnegative matrices can be bounded by the eigenvalue of a non-linear operator. This eigenvalue coincides with the ergodic constant of a risk-sensitive control problem, or of an entropy game, in which the state space consists of all switching sequences of a given length. We show that, by increasing this length, we arrive at a convergent approximation scheme to compute the joint spectral radius. The complexity of this method is exponential in the length of the switching sequences, but it is quite insensitive to the size of the matrices, allowing us to solve very large scale instances (several matrices in dimensions of order 1000 within a minute). An idea of this method is to replace a hierarchy of optimization problems, introduced by Ahmadi, Jungers, Parrilo and Roozbehani, by a hierarchy of nonlinear eigenproblems. To solve the latter eigenproblems, we introduce a projective version of Krasnoselskii-Mann iteration. This method is of independent interest as it applies more generally to the nonlinear eigenproblem for a monotone positively homogeneous map. Here, this method allows for scalability by avoiding the recourse to linear or semidefinite programming techniques.

On unbounded solutions of ergodic problems for non-local Hamilton–Jacobi equations

We study an ergodic problem associated to a non-local Hamilton–Jacobi equation defined on the whole space λ−L[u](x)+|Du(x)|m=f(x) and determine whether (unbounded) solutions exist or not. We prove that there is a threshold growth of the function f, that separates existence and non-existence of solutions, a phenomenon that does not appear in the local version of the problem. Moreover, we show that there exists a critical ergodic constant, λ∗, such that the ergodic problem has solutions for λ⩽λ∗ and such that the only solution bounded from below, which is unique up to an additive constant, is the one associated to λ∗.

Large time behavior of unbounded solutions of first-order Hamilton–Jacobi equations in R N

We study the large time behavior of solutions of first-order convex Hamilton-Jacobi Equations of Eikonal type $u_t+H(x,Du)=l(x),$ set in the whole space $\R^N\times [0,\infty).$ We assume that $l$ is bounded from below but may have arbitrary growth and therefore the solutions may also have arbitrary growth. A complete study of the structure of solutions of the ergodic problem $H(x,Dv)=l(x)+c$ is provided : contrarily to the periodic setting, the ergodic constant is not anymore unique, leading to different large time behavior for the solutions. We establish the ergodic behavior of the solutions of the Cauchy problem (i) when starting with a bounded from below initial condition and (ii) for some particular unbounded from below initial condition, two cases for which we have different ergodic constants which play a role. When the solution is not bounded from below, an example showing that the convergence may fail in general is provided.

Perturbation problems in homogenization of Hamilton–Jacobi equations

This paper is concerned with the behavior of the ergodic constant associated with convex and superlinear Hamilton–Jacobi equation in a periodic environment which is perturbed either by medium with increasing period or by a random Bernoulli perturbation with small parameter. We find a first order Taylor's expansion for the ergodic constant which depends on the dimension d. When d=1 the first order term is non trivial, while for all d≥2 it is always 0. Although such questions have been looked at in the context of linear uniformly elliptic homogenization, our results are the first of this kind in nonlinear settings. Our arguments, which rely on viscosity solutions and the weak KAM theory, also raise several new and challenging questions.

Long Time Average of First Order Mean Field Games and Weak KAM Theory

We show that the long time average of solutions of first order mean field game systems in finite horizon is governed by an ergodic system of mean field game type. The well-posedness of this later system and the uniqueness of the ergodic constant rely on weak KAM theory.

Isoperimetric inequalities for an ergodic stochastic control problem

In this paper we deal with blow-up solutions to an elliptic equation with a nonlinear gradient term. The problem under consideration can be seen as the ergodic limit of a stochastic control problem with state constraints. It is well known that it has a solution only when a parameter which appears in the equation assumes a particular value known as ergodic constant. For such a constant many properties similar to those of an eigenvalue hold true. We show that a Faber–Krahn inequality can be stated for the ergodic constant and that for the corresponding solution a comparison result in terms of the solution to a symmetrized problem can be proved.

Continuous dependence estimates for the ergodic problem of Bellman-Isaacs operators via the parabolic Cauchy problem

This paper concerns continuous dependence estimates for Hamilton-Jacobi-Bellman-Isaacs operators. We establish such an estimate for the parabolic Cauchy problem in the whole space (0,+∞)× R n and, under some periodicity and either ellipticity or controllability assumptions, we deduce a similar estimate for the ergodic constant associated to the operator. An interesting byproduct of the latter result will be the local uniform convergence for some classes of singular perturbation problems. β∈B max α∈A {− tr (a(x, α, β)X )+ f (x, α, β) · p + � (x, α, β)} (1.2) where ∂t ≡ ∂/∂t, Du and D 2 u stand respectively for the gradient and for the Hessian matrix of the real-valued function u = u(t, x). For instance, problems of this kind naturally arise in zero-sum two-persons stochastic differential games: consider the control system for s> 0 dxs = f (xs ,α s ,β s)ds + √ 2σ(xs ,α s ,β s)dWs ,x 0 = x (1.3)

Ergodic BSDEs and related PDEs with Neumann boundary conditions

We study a new class of ergodic backward stochastic differential equations (EBSDEs for short) which is linked with semi-linear Neumann type boundary value problems related to ergodic phenomena. The particularity of these problems is that the ergodic constant appears in Neumann boundary conditions. We study the existence and uniqueness of solutions to EBSDEs and the link with partial differential equations. Then we apply these results to optimal ergodic control problems.

Ergodic Type Problems and Large Time Behaviour of Unbounded Solutions of Hamilton–Jacobi Equations

We study the large time behavior of Lipschitz continuous, possibly unbounded, viscosity solutions of Hamilton–Jacobi Equations in the whole space ℝ N . The associated ergodic problem has Lipschitz continuous solutions if the analogue of the ergodic constant is larger than a minimal value λmin. We obtain various large-time convergence and Liouville type theorems, some of them being of completely new type. We also provide examples showing that, in this unbounded framework, the ergodic behavior may fail, and that the asymptotic behavior may also be unstable with respect to the initial data.

On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear Neumann boundary conditions

We study nonlinear Neumann type boundary value problems related to ergodic phenomenas. The particularity of these problems is that the ergodic constant appears in the (possibly nonlinear) Neumann boundary conditions. We provide, for bounded domains, several results on the existence, uniqueness and properties of this ergodic constant.

On the problem of the ergodic constant in the theory of classical Green's functions

On the problem of the ergodic constant in the theory of classical Green's functions