Conservation laws arising in the study of forward-forward Mean-Field Games

We consider forward-forward Mean Field Game (MFG) models that arise in numerical approximations of stationary MFGs. First, we establish a link between these models and a class of hyperbolic conservation laws as well as certain nonlinear wave equations. Second, we investigate existence and long-time behavior of solutions for such models

The Hessian Riemannian flow and Newton's method for Effective Hamiltonians and Mather measures

Effective Hamiltonians arise in several problems, including homogenization of Hamilton--Jacobi equations, nonlinear control systems, Hamiltonian dynamics, and Aubry--Mather theory. In Aubry--Mather theory, related objects, Mather measures, are also of great importance. Here, we combine ideas from mean-field games with the Hessian Riemannian flow to compute effective Hamiltonians and Mather measures simultaneously. We prove the convergence of the Hessian Riemannian flow in the continuous setting. For the discrete case, we give both the existence and the convergence of the Hessian Riemannian flow. In addition, we explore a variant of Newton's method that greatly improves the performance of the Hessian Riemannian flow. In our numerical experiments, we see that our algorithms preserve the non-negativity of Mather measures and are more stable than {related} methods in problems that are close to singular. Furthermore, our method also provides a way to approximate stationary MFGs.Comment: 24 page

Exponential decay of correlation for the Stochastic Process associated to the Entropy Penalized Method

In this paper we present an upper bound for the decay of correlation for the stationary stochastic process associated with the Entropy Penalized Method. Let L(x, v):\Tt^n\times\Rr^n\to \Rr be a Lagrangian of the form L(x,v) = {1/2}|v|^2 - U(x) + . For each value of $\epsilon$ and $h$, consider the operator \Gg[\phi](x):= -\epsilon h {ln}[\int_{\re^N} e ^{-\frac{hL(x,v)+\phi(x+hv)}{\epsilon h}}dv], as well as the reversed operator \bar \Gg[\phi](x):= -\epsilon h {ln}[\int_{\re^N} e^{-\frac{hL(x+hv,-v)+\phi(x+hv)}{\epsilon h}}dv], both acting on continuous functions \phi:\Tt^n\to \Rr. Denote by $\phi_{\epsilon,h}$ the solution of \Gg[\phi_{\epsilon,h}]=\phi_{\epsilon,h}+\lambda_{\epsilon,h}, and by $\bar \phi_{\epsilon,h}$ the solution of \bar \Gg[\phi_{\epsilon,h}]=\bar \phi_{\epsilon,h}+\lambda_{\epsilon,h}. In order to analyze the decay of correlation for this process we show that the operator ${\cal L} (\phi) (x) = \int e^{- \frac{h L (x,v)}{\epsilon}} \phi(x+h v) d v,$ has a maximal eigenvalue isolated from the rest of the spectrum

Chitosan-Hyaluronic acid hybrid vectors for retinal gene therapy

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