/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Second‐Order Parabolic Differential Equations

In this paper we consider time-dependent mean-field games with subquadratic Hamiltonians and power-like local dependence on the measure. We establish existence of classical solutions under a certain set of conditions depending on both the growth of the Hamiltonian and the dimension. This is done by combining regularity estimates for the Hamilton-Jacobi equation based on the Gagliardo-Nirenberg interpolation inequality with polynomial estimates for the Fokker-Planck equation. This technique improves substantially the previous results on the regularity of time-dependent mean-field games.

Time-Dependent Mean-Field Games in the Subquadratic Case

Selection problems for a discount degenerate viscous Hamilton–Jacobi equation

Generalized ergodic problems: Existence and uniqueness structures of solutions

We prove that the solution of the discounted approximation of a degenerate viscous Hamilton--Jacobi equation with convex Hamiltonians converges to that of the associated ergodic problem. We characterize the limit in terms of stochastic Mather measures by naturally using the nonlinear adjoint method, and deriving a commutation lemma. This convergence result was first achieved by Davini, Fathi, Iturriaga, and Zavidovique for the first order Hamilton--Jacobi equation.

Selection problems for a discounted degenerate viscous Hamilton--Jacobi equation

https://cvgmt.sns.it/media/doc/paper/310/Numerics2012-02-20.pdf

Convergence of a semi-discretization scheme for the Hamilton-Jacobi equation: A new approach with the adjoint method

We develop a variational approach to the vanishing discount problem for fully nonlinear, degenerate elliptic, partial differential equations. Under mild assumptions, we introduce viscosity Mather measures for such partial differential equations, which are natural extensions of the Mather measures. Using the viscosity Mather measures, we prove that the whole family of solutions $v^\lambda$ of the discount problem with the factor $\lambda>0$ converges to a solution of the ergodic problem as $\lambda \to 0$.

The vanishing discount problem and viscosity Mather measures. Part 1: the problem on a torus

We study the qualitative homogenization of second-order Hamilton–Jacobi equations in space-time stationary ergodic random environments. Assuming that the Hamiltonian is convex and superquadratic in the momentum variable (gradient), we establish a homogenization result and characterize the effective Hamiltonian for arbitrary (possibly degenerate) elliptic diffusion matrices. The result extends previous work that required uniform ellipticity and space-time homogeneity for the diffusion.

/pdf/stochastic-homogenization-of-viscous-superquadratic-hamilton-3kd49md7rs.pdf

Hung V. Tran

Papers

Generalized ergodic problems: Existence and uniqueness structures of solutions

Selection problems for a discounted degenerate viscous Hamilton--Jacobi equation

Convergence of a semi-discretization scheme for the Hamilton-Jacobi equation: A new approach with the adjoint method

The vanishing discount problem and viscosity Mather measures. Part 1: the problem on a torus

Stochastic homogenization of viscous superquadratic Hamilton–Jacobi equations in dynamic random environment