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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

In arXiv:1603.01051 (Part 1 of this series), we have introduced a variational approach to studying the vanishing discount problem for fully nonlinear, degenerate elliptic, partial differential equations in a torus. We develop this approach further here to handle boundary value problems. In particular, we establish new representation formulas for solutions of discount problems, critical values, and use them to prove convergence results for the vanishing discount problems.

The vanishing discount problem and viscosity Mather measures. Part 2: boundary value problems

We present stochastic homogenization results for viscous Hamilton-Jacobi equations using a new argument which is based only on the subadditive structure of maximal subsolutions (solutions of the "metric problem") This permits us to give qualitative homogenization results under very general hypotheses: in particular, we treat non-uniformly coercive Hamiltonians which satisfy instead a weaker averaging condition As an application, we derive a general quenched large deviations principle for diffusions in random environments and with absorbing random potentials

Stochastic homogenization of viscous Hamilton-Jacobi equations and applications

The adjoint method, introduced in [L. C. Evans, Arch. Ration. Mech. Anal., 197 (2010), pp. 1053–1088] and [H. V. Tran, Calc. Var. Partial Differential Equations, 41 (2011), pp. 301–319], is used to construct analogues to the Aubry–Mather measures for nonconvex Hamiltonians. More precisely, a general construction of probability measures, which in the convex setting agree with Mather measures, is provided. These measures may fail to be invariant under the Hamiltonian flow and a dissipation arises, which is described by a positive semidefinite matrix of Borel measures. However, in the case of uniformly quasiconvex Hamiltonians the dissipation vanishes, and as a consequence the invariance is guaranteed.

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Aubry-mather measures in the nonconvex setting ∗

We investigate the large-time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton--Jacobi equations on the $n$-dimensional torus. We establish a convergence result to asymptotic solutions as time goes to infinity under rather restricted assumptions.

Remarks on the large time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton--Jacobi equations

We perform a qualitative analysis of a class of weakly coupled Hamilton--Jacobi systems in the spirit of weak KAM theory. We define a family of related action functionals containing the Lagrangians associated with the Hamiltonians of the system. We use them to characterize the subsolutions of the system and to provide explicit representation formulae for subsolutions enjoying an additional maximality property. A crucial step for our analysis is to put the problem in a suitable random frame. The presentation is accessible to readers without a background in probability; only some basic knowledge of measure theory is required.

Hung V. Tran

Papers

The vanishing discount problem and viscosity Mather measures. Part 2: boundary value problems

Stochastic homogenization of viscous Hamilton-Jacobi equations and applications

Aubry-mather measures in the nonconvex setting ∗

Remarks on the large time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton--Jacobi equations

A Lagrangian Approach to Weakly Coupled Hamilton--Jacobi Systems