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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

I present two recent research directions in this dissertation. The first direction is on the study of the Nonlinear Adjoint Method for Hamilton-Jacobi equations, which was introduced recently by Evans [27]. The main feature of this new method consists of the introduction of an additional equation to derive new information about the solutions of the regularized Hamilton-Jacobi equations. More specifically, we linearize the regularizedHamilton-Jacobi equations first and then introduce the corresponding adjoint equations. Looking at the behavior of the solutions of the adjoint equations and using integration byparts techniques, we can prove new estimates, which could not be obtained by previous techniques.We use the Nonlinear Adjoint Method to study the eikonal-like Hamilton-Jacobi equations [79], and Aubry-Mather theory in the non convex settings [10]. The latter one is based on joint work with Filippo Cagnetti and Diogo Gomes. We are able to relax theconvexity conditions of the Hamiltonians in both situations and achieve some new results.The second direction, based on joint work with Hiroyoshi Mitake [69, 68], concerns the study of the properties of viscosity solutions of weakly coupled systems of Hamilton-Jacobi equations. In particular, we are interested in cell problems, large time behavior, and homogenization results of the solutions, which have not been studied much in the literature.We obtain homogenization results for weakly coupled systems of Hamilton-Jacobi equations with fast switching rates and analyze rigorously the initial layers appearing naturally in the systems. Moreover, some properties of the effective Hamiltonian are also derived.Finally, we study the large time behavior of the solutions of weakly coupled systems of Hamilton-Jacobi equations and prove the results under some specific conditions. The general case is still open up to now.

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Some new methods for Hamilton-Jacobi type nonlinear partial differential equations

We are interested in the averaged behavior of interfaces moving in stationary ergodic environments, with oscillatory normal velocity which changes sign. This problem can be reformulated, using level sets, as the homogenization of a Hamilton-Jacobi equation with a positively homogeneous non-coercive Hamiltonian. The periodic setting was earlier studied by Cardaliaguet, Lions and Souganidis (2009). Here we concentrate in the random media and show that the solutions of the oscillatory Hamilton-Jacobi equation converge in $L^\infty$-weak $*$ to a linear combination of the initial datum and the solutions of several initial value problems with deterministic effective Hamiltonian(s), determined by the properties of the random media.

Stochastic homogenization of interfaces moving with changing sign velocity

We study a critical case of Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel. Our method is based on the study of viscosity solutions to a new singular Hamilton-Jacobi equation, which results from applying the Bernstein transform to the original Coagulation-Fragmentation equation. Our results include wellposedness, regularity and long-time behaviors of viscosity solutions to the Hamilton-Jacobi equation in certain regimes, which have implications to wellposedness and long-time behaviors of \emph{mass-conserving} solutions to the Coagulation-Fragmentation equation.

Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel.

We study and characterize the optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form We obtain that the optimal rate of convergence is either $$O(\varepsilon )$$
 or $$O(\varepsilon ^2)$$
 depending on the diffusion matrix A, source term f, and boundary data g Moreover, we show that the set of diffusion matrices A that give optimal rate $$O(\varepsilon )$$
 is open and dense in the set of $$C^2$$
 periodic, symmetric, and positive definite matrices, which means that generically, the optimal rate is $$O(\varepsilon )$$
 

https://link.springer.com/content/pdf/10.1007/s42985-020-00017-z.pdf

Remarks on optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form

. This paper is concerned with the convergence rate of policy iteration for (determin-istic) optimal control problems in continuous time. To overcome the problem of ill-posedness due to lack of regularity, we consider a semi-discrete scheme by adding a viscosity term via ﬁnite diﬀerences in space. We prove that PI for the semi-discrete scheme converges exponentially fast, and provide a bound on the error induced by the semi-discrete scheme. We also consider the discrete space-time scheme, where both space and time are discretized. Convergence rate of PI and the discretization error are studied.

Hung V. Tran

Papers

Some new methods for Hamilton-Jacobi type nonlinear partial differential equations

Stochastic homogenization of interfaces moving with changing sign velocity

Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel.

Remarks on optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form

Policy iteration for the deterministic control problems - a viscosity approach