Toshio Mikami

Optimal Transportation Problem by Stochastic Optimal Control

We solve optimal transportation problem using stochastic optimal control theory. Indeed, for a super linear cost at most quadratic at infinity, we prove Kantorovich duality theorem by a zero noise limit (or vanishing viscosity) argument. We also obtain a characterization of the support of an optimal measure in Monge-Kantorovich minimization problem (MKP) as a graph. Our key tool is a duality result for a stochastic control problem which naturally extends (MKP).

Variational processes from the weak forward equation

In this paper the author constructs Markov diffusion processes from a given system of Borel probability measures on ad-dimensional Euclidean space. He constructs a, so-called, variational process which does not always coincide with a Nelson process. He also discusses Schrödingers problem in quantum mechanics.

Convexified Gauss curvature flow of sets: A stochastic approximation

We construct a discrete stochastic approximation of a convexified Gauss curvature flow of boundaries of bounded open sets in an anisotropic external field. We also show that a weak solution to the PDE which describes the motion of a bounded open set is unique and is a viscosity solution of it.

Equivalent conditions on the central limit theorem for a sequence of probability measures on R

In this paper we give equivalent conditions on the central limit theorem in total variation norm for a sequence of probability measures on R. This generalizes Cacoullos, Papathanasiou and Utevs central limit theorem in L1-norm for a sequence of probability density functions on R. We also give equivalent conditions on the central limit theorem in weak convergence and those on the local limit theorem.

TWO END POINTS MARGINAL PROBLEM BY STOCHASTIC OPTIMAL TRANSPORTATION

We give a sufficient condition under which the stochastic optimal transportation problem is finite, which implies the existence of a semimartingale with given initial and terminal distributions. The idea of the proof is to show the finiteness of the supremum in the duality theorem for the stochastic optimal transportation problem. As a special case, it also gives a new approach for the construction of the h-path process with given initial and terminal distributions. We also consider a problem similar to the above for a class of optimal control problems for a family of solutions to Fokker--Planck equations.

A Characterization of the Knothe--Rosenblatt Processes by a Convergence Result

We introduce a stochastic analogue of the Knothe--Rosenblatt type rearrangements, which we call the Knothe--Rosenblatt processes and which can be considered as a generalization of the h-path processes. We also give a characterization by the convergence result on a class of stochastic control problems, which might be useful for the numerical analysis of the process.

Large deviations and central limit theorems for Eyraud-Farlie-Gumbel-Morgenstern processes

Let {Xn}n = 1[infinity] be a Eyraud-Farlie-Gumbel-Morgenstern process. Put Sn[reverse not equivalent][summation operator]k=1nXk. In this paper we prove the large deviations theorem for Sn/n, and the central limit theorem for Sn/n1/2, as n --> [infinity].

Local action functionals for randomly perturbed dynamical systems on long time intervals

We prove exponential decays of probabilities of randomly perturbed dynamical systems in a d–dimensional Euclidean space Rd on time intervals which go to [0,∞] as the random fluctuation disappears. We also consider the exit problems when unperturbed dynamical systems are attracted to the inside of the domain under consideration

Limit theorems on the exist problems for small random perturbations of dynamical systems I

We consider small random perturbations of dynamical systems on a d-dimensional Euclidean space R d when the origin oϵR d is a hyperbolic equilibrium point of unperturbed dynamical systems. The objects under consideration are empirical measures which are marginal measures of empirical processes at the exit time from a bounded domain and .