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Toshio Mikami

Publications and source records attributed to Toshio Mikami.

9 recordsLinked to original sources
arXiv2025

Stochastic optimal transport for the Langevin dynamics and its zero--mass limit

We introduce a stochastic optimal transport for the Langevin dynamics with positive mass and study its zero--mass limit. The new aspect of this paper is that we only fix the initial and terminal probability distributions of the positions of particles under consideration, but not those of their velocities with Heisenberg's uncertainty principle in mind. In the zero--mass limit, we show that the minimizer of our stochastic optimal transport is tight if and only if the initial momentum of a particle converges to zero. We also show that the limit of a minimizer of our stochastic optimal transport is a minimizer of a standard stochastic optimal transport for continuous semimartingales.

math.PR↗
arXiv2025

A system of Schr\"odinger's problems and functional equations

We propose and study a system of Schr\"odinger's problems and functional equations in probability theory. More precisely, we consider a system of variational problems of relative entropies for probability measures on a Euclidean space with given two endpoint marginals, which can be defined inductively. We also consider an inductively defined system of functional equations, which are Euler's equations for our variational problems. These are generalizations of Schr\"odinger's problem and functional equation. % in probability theory. We prove the existence and uniqueness of solutions to our functional equations, % up to a multiplicative function, from which we show the existence and uniqueness of a minimizer of our variational problem. Our problem gives an approach for a stochastic optimal transport analog of the Knothe--Rosenblatt rearrangement via a variational problem point of view.

math.PR↗
arXiv2023

Stochastic optimal transport with at most quadratic growth cost

We consider a class of stochastic optimal transport, SOT for short, with given two endpoint marginals in the case where a cost function exhibits at most quadratic growth. We first study the upper and lower estimates, the short--time asymptotics, the zero--noise limits, and the explosion rate as time goes to infinity of SOT. We also show that the value function of SOT is equal to zero or infinity in the case where a cost function exhibits less than linear growth. As a by--product, we characterize the finiteness of the value function of SOT by that of the Monge--Kantorovich problem with the same two endpoint marginals. As an application, we show the existence of a continuous semimartingale, with given initial and terminal distributions, of which the drift vector is $r$th integrable for $r\in [1,2)$. We also consider the same problem for Schrödinger's problem where $r=2$. This paper is a continuation of our previous work.

math.PR↗
arXiv2022

A remark on the Lagrangian formulation of optimal transport with a non-convex cost

We study the Lagrangian formulation of a class of the Monge-Kantorovich optimal transportation problem. It can be considered a stochastic optimal transportation problem for absolutely continuous stochastic processes. A cost function and stochastic processes under consideration is not convex and have essentially bounded time derivatives almost surely, respectively. This paper is a continuation of the second author's master thesis.

math.OC↗
arXiv2021

A Hamilton-Jacobi PDE associated with hydrodynamic fluctuations from a nonlinear diffusion equation

We study a class of Hamilton-Jacobi partial differential equations in the space of probability measures. In the first part of this paper, we prove comparison principles (implying uniqueness) for this class. In the second part, we establish the existence of a solution and give a representation using a family of partial differential equations with control. A large part of our analysis exploits special structures of the Hamiltonian, which might look mysterious at first sight. However, we show that this Hamiltonian structure arises naturally as limit of Hamiltonians of microscopical models. Indeed, in the third part of this paper, we informally derive the Hamiltonian studied before, in a context of fluctuation theory on the hydrodynamic scale. The analysis is carried out for a specific model of stochastic interacting particles in gas kinetics, namely a version of the Carleman model. We use a two-scale averaging method on Hamiltonians defined in the space of probability measures to derive the limiting Hamiltonian.

math.AP↗
arXiv2020

Stochastic optimal transport revisited

We prove the Duality Theorems for the stochastic optimal transportation problems with a convex cost function without a regularity assumption that is often supposed in the proof of the lower semicontinuity of an action integral. In our new approach, we prove that the stochastic optimal transportation problems with a convex cost function are equivalent to a class of variational problems for the Fokker-Planck equation, which lets us revisit them. It is done by the so-called superposition principle and by an idea from the mather theory. The superposition principle is the construction of a semimartingale from the Fokker-Planck equation and can be considered a class of the so-called marginal problems that construct stochastic processes from given marginal distributions. It was first considered in stochastic mechanics by E. Nelson, called Nelson's problem, and was proved by E. Carlen first. The semimartingale is called the Nelson process, provided it is Markovian. We also consider the Markov property of a minimizer of the stochastic optimal transportation problem with a nonconvex cost in a one-dimensional case. In the proof, the superposition principle and the minimizer of an optimal transportation problem with a concave cost function play crucial roles. Lastly, we prove the semiconcavity and the Lipschitz continuity of Schrodinger's problem that is a typical example of the stochastic optimal transportation problem.

math.PR↗
arXiv2020

Regularity of Schrödinger's functional equation in the weak topology and moment measures

We study the continuity and the measurability of the solution to Schrödinger's functional equation, with respect to space, kernel and marginals, provided the space of all Borel probability measures is endowed with the weak topology. This is a continuation of our previous result where the space of all Borel probability measures was endowed with the strong topology. As an application, we construct a convex function of which the moment measure is a given probability measure, by the zero noise limit of a class of stochastic optimal transportation problems.

math.PR↗
arXiv2019

Regularity of Schroedinger's functional equation and mean field PDEs for h-path processes

E. Schroedinger proposed the equation to find the statistical property of a quantum particle on a finite time interval. It is called "Schroedinger's functional equation". Given probability distributions of a particle at initial and terminal times, it determines the joint distribution of a quantum particle at initial and terminal times so that a particle is Markovian. S. Bernstein generalized Schroedinger's idea and introduced the so-called Bernstein processes which are also called reciprocal processes or one-dimensional Markov random fields. The theory of stochastic differential equation for Schroedinger's functional equation was given by B. Jamison. The solution is Doob's h-path process with given two end point marginals. We show that the solution of Schroedinger's functional equation is measurable in space, kernel and marginals. As an application, we show that the drift vector of the h-path process with given two end point marginals is a measurable function of space, time and marginal at each time. In particular, we show that the marginals satisfy a class of mean field PDE systems of which the coefficients are measurable function of space, time and marginal. We also show that Schroedinger's functional equation is the Euler equation of a stochastic optimal transportation problem.

math.PR↗