Shigeki Aida

On the Small Time Asymptotics of Diffusion Processes on Path Groups

In this paper, we obtain the Varadhan type small time asymptotics for diffusion processes on path groups.

Short Time Asymptotics of a Certain Infinite Dimensional Diffusion Process

The main objective of this contribution is to prove the Varadhan type short-time asymptotics of an infinite dimensional diffusion process associated with a certain Dirichlet form. This paper gives a generalization of Fang’s results of the Ornstein-Uhlenbeck process on an abstract Wiener space.

Equivalence of heat kernel measure and pinned Wiener measure on loop groups

Abstract We show that heat kernel measure and pinned Wiener measure on loop groups over simply connected compact Lie groups are equivalent.

Vanishing of one-dimensional L2-cohomologies of loop groups☆

Abstract Let G be a simply connected compact Lie group. Let L e ( G ) be the based loop group with the base point e which is the identity element. Let ν e be the pinned Brownian motion measure on L e ( G ) and let α ∈ L 2 ( ⋀ 1 T ⁎ L e ( G ) , ν e ) ∩ D ∞ , p ( ⋀ 1 T ⁎ L e ( G ) , ν e ) ( 1 p 2 ) be a closed 1-form on L e ( G ) . Using results in rough path analysis, we prove that there exists a measurable function f on L e ( G ) such that d f = α . Moreover we prove that dim ker □ = 0 for the Hodge–Kodaira type operator □ acting on 1-forms on L e ( G ) .

Semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space

Abstract We study a semiclassical limit of the lowest eigenvalue of a Schrodinger operator on a Wiener space. Key results are semiboundedness theorem of the Schrodinger operator, Laplace-type asymptotic formula and IMS localization formula. We also make a remark on the semiclassical problem of a Schrodinger operator on a path space over a Riemannian manifold.

Weak Poincaré inequalities on domains defined by Brownian rough paths

We prove weak Poincare inequalities on domains which are inverse images of open sets in Wiener spaces under continuous functions of Brownian rough paths. The result is applicable to Dirichlet forms on loop groups and connected open subsets of path spaces over compact Riemannian manifolds.

Precise Gaussian estimates of heat kernels on asymptotically flat Riemannian manifolds with poles

We give precise Gaussian upper and lower bound estimates on heat kernels on Riemannian manifolds with poles under assumptions that the Riemannian curvature tensor goes to 0 sufficiently fast at infinity. Under additional assumptions on the curvature, we give estimates on the logarithmic derivatives of the heat kernels. The proof relies on the Elworthy-Truman’s formula of heat kernels and Elworthy and Yor’s observation on the derivative process of certain stochastic flows. As an application of them, we prove logarithmic Sobolev inequalities on pinned path spaces over such Riemannian manifolds.

WITTEN LAPLACIAN ON PINNED PATH GROUP AND ITS EXPECTED SEMICLASSICAL BEHAVIOR

A right invariant Riemannian metric is defined on a pinned path group over a compact Lie group G. The energy function of the path is a Morse function and the critical points are geodesics. We calculate the eigenvalues of the Hessian at the critical points when G=SU(n). On the other hand, there exists a pinned Brownian motion measure νλ with a variance parameter 1/λ on the pinned path group and we can define a Hodge-Kodaira-Witten type operator □λ on L2(νλ)-space of p-forms on the pinned path group. By using the explicit expression of eigenvalues of the Hessian of the energy function, we discuss the asymptotic behavior of the botton of the spectrum of □λ as λ→∞ by a formal semiclassical analysis.

D∞-cohomology groups and D∞-maps on submanifolds in Wiener spaces

Abstract We introduce the notion of a D ∞ -map between two submanifolds in a Wiener space and their pull-back of differential forms in the framework of Malliavin calculus. Also we show the vanishing of the cohomology of certain quadratic hypersurfaces in a Wiener space.

Precise Gaussian Lower Bounds on Heat Kernels

Let g be a Riemannian metric on a Euclidean space. The Levi-Civita Laplace-Beltrami operator Δ generates a diffusion semi-group. We denote the heat kernel by p(t,x, y). The following estimate is called a Gaussian bound on the heat kernel: there exist 0 0 and C1, C2 > 0 such that for any t > 0, here d(x, y) denotes the Riemannian distance between x and y. In this article, we will study more precise estimates on the lower bound for a fixed x as follows: there exists a positive constant C such that for any 0t 0, except for Li-Yau’s result in [7] which asserts that (1.2) holds with under the assumption that Ric > 0. To my knowledge, there seems to be no other criteria.