Naomasa Ueki

Classical and Quantum Behavior of the Integrated Density of States for a Randomly Perturbed Lattice

The asymptotic behavior of the integrated density of states for a randomly perturbed lattice at the infimum of the spectrum is investigated. The leading term is determined when the decay of the single site potential is slow. The leading term depends only on the classical effect from the scalar potential. To the contrary, the quantum effect appears when the decay of the single site potential is fast. The corresponding leading term is estimated and the leading order is determined. In the multidimensional cases, the leading order varies in different ways from the known results in the Poisson case. The same problem is considered for the negative potential. These estimates are applied to investigate the long time asymptotics of Wiener integrals associated with the random potentials.

Moment asymptotics for the parabolic Anderson problem with a perturbed lattice potential

Abstract The parabolic Anderson problem with a random potential obtained by attaching a long tailed potential around a randomly perturbed lattice is studied. The moment asymptotics of the total mass of the solution is derived. The results show that the total mass of the solution concentrates on a small set in the space of configuration.

A probabilistic proof of the Gauss-Bonnet-Chern theorem for manifolds with boundary

A probabilistic method to solve a heat equation on differential forms on a closed compact Riemannian manifold is now well-known. In the case of a Riemannian manifold with boundary, Conner investigated the initial value problems for differential forms under absolute or relative boundary conditions. Further Gilkey proved the Gauss-Bonnet-Chern theorem for a manifold with boundary. We will prove it by a probabilistic method. To construct the fundamental solution, we adopt a probabilistic approach to the initial value problem. Combining this result with a modified Malliavin calculus, we can express the fundamental solution for the initial value problem as a generalized Wiener functional expectation, which serves to compute directly our result

A stochastic approach to the Poincaré-Hopf theorem

SummaryIn this paper, the Poincaré-Hopf theorem (Hopf [8]) is proved by a probabilistic method. The proof of this theorem is reduced to an estimate of a supertrace of the fundamental solution. We use the Malliavian calculus to estimate the fundametal solution. Our proof can be applied to the case that the zero point set is the union of the submanifolds under some conditions.

Asymptotic expansion of stochastic oscillatory integrals with rotation invariance

Abstract Asymptotics of oscillatory integrals on the classical 2-dimensional Wiener space, whose phase functional is the stochastic line integral of a 1-form, is considered. Under the assumption that the exterior derivative of the 1-form is rotation invariant, an asymptotic expansion is obtained. This result is extended to the process associated to a general rotation invariant metric.

Wegner estimates, Lifshitz tails, and Anderson localization for Gaussian random magnetic fields

The Wegner estimate for the Hamiltonian of the Anderson model for the special Gaussian random magnetic field is extended to more general magnetic fields. The Lifshitz tail upper bounds of the integrated density of states as analyzed by Nakamura are reviewed and extended so that Gaussian random magnetic fields can be treated. By these and multiscale analysis, the Anderson localization at low energies is proven.