Satoshi Takanobu

Estimate of the difference between the Kac operator and the Schrödinger semigroup

An operator norm estimate of the difference between the Kac operator and the Schrödinger semigroup is proved and used to give a variant of the Trotter product formula for Schrödinger operators in theLp operator norm. This extends Helffer’s result in theL2 operator norm to the case in theLp operator norm for more general scalar potentials and with vector potentials. The method of the proof is probabilistic based on the Feynman—Kac and Feynman—Kac—Itô formula.

Random Weyl sampling for robust numerical integration of complicated functions

— We introduce the random Weyl sampling. It is a sampling method that uses a sequence {x + na}^=Q on fc-dimensional torus T* = [0, l) where the pair of parameters (z, α) is considered to be random governed by the uniform measure on T*. It turns out that the random Weyl sampling has almost the same probabilistic error estimate for all square-integrable functions s the i.i.d.-sampling does. A remarkable point is that the random Weyl sampling requires much fewer pseudo-random numbers than the i.i.d.sampling does, and hence it is much less sensitive to their quality. We characterize the desirable pair (χ, α) for the purpose in terms of computational complexity.