Asymptoic theory of the spectral functions of one-dimensional second order differential operators and its applications to diffusion processes

2015 Fiscal Year Final Research Report

• Project/Area Number: 24540110
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: University of Tsukuba
• Principal Investigator: Kasahara Yuji 筑波大学, 数理物質系, 教授 (60108975)
• Co-Investigator(Kenkyū-buntansha): LIANG Song 筑波大学, 数理物質系, 准教授 (60324399)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: 拡散過程 / スペクトル関数 / 正則変動関数 / タウバー型定理 / 推移確率密度 / ２階微分作用素

Outline of Final Research Achievements

Stochastic processes are mathematical models which describe various quantities varying randomly. The diffusion process is a basic stochastic process that varies without jumps, and it has applications in various fields. A diffusion corresponds to a second-order differential operator (generator) and its spectral function plays an important role to describe various properties and quantities of the stochastic process.
In our research we clarified the relation between the asymptotic behavior of the spectral function and that of the drift-coefficient of the operator. We also obtained some Tauberian theorems which are tools to extend results on the half line to those of the full line.

Free Research Field

数学（確率論）