Sobolev-Orlicz embedding theorems for Dirichlet spaces and their probabilistic interpretation

2023 Fiscal Year Final Research Report

• Project/Area Number: 21K20326
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: Kyoto Institute of Technology
• Principal Investigator: Mori Takahiro 京都工芸繊維大学, 基盤科学系, 助教 (80909911)
• Project Period (FY): 2021-08-30 – 2024-03-31
• Keywords: Dirichlet形式 / Sobolev不等式 / intersection measure

Outline of Final Research Achievements

Brownian motion (BM) is one of the most fundamental stochastic processes. One of its properties is the phenomenon of trajectory crossing. This is the property that a single BM has a point that is reached multiple times (self-intersection), or a point where all trajectories intersect (mutual intersection) when multiple independent BMs run.
In this study, in order to investigate the property of intersection, we consider BMs, which are probabilistic objects, and Dirichlet spaces, which are corresponding functional analytic objects. In particular, we attempted to clarify the property of infinite crossing for 2-dimensional BM from the theory of Dirichlet forms.

Free Research Field

確率論

Academic Significance and Societal Importance of the Research Achievements

ブラウン運動は花粉内の微粒子の運動のモデルとして提唱され, 物理学に留まらず株価の変動と言った経済学等にも応用されている. ブラウン運動の軌跡を一本の紐とみなすことでランダムポリマーへの応用が考えられ, ブラウン運動の軌跡の交差現象は高分子の屈曲性に相当する. 本研究はその数学的基礎づけを行っているという点で社会的意義がある.