1999 Fiscal Year Final Research Report Summary
Brownian Motion and Nonlinear Boundary Value Problems
| Project/Area Number |
10440050
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Basic analysis
|
|---|
| Research Institution | University of Tsukuba |
|---|
Principal Investigator |
TAIRA Kazuaki University of Tsukuba, Institute of Mathematics, Professor, 数学系, 教授 (90016163)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
UMEZU Kenichiro Maebashi Institute of Technology, Lecturer, 工学部, 講師 (00295453)
KINOSHITA Tamotu University of Tsukuba, Institute of Mathematics, Assistant, 数学系, 助手 (90301077)
MINAMI Nariyuki University of Tsukuba, Institute of Mathematics, Associate Professor, 数学系, 助教授 (10183964)
SAKAMOTO Kunimochi Hiroshima University, Faculty of Science, Associate Professor, 理学部, 助教授 (40243547)
KUBO Izumi Hiroshima University, Faculty of Science, Professor, 理学部, 教授 (70022621)
|
|---|
| Project Period (FY) |
1998 – 1999
|
| Keywords | Brownian Motion / Boundary Value Problem / Feller Semigroup / Population Dynamics / Chemical Reactor Theory / Harmonic Integral |
|---|
| Research Abstract |
Our results may be summarized as follows :
1. First we studied from the viewpoint of functional analysis the problem of construction of Markov processes with boundary conditions in probability theory. Our approach is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. We constructed a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it dies at the time when it reaches the set where the particle is definitely absorbed.
2. Secondly we studied existence and uniqueness problems of positive solutions of diffusive logistic equations with indefinite weights which model population dynamics in environments with strong spatial heterogeneity. We proved that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) loca
… More
ted some distance away from the boundary of the environment. Moreover we discuss the stability properties for positive steady states.
3. Thirdly we studied semilinear elliptic boundary value problems arising in chemical reactor theory which obey the simple Arrhenius rate law and Newtonian cooling. We proved that ignition and extinction phenomena occur in the stable steady temperature profile at some critical values of a dimensionless heat evolution rate.
4. Finally we gived an analytic proof of an index formula for the relative de Rham cohomology groups which may be considered as a generalization of the celebrated Hodge--Kodaira theory for the absolute de Rham cohomology groups. In deriving our index formula, the theory of harmonic forms satisfying an interior boundary condition plays a fundamental role. Our approach has a great advantage of intuitive interpretation of the index formula in terms of Brownian motion from the point of view of probability theory. Our result may be stated as follows : Brownian motion describes the topology of a compact Riemannian manifold through its Euler--Poincare characteristic. Less
|
