Project/Area Number 
19K14555

Research Category 
GrantinAid for EarlyCareer Scientists

Allocation Type  Multiyear Fund 
Review Section 
Basic Section 12010:Basic analysisrelated

Research Institution  Kyushu University 
Principal Investigator 
タ・ビィエ トン 九州大学, 農学研究院, 准教授 (30771109)

Project Period (FY) 
20190401 – 20230331

Project Status 
Granted (Fiscal Year 2019)

Budget Amount *help 
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2022: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2021: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)

Keywords  Evolution equations / Strict solutions / Wiener process / Evolution Equations / Forest Ecosystems / Coat Patterns 
Outline of Research at the Start 
Many phenomena including noise can be described by rough evolution equations (REEs). Once these equations are solved, it will give new understanding on the phenomena.
This project will construct a new theory to solve a wide class of REEs. The theory will be then used to study the effect of noise on forest kinematic ecosystem and on the coat pattern of animals.

Outline of Annual Research Achievements 
We considered a semilinear evolution equation with additive noise of the form dX+AXdt=[F_1(t)+F_2(X)]dt+G(t)dW(t) in a Banach space. Here, we assume that the linear operator A is a sectorial operator generating an analytical semigroup. And, W is a cylindrical Wiener process. By using the semigroup approach and fixed point arguments, under some conditions on the coefficients F_1, F_2, we proved existence of strict solutions to the equation. In addtion, the regularity of the solutions is also obtained.

Current Status of Research Progress 
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
I do not use the Young integral approach but the semigroup approach. The latter approach is effective for the equation considered in the Summary of Research Achievements.

Strategy for Future Research Activity 
I will be in the plan stated in the original proposal. Now I consider a semilinear equation with multiple noise: dX+AXdt=[F_1(t)+F_2(X)]dt+G(t,X)dW(t).
I will try to use the Young integral approach as stated in the original proposal but also the semigroup approach. The final goal is to construct a solution to the equation and then show its regularity.
For the semigroup approach, the variable appearing in stochastic convolutions will be explained as as a multiplication operator. Roughly speaking, any element U in L2 space can be explained as a linear operator from L2 to itself by U(v)=Uv if the product between U and v is still an element of L2. In this way, we may obtain a meaningful stochastic convolution, and therefore a solution to the equation.
