Building a Theory of Regular Structures for Non-Autonomous and Quasi-Linear Rough Evolution Equations, and Applying the Theory to Forest Kinematic Ecosystems

2019 Fiscal Year Research-status Report

• Project/Area Number: 19K14555
• 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): 2019-04-01 – 2023-03-31
• Keywords: Evolution equations / Strict solutions / Wiener process

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

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.

Causes of Carryover

I canceled some business trips due to coronavirus outbreak in this fiscal year. I would like to carry the amount to the next fiscal year. I will buy books, a PC, and make business trips that I could not do in the current fiscal year.