主講人：Fukushima
講座題目：BMD applied to Komatu-Loewner and Bauer-Fridrich equations for planer slit domains
時　　間：2012年7月26日下午1-2
地　　點：研究生樓209A
主講人介紹
　　Masatoshi Fukushima received the B.S. degree in mathematics in1959 and M.S. degree in Mathemtics in 1961, both from Kyoto University, Japan. He received his Ph.D. degree in　Mathematics from the OsakaUniversity, Japan in 1967 under the direction of K. Ito. He has held positions at Nagoya University, Kyoto University, Tokyo University of Education, Osaka University and Kansai University. He is now a Professor Emeritus from Osaka University.
　　Masatoshi Fukushima is a world renowned mathematician, His fundamental work together with that of Martin L. Silverstein on Dirichlet forms and Markov processes established a profound connection between probabilistic and analytic potential theory. It provides an effective tool to study objects in both fields,especially in infinite dimensional spaces and in spaces with non-smooth structure. He has authored or co-authored more than 100 research papers and 3 books. His book on Dirichlet forms and Symmetric Markov Processes is one of the most cited books in the field.He is an invited speaker at the International Congress of Mathematicians held at Helsinki in 1978. He received 2003 Analysis Prize of the Mathematical Society of Japan. He is an invited Lecturer of the London Mathematical Society in 2006.
主要內容：
　　BMD is an abbreviation of Brownian motion with darning, which behaves like a reflecting Brownian motion on a planar regoin with many holes but by regarding each hole as a one-point set. BMD is quite useful in the study of conformal mappings from multiply connected domains onto standard slit domains and a related differential equation which was first derived by Y. Komatu in 1950 and then by R.O. Bauer and R.M. Friedrich in 2008, extending the well-known Loewner differential equation for simply connected domains. However the Komatu-Loewner differential equation has been established only in the sense of the left derivative.
　　It can be shown that the kernel appearing in the Komatu-Loewner equation is just a complex Poisson kernel of BMD and the stated conformal maps are expressed probabilistically in terms of BMD. Combining these properties with a PDE method of variations of the classical Green function, we can show that the Komatu-Loewner equation is a genuine differential equation. We can further derive a differential equation for the induced motion of the slits first observed by Bauer-Friedrich and thereby recover the original family of conformal maps from a given motion on the boundary.
　　This is a joint work with Zhen-Qing Chen and Steffen Rohde.