- Date: August 17, 2017, 10:00 - 16:50
- Venue: Room B301, Faculty of Science Building B, Kobe University（神戸大学理学部B棟3階B301号室）

Transportation and maps (English) 交通案内と周辺地図（日本語）

To go to Room B301, please go upstairs from the main entrance of Building B to the 3rd floor and then turn right.

（B301号室はB棟入口脇にある階段を上がり，そこから右に進んだところにあります．） - Organizer: Naotaka Kajino (Kobe University)

- 10:00 - 10:50 KUBOTA, Naoki (Nihon University)

Continuity results for the frog model in random initial configurations - 11:10 - 12:00 NAMBA, Ryuya (Okayama University)

Central limit theorems for non-symmetric random walks on nilpotent covering graphs

- 13:20 - 14:20 ANDRES, Sebastian (University of Cambridge)

Diffusion processes on branching Brownian motion - 14:40 - 15:30 MATSUURA, Kouhei (Tohoku University)

Compactness of Brownian semigroups associated with Robin boundary condition - 15:50 - 16:40 ABE, Yoshihiro (Kobe University)

Second order term of cover time for planar simple random walk

**KUBOTA, Naoki (Nihon University)**

**Continuity results for the frog model in random initial configurations**

We consider the frog model in random initial configurations. The dynamics of this model is as

follows: Assign randomly simple random walks to sites of the multidimensional cubic lattice,

and these simple random walks are regarded as ``frogs.'' Suppose that at least one frog exists

at the origin and only frogs sitting on the origin are active (the other frogs are sleeping and do

not move at first). When sleeping frogs are attacked by an active one, those become active and

start moving. Then, a fundamental object of study is the first passage time at which an active

frog reaches a site. It is known that this first passage time is asymptotically equal to the so-called

time constant, which depends on the law of the random initial configuration. In this talk we present

the continuity for the time constant with respect to the law of the random initial configuration.**NAMBA, Ryuya (Okayama University)**

**Central limit theorems for non-symmetric random walks on nilpotent covering graphs**

The long time asymptotics for random walks on infinite graphs is a principal topic for both

geometers and probabilists. A covering graph of a finite graph with a nilpotent covering

transformation group is called a nilpotent covering graph, regarded as a generalization of

a crystal lattice or the Cayley graph of a finite generated group of polynomial growth. We

discuss non-symmetric random walks on nilpotent covering graphs from a view point of the

theory of discrete geometric analysis developed by Kotani and Sunada. Our main purpose of

this talk is to give a functional central limit theorem for them. We also discuss an example of

non-symmetric random walks on nilpotent covering graphs with several figures and animations.

This talk is based on joint work with Satoshi Ishiwata (Yamagata University) and Hiroshi Kawabi

(Okayama University).**ANDRES, Sebastian (University of Cambridge)**

**Diffusion processes on branching Brownian motion**

Branching Brownian motion (BBM) is a classical process in probability, describing a population

of particles performing independent Brownian motion and branching according to a Galton

Watson process. In this talk we present a one-dimensional diffusion process on BBM particles

which is symmetric with respect to a certain random martingale measure. This process is obtained

by a time-change of a standard Brownian motion in terms of the associated positive continuous

additive functional. In a sense it may be regarded as an analogue of Liouville Brownian motion

which has been recently constructed in the context of a Gaussian free field. This is joint work

with Lisa Hartung (New York University).**MATSUURA, Kouhei (Tohoku University)**

**Compactness of Brownian semigroups associated with Robin boundary condition**

In this talk, we consider the first order $L^2$-Sobolev space with Robin boundary condition.

Moreover, we give a sufficient condition for the semigroup associated with this space is an

$L^2$-compact operator.**ABE, Yoshihiro (Kobe University)**

**Second order term of cover time for planar simple random walk**

I will consider the cover time for a simple random walk on the two-dimensional discrete

torus of side length n. Dembo-Peres-Rosen-Zeitouni (2004) identified the leading term in

the asymptotics for the cover time as n goes to infinity. I will talk about the exact second

order term. This is a discrete analogue of the work on the cover time for planar Brownian

motion by Belius and Kistler (2017).

- JSPS Grant-in-Aid for Young Scientists (B) Grant Number 15K17554

「特異的幾何学構造に由来する微分作用素とその確率論的対応物に対する数学解析」

(Principal Investigator: Naotaka Kajino (Department of Mathematics, Kobe University))

Naotaka Kajino

Department of Mathematics,
Graduate School of Science,
Kobe University,

Rokkodai-cho 1-1, Nada-ku, 657-8501 Kobe, Japan

Office: Faculty of Science Building B, Room B426

Tel: +81-78-803-5616

Fax: +81-78-803-5610

E-mail: nkajino ``at" math.kobe-u.ac.jp