Date: 6 Nov. (Thu.)
Place: West Wing, 6th floor, Colloquium Room and on the Web (zoom)
Time: 15:40-19:00
- Speaker 1: Yoshihiro Abe (Tohoku University）
- Time:15:40-16:40
- Title: 正則木上の単純ランダムウォークが頻繁に訪問する点について
- Abstract:
グラフ上の単純ランダムウォーク(SRW)が頻繁に訪問する点(thick point)の研究は古くからなされています．
特に2次元格子の場合は解析が困難であることが知られていますが，局所時間の最大値に対する大数の法則やthick pointの個数に関する極限定理などが既に得られています．しかし，局所時間の最大値の法則収束や対応する極値過程の収束などは未解決のままです．
Biskup-Louidor (2024)は，2次元格子の場合よりも解析しやすい正則木上のSRWの局所時間を考え，その最大値が法則収束することを示しました．それに続く自然な問いとして，「対応する極値過程は収束するか？」が考えられます．
本講演では，その極値過程があるPoisson点過程に収束する，という結果を紹介します．
本講演はMarek Biskup氏 (UCLA)との共同研究にもとづきます．
- Speaker 2: Masaki Wada (Fukushima University)
- Time:16:50-17:50
- Title: Elephant random walks with increasing memory of the very recent past
- Abstract:
Nakano and Takei considered the limit theorem for elephant random
walks remembering the very recent past (ERWVRP in abbreviation)
with fixed memory parameter $0 < p < 1$ (arXiv:2505.08285v2).
In this talk, we consider the variance of ERWVRP with time dependent
memory parameter $p_n$ satisfying $p_n \to 1 (n \to \infty)$.
The decay order of $1-p_n$ plays a crucial role for the order of the variance.
- Speaker 3: Yuu Hariya (Tohoku University)
- Time:18:00-19:00
- Title: Invariance of three-dimensional Bessel bridges in terms of time reversal
- Abstract:
Given three real numbers $a, b$ and $t$ with $t$ positive, let $\beta$ be a
one-dimensional Brownian bridge of length $t$ from $a$ to $b$. In this talk,
based on a conditional identity in law between Brownian bridges stemming from
Pitman’s theorem, we show that the process given by
\[
\beta_{t-s}+\biggl| b-a+
\min _{0\le u\le t-s}\beta_{u}-\min _{t-s\le u\le t}\beta_{u}
\biggr|
-\biggl|
\min _{0\le u\le t-s}\beta_{u}-\min _{t-s\le u\le t}\beta_{u}
\biggr|
\]
for $0 \le s \le t$, has the same law as $\beta$. The path transformation
that describes the above process is proven to be an involution, commute with
time reversal, and preserve a Pitman-type transformation in conjunction with
time reversal. Since it does not change the minimum value in particular,
the transformation also preserves the law of a three-dimensional Bessel bridge
of length $t$. As an application, some distributional invariances of three-dimensional
Bessel processes are derived. This talk is based on arXiv:2503.06813.
日本語
English