### Abstract

Concentration for First Passage Percolation.
Shuta Nakajima (Kyoto University) /
We can regard First Passage Percolation(F.P.P) as the minimal time problem in random medium. I studied about the concentration for oriented F.P.P and got the limiting results connected with this model. In the proof, we can see the interesting property of minimizing path and the relationship between the length of minimizing path and concentration.

1st Lecture by David Brydges
David Brydges (USA) /

Lamplighter random walks on fractals
Chikara Nakamura (Kyoto University) /
In this talk, I will present some works on switch-walk-switch random walks on lamplighter graph $\mathbb{Z}_2 \wr G$. The lamplighter graph $\mathbb{Z}_2 \wr G$ consists of the graph $G$ and additional lamps on each vertex of $G$, and we take into account which lamps are on or off. We obtain the sharp on-diagonal heat kernel estimates and the law of the iterated logarithms for the random walks on $\mathbb{Z}_2 \wr G$ under the assumption that the random walks on the underlying graph $G$ enjoy the sub-Gaussian heat kernel estimates. This is a joint work with Prof. T. Kumagai.

2nd Lecture by David Brydges
David Brydges (USA) /

Martin Boundary for Gaussian Diffusion Processes
Wei-Da Chen (National Central University) /
In this note, we study the Martin boundary for some Gaussian diffusion processes $X_{t}$. General theory of Martin boundary for Markov process has been well developed in the literature. See Kunita-Watanabe (1965), Dynkin (1969), Salminen (1981). One of its important applications is to give the unique representation of harmonic functions in terms of minimal Martin functions. There are very few concrete examples that properties of Martin boundary are studied, such as Brownian motion. Martin bounday for $2D$ Gaussian transient diffusion process were studied in Cranston-Orey-Rosler (1983). They studied the space time Martin boundary for such diffusion process. As a result, they obtained the representaion for positive harmonic functions. The space-time Martin boundary is to consider the Martin boundary for $\widetilde{X}_{t}=(t,X_{t})\in(0,\infty)\times\mathbb{R}^{d}$ as a Markov process. In this note, we show how to generalize this idea to the high dimensional space. In this case, we first observe that the space-time Martin topology becomes concrete and can be described in terms of Euclidean topology. The Martin functions can be calculated using Riccati's equation. We will discuss the convergence of some h-diffusion process, the diffusion process under h-transform, where h is a positive space time harmonic function. As a result, Hitting'' probability measure will be calculated for some examples.

3rd Lecture by David Brydges
David Brydges (USA) /
We consider the fundamental solution $p(t,x)$ of the following fractional partial differential equation $$\partial_{t}^{\alpha}u(t,x)=-(-\Delta)^{\beta}u(t,x),\quad(t,x)\in(0,\infty)\times\mathbb{R}^{d} \qquad \qquad \text{(1)}$$ where $\partial_{t}^{\alpha}$ denotes the Caputo fractional derivative and $-(-\Delta)^{\beta}$ is the fractional Laplacian. If $\alpha=1$ and $\beta=1$ then $(1)$ is the classical heat equation and we can interpret $p(t,x)$ as the probability density function of Brownian motion. Similarly, for $\alpha\in(0,1)$ and $\beta\in(0,1)$, there is a connection between equation $(1)$ and some non-Gaussian stochastic processes which describe anomalous diffusion, for instance, cohesive movement (sub-diffusion) and long-range jump movement (super-diffusion). In this talk we provide asymptotic behaviors and sharp upper bounds of $p(t,x)$ and its derivatives. This is joint work with Kyeong-Hun Kim.