2015-06-15

2015-06-16

2015-06-17

Concentration for First Passage Percolation.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

Please click here for the lecture note.

Lamplighter random walks on fractals

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.

2015-06-18

2nd Lecture by David BrydgesPlease click here for the lecture note.

Martin Boundary for Gaussian Diffusion Processes

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

Please click here for the lecture note.

Asymptotic behaviors of fundamental solution and its derivatives related to anomalous diffusion

We consider the fundamental solution $p(t,x)$ of the following fractional partial differential equation
\begin{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)}
\end{equation}
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.

2015-06-19

Sooner waiting time problems in relation to the k-th generalized Fibonacci sequencesTraditionally, the distributions of runs and patterns were studied via combinatorial analysis. For example, Mood (1940) wrote: "The distribution problem is, of course, a combinatorial one, and the whole development depends on some identities in combinatory analysis" and Wald and Wolfowitz, 1940; David, 1947; Hirano, 1986; Godbole, 1990. However, finding the appropriate combinatorial identities to derive the probability distributions can be difficult, if not impossible, for complex runs, and this perhaps is the reason why the exact distributions of many common runs remain unknown especially for non-i.i.d. cases. Furthermore, the required identities often differ even for similar runs, and hence, even in the simplest case of independent and identically distributed (i.i.d.) two-state trials (so-called "Bernoulli trials"), each new distribution. In this talk, I want to introduce two kinds of waiting time problem First, Consider an infinite sequence of Bernoulli trials {Xi|i = 1, 2, . . .}. Let W(k) denote the waiting time, the number of trials needed, to get either consecutive k ones or k zeros for the first time. The probability distribution of W(k) is derived for both independent and homogeneous two-state Markovian Bernoulli trials, using a generalized Fibonacci sequence of order k. For independent Bernoulli trials, a special case of symmetric trial with p = 1/2 is considered. Second, Let V (k) denote the waiting time, the number of trials needed to get a consecutive k ones. We propose recurrence algorithms for the probability distribution function (pdf) and the probability generating function (pgf) of V (k) in sequences of independent and Markov dependent Bernoulli trials using generalized Fibonacci sequences of order k.