機率研討會
主講者: Prof. Kiyoshi Kawazu (Alethia Univ.)
講題: Recurrence of Multi-dimensional Brownian Motion in Random Environment
時間: 2008-05-19 (Mon.)  15:30- 17:00
地點:
Abstract: Let $\{W(x), x\in R\}$ be a one –dimensional Wiener process with $W(0)=0$ and $\{B(t), t\geq 0\}$ be a standard d-dimensional Brownian motion. We assume they are independent. Let ${X(t), t\geq 0}$ be a diffusion process defined by the formal stochastic differential equations, $X(t)=(X_1(t), X_2(t), \cdots, X_d(t)), t\geq 0,$
$dX_i(t)=dB_i(t) -1/2 W^{\prime} (X_i(t)) dt$,$ X_(0)=0, i=1,2,\cdots, d$. We call $X(t)$ a multi-dimensional Brownian motion in random environment $W(x)$. Each $X_i(t)$ is a diffusion process with a scale function $S_i(x)=\int_0^x exp(W(y)) dy$ and a speed measure $m_i(x)=2 \exp(W(x)) dx$. So we can say that $X(t)$ is a direct product process of one-dimensional diffusion processes $X_i(t)$ in some sense. We call it a d-dimensional Brox diffusion. We consider the recurrence property of Brox diffusion. We also consider the diffusion $X_i(t)$
with drift $-1/2 W^{\prime}_i(X_i(t))$, where $W(x_1,x_2, \cdots, x_d)=(W_1(x_1), W_2(x_2),\cdots, W_d(x_d))$ is the d-dimensional Wiener process.
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