Probability of hitting a distant point for the voter model started with a single one

击中概率与一致$S$-瞬态子图

Hitting probabilities and uniformly $S$-transient subgraphs

该研究围绕图上随机游走的击中概率展开，提出$S$-瞬态概念，在热核满足双侧高斯估计等条件下证明相关概率的上下界，还给出调和测度等相关概率的双侧估计。

Tightness of voter model interfaces

Consider a long-range, one-dimensional voter model started with all zeroes on the negative integers and all ones on the positive integers. If the process obtained by identifying states that are translations of each other is positively recurrent, then it is said that the voter model exhibits interface tightness. In 1995, Cox and Durrett proved that one-dimensional voter models exhibit interface tightness if their infection rates have a finite third moment. Recently, Belhaouari, Mountford, and Valle have improved this by showing that a finite second moment suffices. The present paper gives a new…

Distribution of the Brownian motion on its way to hitting zero

For the one-dimensional Brownian motion $B=(B_t)_{t\ge 0}$, started at $x>0$, and the first hitting time $τ=\inf\{t\ge 0:B_t=0\}$, we find the probability density of $B_{uτ}$ for a $u\in(0,1)$, i.e. of the Brownian motion on its way to hitting zero.

On hitting times of affine boundaries by reflecting Brownian motion and Bessel processes

Firstly, we compute the distribution function for the hitting time of a linear time-dependent boundary $t\mapsto a+bt,\ a\geq 0,\,b\in \R,$ by a reflecting Brownian motion. The main tool hereby is Doob's formula which gives the probability that Brownian motion started inside a wedge does not hit this wedge. Other key ingredients are the time inversion property of Brownian motion and the time reversal property of diffusion bridges. Secondly, this methodology can also be applied for the three dimensional Bessel process. Thirdly, we consider Bessel bridges from 0 to 0 with dimension parameter $δ>…

Density of space-time distribution of Brownian first hitting of a disc and a ball

We compute the joint distribution of the site and the time at which a $d$-dimensional standard Brownian motion $B_t$ hits the surface of the ball $ U(a) =\{|{\bf x}|<a\}$ for the first time. The asymptotic form of its density is obtained when either the hitting time or the starting site $B_0$ becomes large. Our results entail that if Brownian motion is started at ${\bf x}$ and conditioned to hit $U(a)$ at time $t$ for the first time, the distribution of the hitting site approaches the uniform distribution or the point mass at $a{\bf x}/|{\bf x}|$ according as $|{\bf x}|/t$ tends to zero or inf…

Levy processes: Hitting time, overshoot and undershoot II - Asymptotic behaviour

Let (X_t, t>=0) be a Levy process started at 0, with Levy measure nu and T_x the first hitting time of level x>0: T_x:=inf{t>=0; X_t>x}. Let $F(theta, mu, rho,.) be the joint Laplace transform of (T_x, K_x, L_x): F(theta,mu,rho,x) :=E(e^(-theta T_x - mu K_x ρL_x) 1_(T_x<+infinity)), where theta>=0, mu>=0, rho>=0, x>=0, K_x:=X_(T_x)-x and L_x:=x-X_(T_(x^-)). If we assume that nu has finite exponential moments we exhibit an asymptotic expansion for F(theta,mu,rho,x), as x -> +infinity. A limit theorem involving a normalization of the triplet (T_x,K_x,L_x) as x -> +infinity, may be deduced. At la…