Y. Lacroix | Searcharxiv

arXiv2005

Asymptotics for hitting times

In this paper we characterize possible asymptotics for hitting times in aperiodic ergodic dynamical systems: asymptotics are proved to be the distribution functions of subprobability measures on the line belonging to the functional class {6pt} {-3mm}(A){6mm}F={F:R\to [0,1]:\left\lbrack \matrixF is increasing, null on ]-\infty, 0]; \noalignF is continuous and concave; \noalignF(t)\le t for t\ge 0.\right.}. {6pt} Note that all possible asymptotics are absolutely continuous.

math.PR↗

arXiv2004

Hitting and return times in ergodic dynamical systems

Given an ergodic dynamical system $(X,T,μ)$, and $U\subset X$ measurable with $μ(U)>0$, let $μ(U)τ_U(x)$ denote the normalized hitting time of $x\in X$ to $U$. We prove that given a sequence $(U_n)$ with $μ(U_n)\to 0$, the distribution function of the normalized hitting times to $U_n$ converges weakly to some sub-probability distribution $F$ if and only if the distribution function of the normalized return time converges weakly to some distribution function $\tilde F$, and that in the converging case, $$ F(t)=\int_0^t(1-\tilde F(s))ds, t\ge 0.\tag$\diamondsuit$ $$ This in particular characterizes asymptotics for hitting times, and shows that the asymptotics for return times is exponential if and only if the one for hitting times is too.

math.DS↗