We investigate the cutoff phenomenon for Markov processes under information divergences such as $f$-divergences and R\'enyi divergences. We classify most common divergences into four types, namely $L^2$-type, $\mathrm{TV}$-type, separation-type and $\mathrm{KL}$ divergence, in which we prove that the cutoff phenomenon are equivalent and relate the cutoff time and window among members within each type. To justify that this classification is natural, we provide examples in which the family of Markov processes exhibit cutoff in one type but not in another. We also establish new product conditions in these settings for the processes to exhibit cutoff, along with new results in non-reversible or non-normal situations. The proofs rely on a functional analytic approach towards cutoff.

翻译：我们研究了马尔可夫过程在$f$-散度与Rényi散度等信息散度下的截断现象。我们将最常见的散度划分为四种类型：$L^2$型、$\mathrm{TV}$型、分离型与$\mathrm{KL}$散度，并证明了每种类型内部成员的截断现象具有等价性，且其截断时间与窗口存在关联。为论证该分类的自然性，我们给出了若干马尔可夫过程族在某类型中呈现截断而在另一类型中不呈现的实例。我们还在这些设定中建立了过程呈现截断的新乘积条件，并给出了非可逆或非正规情形下的新结论。证明过程依赖于研究截断现象的泛函分析方法。