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Brownian motion and Random Walk above Quenched Random Wall. (arXiv:1507.08578v3 [math.PR] UPDATED)
来源于:arXiv
We study the persistence exponent for the first passage time of a random walk
below the trajectory of another random walk. More precisely, let $\{B_n\}$ and
$\{W_n\}$ be two centered, weakly dependent random walks. We establish that
$\mathbb{P}(\forall_{n\leq N} B_n \geq W_n|W) = N^{-\gamma + o(1)}$ for a
non-random $\gamma\geq 1/2$. In the classical setting, $W_n \equiv 0$, it is
well-known that $\gamma = 1/2$. We prove that for any non-trivial $W$ one has
$\gamma>1/2$ and the exponent $\gamma$ depends only on
$\text{Var}(B_1)/\text{Var}(W_1)$.
Our result holds also in the continuous setting, when $B$ and $W$ are
independent and possibly perturbed Brownian motions or Ornstein-Uhlenbeck
processes. In the latter case the probability decays at exponential rate. 查看全文>>