Abstract
We study the rst exit time from an arbitrary cone with apex at the origin
by a non-homogeneous random walk (Markov chain) on Zd (d 2) with mean drift
that is asymptotically zero. Specically, if the mean drift at x 2 Zd is of magnitude
O(kxk􀀀1), we show that < 1 a.s. for any cone. On the other hand, for an appropriate drift eld with mean drifts of magnitude kxk􀀀, 2 (0; 1), we prove that our random walk has a limiting (random) direction and so eventually remains in an arbitrarily narrow cone. The conditions imposed on the random walk are minimal: we assume only a uniform bound on 2nd moments for the increments and a form of weak isotropy. We give several illustrative examples, including a random walk in random environment model.
by a non-homogeneous random walk (Markov chain) on Zd (d 2) with mean drift
that is asymptotically zero. Specically, if the mean drift at x 2 Zd is of magnitude
O(kxk􀀀1), we show that < 1 a.s. for any cone. On the other hand, for an appropriate drift eld with mean drifts of magnitude kxk􀀀, 2 (0; 1), we prove that our random walk has a limiting (random) direction and so eventually remains in an arbitrarily narrow cone. The conditions imposed on the random walk are minimal: we assume only a uniform bound on 2nd moments for the increments and a form of weak isotropy. We give several illustrative examples, including a random walk in random environment model.
| Original language | English |
|---|---|
| Pages (from-to) | 351-388 |
| Number of pages | 38 |
| Journal | Markov Processes and Related Fields |
| Volume | 16 |
| Issue number | 2 |
| Publication status | Published - Jul 2010 |
Keywords
- asymptotic direction
- exit from cones
- inhomogeneous random walk
- perturbed random walk
- random walk in random environment