Abstract
For normally reflected Brownian motion and for simple random walk on independently growing in time $d$-dimensional domains, $d\ge3$, we establish a sharp criterion for recurrence versus transience in terms of the growth rate.
Highlights
There has been much interest in studies of random walks in random environment
Moving to Zd, d ≥ 3, it was conjectured by the last author that recurrence of once reinforced random walk is sensitive to the strength of the reinforcement
The motion of walker excited towards the origin on the boundary of its range is recurrent in any dimension regardless of the strength of the excitation, see [K1, Section 2] and [K2], while as shown in [BW], excitation by means of a drift in e1 direction results in transience for any strength of the drift in any dimension d ≥ 2 (Cf. [KZ] for results in one dimension, related excitation models, and open problems)
Summary
There has been much interest in studies of random walks in random environment (see [HMZ]). In proving Lemma 3.1 we rely on the following invariance principle in bounded uniform domains, which allows us to transform hitting probabilities of srw to the corresponding probabilities for an rbm.
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