Abstract
We consider reflecting random walks on the nonnegative integers with drift of order $1/x$ at height $x$. We establish explicit asymptotics for various probabilities associated to such walks, including the distribution of the hitting time of $0$ and first return time to $0$, and the probability of being at a given height at a given time (uniformly in a large range of heights.) In particular, for certain drifts inversely proportional to $x$ up to smaller-order correction terms, we show that the probability of a first return to $0$ at time $n$ decays as a certain inverse power of $n$, multiplied by a slowly varying factor that depends on the drift correction terms.
Highlights
We consider random walks on + = {0, 1, 2, . . . }, reflecting at 0, with steps ±1 and transition probabilities of the form δ p(x, x + 1) = px = 21− +o 2x x as x → ∞, p(x, x − 1) = qx = 1 − px, (1.1)for x ≥ 1
Bessel-like walks are a special case of what is called the Lamperti problem—random walks with asymptotically zero drift
Bessel-like walks arise for example when symmetric simple random walk (SSRW) is modified by a potential proportional to log x
Summary
(1 − θ )P0(τ0 = m)Mk ≤ Pk(τ0 = m) ≤ (1 + θ )P0(τ0 = m)Mk. We will see below that the left and right sides of (2.15) represent approximately the probabilities for a Bessel process, with the same drift parameter δ and starting height k, to hit 0 in [m − 1, m + 1]. (We need only consider this process until the time, if any, that it hits 0, which avoids certain technical complications.) The imbedded walk is defined in the standard way: we start both the RW and the Bessel process at the same integer height k.
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