Conditioned Two-Dimensional Simple Random Walk: Green’s Function and Harmonic Measure

Abstract

We study the Doob's h-transform of the two-dimensional simple random walk with respect to its potential kernel, which can be thought of as the twodimensional simple random walk conditioned on never hitting the origin. We derive an explicit formula for the Green's function of this random walk, and also prove a quantitative result on the speed of convergence of the (conditional) entrance measure to the harmonic measure (for the conditioned walk) on a finite set.

Transience of conditioned walks on the plane: encounters and speed of escape

“…Step 2 with (16) we conclude the proof of the lower bound in the case n 1/3 |x|, |y| √ M n. Since we already have the upper bound for this case, all that remains is to extend (13) to the other cases using the decomposition in (15).…”

“…Item (vi) is a consequence of (v). A straightforward derivation of (vi) without the aid of (v) and further potential-theoretic results about the S-walk can be found in [13].…”

“…Now, let us focus on proving (13). Notice that (14) already proves the upper bound in (13) for the case |y|, |x| n 1/3 .…”

Rate of escape of conditioned Brownian motion

“…Though we do not know the actual value of K * we can see that both theorems are much finer than the corresponding Theorem 1.2 of [22]. These two theorems together yield a precise version of the observation from [20] that the pathwise divergence of R to infinity occurs in a highly irregular way. The future minima process has been considered earlier, e.g.…”

“…is "big enough and well distributed in space", then the proportion of visited sites is approximately uniformly distributed on [0,1]. In [20] the explicit formula for the Green function is obtained, and a survey is given in Chapter 4 of [21].…”