Recurrence of Drifted Brownian Motion Conditioned to not hit Moving Barrier

Question

Suppose we have a Brownian motion $X$ with $X_0>0$ and drift $\mu$ conditioned to be less than a barrier $R$ which has behaviour $R_0 = r$, $dR_s = \nu \, ds$, where $\mu > \nu > 0$.

Can we show that $X$ is positive recurrent in the sense that: $$ \mathbb{E}[\inf\{t>0: X_t \notin (0,\infty)\}] < \infty $$

My instinct is that it should be true, but I truly don't know where I would begin proving this. This paper https://arxiv.org/pdf/1710.02350.pdf provides the distribution of the maximum of a Brownian motion conditioned to not hit some fixed lower boundary, but it's not clear to me how I can utilise this result.

Answer 1

It is transient. Consider the process $Y_t=R_t-X_t$; it is a Brownian motion on the positive half-line with a constant negative drift $\nu-\mu$. The probability that it does not hit the origin up to time $T$ can be calculated from the explicit formula found e.g. in this answer. As $T\to\infty$, this probability, given the process up to time $t$, is proportional to $Y_te^{(\mu-\nu)Y_t}.$ This is the Radon-Nikodym derivative of the conditioned process with respect to the unconditioned one. So, Girsanov theorem gives that $Y_t$ conditioned to never hit the origin satisfies the SDE $$ dY_t=\frac{1}{Y_t}dt+dB_t, $$ the constant drifts cancelling out exactly. I.e., $Y_t$ is a Bessel process of dimension $d=3$, $Y_t\stackrel{\mathcal{D}}{=}|Z_t|$, where $Z_t$ is the 3D standard Brownian motion (see e.g. these notes). In these terms, you are asking about the first time $t$ such that $|Z_t|=C+\nu t$, with $C=X_0$. It is clear by Brownian scaling that with positive probability, there's no such $t$.