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.

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.