One-Dimensional Diffusions That Eventually Stop Down-Crossing

Consider a diffusion process corresponding to the operator $L=\frac{1}{2}a\frac{d^2}{d{x}^2}+b\frac{d}{dx}$ and which is transient to $+\mathrm{\infty }$. For $c>0$, we give an explicit criterion in terms of the coefficients $a$ and $b$ which determines whether or not the diffusion almost surely eventually stops making down-crossings of length $c$. As a particular case, we show that if $a=1$, then the diffusion almost surely stops making down-crossings of length $c$ if $b(x)\ge \frac{1}{2c}\log x+\frac{\gamma }{c}\log \log x$, for some $\gamma >1$ and for large $x$, but makes down-crossings of length $c$ at arbitrarily large times if $b(x)\le \frac{1}{2c}\log x+\frac{1}{c}\log \log x$, for large $x$.