How to obtain transience from bounded radial mean curvature

We show that Brownian motion on any unbounded submanifold $P$ in an ambient manifold $N$ with a pole $p$ is transient if the following conditions are satisfied: The $p$-radial mean curvatures of $P$ are sufficiently small outside a compact set and the $p$-radial sectional curvatures of $N$ are sufficiently negative. The `sufficiency' conditions are obtained via comparison with explicit transience criteria for radially drifted Brownian motion in warped product model spaces.