Transience or recurrence is the fundamental property of a Brownian on a given manifold but the definition requires infinite time limit.
In practice, one can not wait until $t\uparrow \infty$ to conclude whether a time series, a realization of a Brownian motion, is recurrent or transient.
In theory, one may wish $t<1$ so that the support of the probability density is within a coordinate patch.
Since transience or recurrence is intrinsic to the Brownian motion and the underlying manifold (dimension, compactness, curvature, etc.), will it be evident in short time limit $t<1$? Are there equivalent tests for transience and recurrence without taking $t\uparrow \infty$?
In particular, can I approximate a recurrent Brownian motion on a manifold by a transient Brownian motion (say in Euclidean space $d>2$)?
Brownian motion $B_t$ in $\Bbb R^3$ is transient. One can construct a (positive) recurrent diffusion process $X_t$ that is identical to $B_t$ up until the first time $\tau$ that $B_t$ exits the ball of radius $R$, centered at the origin, where $R>0$ is chosen so large that $\Bbb P[\tau<1]$ is as small as you please. [For example introduce a strong drift toward the origin that is active only when the process is outside the ball of radius $R$.] This makes it seem a hopeless task to provide a short time test for transience/recurrence.