# Does 0-1 law extend to Brownian motions in short times?

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↑∞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<1t<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<1t<1? Are there equivalent tests for transience and recurrence without taking t↑∞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>2d>2)?

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