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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1 Answer

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Brownian motion BtB_t in R3\Bbb R^3 is transient. One can construct a (positive) recurrent diffusion process XtX_t that is identical to BtB_t up until the first time τ\tau that BtB_t exits the ball of radius RR, centered at the origin, where R>0R>0 is chosen so large that P[τ<1]\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 RR.] This makes it seem a hopeless task to provide a short time test for transience/recurrence.