Bessel Processes

The distance |B(t)| from home: an outward drift d-1/2R made of pure geometry, and the dimension that decides whether the wanderer returns.

The idea

Definition (Bessel process).

Let $B(t) = \big(B_{1}(t), \dots, B_{d}(t)\big)$ consist of $d$ independent standard Brownian motions, one per coordinate. The Bessel process of dimension $d$ is the radial part of $B$, $R(t) = |B(t)| = \sqrt{B_{1}(t)^{2} + \cdots + B_{d}(t)^{2}},$ the distance of the Brownian motion from the origin at time $t$.

This single number carries the question of return. The motion revisits every neighbourhood of its starting point exactly when $R$ becomes small at arbitrarily large times, and it escapes for good exactly when $R$ grows without bound.

No coordinate of $B$ has a drift, so it is natural to expect that $R$ has none either. In fact it does: distance is a curved function of position, and the curvature produces a drift. How strongly that drift competes with the noise depends on the dimension $d$, and the comparison decides, in each dimension, whether the motion returns.

Ways to work on it

• Walkthrough. The distance of a Brownian motion from the origin: its geometric drift, and how dimension decides recurrence.
• Proof. Deriving dR = dW + d-1/2R dt: the Laplacian of the distance function, and why the noise is one Brownian motion.
• Practice. Compute drifts, recurrence behavior, and moments for Bessel processes across dimensions.
• Hardest. Decide whether the reciprocal distance of a three-dimensional Brownian motion is a true martingale.

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