a Bessel process
/ BESS-ell /
A Bessel process is the distance of a multidimensional Brownian motion from the origin — or, more generally, the diffusion you get by allowing that 'dimension' to be any real number, not just an integer. It answers a very concrete question: if a particle diffuses in d-dimensional space, how does its radius |B_t| behave on its own? The radius is itself a one-dimensional diffusion, and studying it directly reveals deep facts about Brownian motion: recurrence versus transience, polarity of points, and the behaviour near the origin.
For integer dimension d, the Bessel process of dimension d, written BES(d), is R_t = |B_t| where B_t is a d-dimensional Brownian motion. Applying Ito's formula to the Euclidean norm shows R_t solves the SDE dR_t = ((d-1)/(2 R_t)) dt + dW_t for a one-dimensional Brownian motion W (valid away from 0). The drift (d-1)/(2R) is a repulsion from the origin that grows as the dimension grows; this single coefficient is what makes the process more and more transient as d increases. One then DEFINES BES(d) for any real d >= 0 by the same SDE (or via its square, the squared Bessel process BESQ(d), dY = d dt + 2 sqrt(Y) dW, which is well-defined and additive in d). The key dichotomy: for d >= 2 the origin is polar — the process started away from 0 never hits 0 — while for 0 <= d < 2 it does reach 0 (and the boundary behaviour, absorbing or reflecting, must be specified). At d = 2 the process is neighbourhood-recurrent but never actually hits the point 0; for d >= 3 it is transient (R_t -> infinity).
Bessel processes are the natural language for the fine structure of Brownian motion and appear throughout: the radial part of Brownian motion explains why planar Brownian motion is neighbourhood-recurrent but does not hit points (d=2 polarity) and why spatial Brownian motion is transient (d>=3); BES(3) is famous as a Brownian motion conditioned to stay positive and appears in the Ray-Knight theorems and in the study of Brownian local time; and the squared Bessel family is exactly the driverless cousin of the CIR process, which is a mean-reverting, time-changed squared Bessel. The honest subtleties: the SDE form has a singular drift at 0, so the equation is only literally valid away from the origin and the behaviour at 0 must be handled separately (the squared-Bessel formulation is cleaner there); and 'dimension' d is a continuous parameter — non-integer Bessel processes are perfectly good diffusions even though there is no R^d to take a norm in.
Planar Brownian motion (d = 2) gives the BES(2) radial process dR = (1/(2R)) dt + dW. The repulsive drift is just strong enough that the process never hits 0 (points are polar in the plane), yet weak enough that it keeps returning to every neighbourhood of the origin — neighbourhood-recurrence. Raise the dimension to 3 and the stronger drift (1/R) makes R_t -> infinity: spatial Brownian motion is transient.
The radial process encodes Brownian recurrence/transience: neighbourhood-recurrent in d=2, transient in d>=3, points polar in d>=2.
The SDE drift (d-1)/(2R) is singular at 0, so the equation only holds away from the origin; behaviour at 0 (and existence for d < 2, where 0 is reached) needs the squared-Bessel formulation or an explicit boundary condition. Dimension d may be any nonnegative real, not just an integer.