Let B be a d-dimensional standard Brownian motion and R_t = |B_t|. Derive dR_t = d beta_t + ((d-1)/(2 R_t)) dt for some one dimensional Brownian motion beta, and say for which d the radial process is transient.

Let B be a d-dimensional standard Brownian motion and R_t = |B_t|. Derive dR_t = d beta_t + ((d-1)/(2 R_t)) dt for some one dimensional Brownian motion beta, and say for which d the radial process is transient.

Approach: Apply Ito to the square root of the sum of squares, collect the martingale part, and identify it as a Brownian motion by computing its quadratic variation.

dR_t = d beta_t + ((d-1)/(2 R_t)) dt with beta_t = ∫_0^t sum_i (B_i(s)/R_s) dB_i(s), and R is recurrent for d ≤ 2 and transient for d ≥ 3. Ito's lemma applied to R = sqrt(sum_i B_i^2) gives first order terms sum_i (B_i/R) dB_i and second order terms (1/2) sum_i (1/R - B_i^2 R^{-3}) dt = (1/2)(d/R - 1/R) dt = ((d-1)/(2R)) dt, since sum_i B_i^2 = R^2. The candidate beta is a continuous local martingale with quadratic variation ∫_0^t sum_i (B_i/R)^2 ds = t, so Levy's characterisation makes it a standard Brownian motion, and R is the Bessel process of dimension d. The scale function for the drift (d-1)/(2r) is log r for d = 2 and r^{2-d} for d ≥ 3, and the latter is bounded near infinity, which gives transience for d ≥ 3 and recurrence below it.

Follow-up: For d = 3 and R_0 = 1, what is the probability that the process ever returns within distance 1/2 of the origin?

Key concepts: Bessel process, Levy characterisation, Ito's lemma, transience.