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For an integer $\delta \ge 1$, a Bessel process is the radial part $R_t=|W_t|$ of $\delta$-dimensional Brownian motion. The definition extends to every real $\delta \ge 0$ by taking the nonnegative square root of a squared Bessel process:Before its first hit of zero, the Itô formula givesThis singular equation describes the process away from zero; continuation at zero must be specified. At dimension one the process is Reflected Brownian motion and the global equation includes local time of a semimartingale. For dimensions below one the usual singular-drift equation cannot simply be asserted across zero. The squared Bessel process definition avoids that ambiguity.^[ref]

Gregory F. Lawler studies random walks, Brownian motion and Schramm–Loewner evolution, with connections to statistical physics and complex analysis. His University of Chicago research profile emphasizes paths such as loop-erased random walks and their relation to uniform spanning trees.^[ref]

Lawler and Wendelin Werner developed the Brownian loop soup, using a conformally invariant measure on planar Brownian loops to build a Poisson point process of loops.^[ref] His Notes on the Bessel Process by Gregory F. Lawler provide another route from paths to geometry: a Bessel process describes radial diffusion, with its dimension parameter controlling whether the origin is reached. The boundary behavior matters when extending the process beyond its first zero.^[ref]