the Ray-Knight theorems
/ ray-nite /
Fix a Brownian path and look at its local time L_t^a not as a function of time t but as a function of the LEVEL a — the profile a -> L_t^a recording how much time the path spent near each height. The Ray-Knight theorems make the astonishing statement that this random profile, as a process in the space variable a, is itself a recognisable diffusion: a squared Bessel process. They turn a question about Brownian local time into a clean statement about Bessel-squared processes.
Two canonical versions. First Ray-Knight: stop at tau_1 = inf{ t : B_t = 1 }, the first hit of level 1, with B_0 = 0. Then the local-time profile a -> L_(tau_1)^a is, for a running from 1 down to 0, a squared Bessel process of dimension 2 (BESQ(2)) started at 0; continued below 0, for a < 0 it is a BESQ(0) (which is absorbed at 0). Second Ray-Knight: stop at the inverse local time tau_l = inf{ t : L_t^0 > l }, when local time at 0 first exceeds l. Then a -> L_(tau_l)^a is, for a >= 0, a BESQ(0) started at l, absorbed when it hits 0. A squared Bessel process of dimension delta solves dZ = delta dt + 2 sqrt(Z) dB and is the square of the radial part of delta-dimensional Brownian motion (for integer delta); the change of variable from delta-dimensional radial motion to local-time profile is the content of the theorems, provable via excursion theory or via the additivity of squared Bessel laws.
Why it matters: the Ray-Knight theorems are the precise dictionary between local time and Bessel processes, and they are the engine for hard computations — the law of total local time, the supremum of local time, and the deep connections to random trees (via the Brownian excursion = normalised BESQ), to the continuum random tree, and to the Gaussian free field (Dynkin isomorphism, where squared local time pairs with the square of a Gaussian field). The honest caveat: these are theorems about a SPECIFIC stopping rule (a hit of a level, or an inverse-local-time), and the dimension of the squared Bessel process changes with the rule and the region (2 above vs 0 below the start); quoting 'local time is Bessel-squared' without the exact stopping time and the dimension bookkeeping is meaningless. They require the strong Markov property and the continuity of local time established earlier.
Run Brownian motion from 0 until it first hits 1. Plot, for each height a between 0 and 1, the total time the path spent near a. The first Ray-Knight theorem says this graph is a BESQ(2) sample — exactly the squared length of a 2-dimensional Brownian motion read along the height axis.
Local time read across LEVELS (not time) is a squared Bessel process — the dimension depends on the stopping rule and region.
The squared-Bessel dimension is not universal: it depends on the exact stopping time (hit a level vs inverse local time) and on whether you are above or below the start; state both, or the theorem is empty.