Abstract
The Ray–Knight theorems show that the local time processes of various path fragments derived from a one-dimensional Brownian motion $B$ are squared Bessel processes of dimensions $0$, $2$, and $4$. It is also known that for various singular perturbations $X= |B| + \mu \ell $ of a reflecting Brownian motion $|B|$ by a multiple $\mu $ of its local time process $\ell $ at $0$, corresponding local time processes of $X$ are squared Bessel with other real dimension parameters, both positive and negative. Here, we embed squared Bessel processes of all real dimensions directly in the local time process of $B$. This is done by decomposing the path of $B$ into its excursions above and below a family of continuous random levels determined by the Harrison–Shepp construction of skew Brownian motion as the strong solution of an SDE driven by $B$. This embedding connects to Brownian local times a framework of point processes of squared Bessel excursions of negative dimension and associated stable processes, recently introduced by Forman, Pal, Rizzolo and Winkel to set up interval partition evolutions that arise in their approach to the Aldous diffusion on a space of continuum trees.
Highlights
Introduction and statement of main resultsA remarkable duality between BESQ(δ) processes of dimensions δ = 2 ± 2α was pointed out in [22, Theorem (3.3) and Remark (4.2)(ii)] and [23, Section 3]:
Squared Bessel processes are a family of one-dimensional diffusions on [0, ∞), defined by continuous solutions Y = (Y (x), 0 ≤ x ≤ ζ) of the stochastic differential equation dY (x) = δ dx + 2 Y (x)dB(x), Y (0) = y ≥ 0, 0 < x < ζ where δ is a real parameter, B = (B(x), x ≥ 0) is standard Brownian motion, and ζ is the lifetime of Y, defined by if δ > 0 ζ :=
That Wγ± =d Rμ± := |B| ± μ± as in Lemmas 5 and 6 for μ± := 2γ/(1 − ±γ). In this framework of skew Brownian motion, we give a more explicit statement of the local times decomposition claimed in Theorem 3
Summary
A remarkable duality between BESQ(δ) processes of dimensions δ = 2 ± 2α was pointed out in [22, Theorem (3.3) and Remark (4.2)(ii)] and [23, Section 3]:. The C[0, ∞)-valued process (Yy(δ), y ≥ 0, δ ≥ 0) has stationary independent increments in both y ≥ 0 and δ ≥ 0 This construction, and the duality between dimensions 0 and 4, explained the multiple appearances of BESQ(δ) processes and their bridges for δ = 0, 2 and 4 in the Ray–Knight descriptions of Brownian local time processes. According to one of the Ray–Knight theorems, the process (L(x, τ (v)), x ≥ 0) is a BESQv(0) This raises the following question: Can we find the pair (Y, Y ) of Corollary 2 embedded in the local times of B?.
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