Abstract
Consider a spectrally positive Stable$(1\!+\!\alpha )$ process whose jumps we interpret as lifetimes of individuals. We mark the jumps by continuous excursions assigning “sizes” varying during the lifetime. As for Crump–Mode–Jagers processes (with “characteristics”), we consider for each level the collection of individuals alive. We arrange their “sizes” at the crossing height from left to right to form an interval partition. We study the continuity and Markov properties of the interval-partition-valued process indexed by level. From the perspective of the Stable$(1\!+\!\alpha )$ process, this yields new theorems of Ray–Knight-type. From the perspective of branching processes, this yields new, self-similar models with dense sets of birth and death times of (mostly short-lived) individuals. This paper feeds into projects resolving conjectures by Feng and Sun (2010) on the existence of certain measure-valued diffusions with Poisson–Dirichlet stationary laws, and by Aldous (1999) on the existence of a continuum-tree-valued diffusion.
Highlights
We define interval partitions, following Aldous [3, Section 17] and Pitman [43, Chapter 4].Construction of interval partition diffusions (N, X) X (fj(z), z ≥ 0) y$tt_jj$ z skewer(y, N, X)X fj(y − X(tj−))Definition 1.1
An interval partition is a set β of disjoint, open subintervals of some interval [0, M ], that cover [0, M ] up to a Lebesgue-null set
To define metrics on IH and on Iα, we say that a correspondence from β ∈ IH to γ ∈ IH is a finite sequence of pairs of intervals (U1, V1), . . . , (Un, Vn) ∈ β × γ, n ≥ 0, where the sequences (Uj )j∈[n] and (Vj )j∈[n] are each strictly increasing in the left-to-right ordering of the interval partitions
Summary
We define interval partitions, following Aldous [3, Section 17] and Pitman [43, Chapter 4]. This is the setting in which [17, Corollary 7] established the continuity of what in our notation is the total mass process ( skewer(y, N, X) , y ≥ 0) We strengthen this one-dimensional result to our interval-partition setting, as follows. We will assume that the spindles are associated with a [0, ∞)-valued diffusion process It is well-known that positive self-similar diffusions (absorbed when hitting the boundary at zero) form a threeparameter family, which we will call (α, q, c)-block diffusions, and we find an appropriate excursion measure ν = νq(−,c2α) when α ∈ (0, 1), q > α, c > 0. 0≤t≤T : H(t)=y,R(t)=0 is path-continuous (with respect to the vague topology for σ-finite measures on (0, ∞)) and Markovian for suitable stopping times T such as any inverse local time of (R, H) at (0, 0) He further showed in [5, Corollary II.4] that y → λy(T ) := 2 xμy[0,T ](dx) = 2. Our construction here gives new insights into Petrov’s diffusions and provides an approach to measure-valued processes, which we intend to explore in future work
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