The Height Process of a Continuous-State Branching Process with Interaction
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The Height Process of a Continuous-State Branching Process with Interaction Zenghu Li1 · Etienne Pardoux2
· Anton Wakolbinger3
Received: 4 October 2019 / Revised: 6 September 2020 / Accepted: 27 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract For a generalized continuous-state branching process with non-vanishing diffusion part, finite expectation and a directed (“left-to-right”) interaction, we construct the height process of its forest of genealogical trees. The connection between this height process and the population size process is given by an extension of the second Ray– Knight theorem. This paper generalizes earlier work of the two last authors which was restricted to the case of continuous branching mechanisms. Our approach is different from that of Berestycki et al. (Probab Theory Relat Fields 172:725–788, 2018). There the diffusion part of the population process was allowed to vanish, but the class of interactions was more restricted. Keywords Continuous-state branching process · Population dynamics with interaction · Genealogy · Height process of a random tree Mathematics Subject Classification (2020) Primary 60J80 · 60J25 · 60H10 · Secondary 92D25
B
Etienne Pardoux [email protected] Zenghu Li [email protected] Anton Wakolbinger [email protected]
1
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
2
CNRS, Centrale Marseille, I2M, Aix Marseille Univ, Marseille, France
3
Institute of Mathematics, FB 12, Goethe-University, 60629 Frankfurt, Germany
123
Journal of Theoretical Probability
1 Introduction The most general continuous-state branching processes (CSBP’s) are solutions of SDEs of the form Z tx
t
= x +γ 0
+
t 0
Z rx dr
+
∞
Z rx−
0
t
Z rx
2β 0
W (dr , du) +
0
t 0
Z rx−
0
1
, du, dz) z M(dr
0
z M(dr , du, dz), t ≥ 0,
(1.1)
1
where W (dr , du) is a space–time white noise, M(dr , du, dz) is a Poisson ran , du, dz) = dom measure (PRM) on (0, +∞)3 with intensity dr du π(dz) and M(dr M(dr , du, dz) − dr du π(dz). The σ -finite measure π is assumed to be such that (z 2 ∧ 1)π(dz) is a finite measure on (0, ∞). We shall assume in this paper that
∞
β > 0,
(z 2 ∧ z)π(dz) < ∞.
(1.2)
0
The assumption β > 0 will be essential to obtain a new representation of the height process (of a genealogical forest) that underlies (1.1), see Proposition 3.13. This approach, using tools from stochastic analysis, will be the basis for a representation of H also in the case with interaction, see (1.6). The ∞second condition in (1.2) allows us to replace the drift coefficient γ by −α := γ − 1 zπ(dz), and to write the last two integrals in namely equation (1.1) as a single integral with respect to M, Z tx
t
= x −α 0
+
t 0
Z rx dr Z rx−
0
t + 2β 0 ∞
Z rx
W (dr , du)
0
(1.3)
, du, dz), t ≥ 0. z M(dr
0
Moreover, we shall consider a generalized CSBP, where the linear drift −αz is replaced by a nonlinear drift f (z), which in general destro
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