Abstract
We give a new proof for a Ray-Knight representation of Feller's branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion $H$ with a drift that is affine linear in the local time accumulated by $H$ at its current level. In Le et al. (2011) such a representation was obtained by an approximation through Harris paths that code the genealogies of particle systems. The present proof is purely in terms of stochastic analysis, and is inspired by previous work of Norris, Rogers and Williams (1988).
Highlights
The second one of the two classical Ray-Knight theorems establishes a representation of Feller’s branching diffusion in terms of reflected Brownian motion
The local time accumulated by the resulting path at “height” t, viewed as a process indexed by t, is a Feller branching diffusion obeying the SDE d Ztx = 2 Ztx dWtx with Z0x = x
One way to interpret this is to view the reflected Brownian path as an exploration path which codes the genealogy of a continuous state branching process: the local time of the exploration path at height t measures the “width of the genealogical forest” at this level, or equivalently, the mass of the population that is alive at the time corresponding to this height
Summary
The second one of the two classical Ray-Knight theorems (see e.g. [10] or [11]) establishes a representation of Feller’s branching diffusion in terms of reflected Brownian motion. [6]): the local time of the exploration path at height t measures the “width of the genealogical forest” at this level, or equivalently, the mass (or size) of the population that is alive at the time corresponding to this height This mass is Zt , the state of the branching process at time t. Just like Feller’s branching diffusion, the solution of (1), called Feller’s branching diffusion with logistic growth, arises as the diffusion limit of discrete population models, but with an interaction between individuals. In our recent work [5] we obtained the process H as the limit of exploration processes of discrete population models, and in this way provided a Ray-Knight representation of (1). The last section gives two remarks concerning a possible shortcut in the proof of the Theorem, and a general version of the second Ray-Knight theorem in the framework of [7]
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