arXiv subjects

Tibor Mach

Publications and source records attributed to Tibor Mach.

2 recordsLinked to original sources

arXiv2018

Recursive tree processes and the mean-field limit of stochastic flows

Interacting particle systems can often be constructed from a graphical representation, by applying local maps at the times of associated Poisson processes. This leads to a natural coupling of systems started in different initial states. We consider interacting particle systems on the complete graph in the mean-field limit, i.e., as the number of vertices tends to infinity. We are not only interested in the mean-field limit of a single process, but mainly in how several coupled processes behave in the limit. This turns out to be closely related to recursive tree processes as studied by Aldous and Bandyopadyay in discrete time. We here develop an analogue theory for recursive tree processes in continuous time. We illustrate the abstract theory on an example of a particle system with cooperative branching. This yields an interesting new example of a recursive tree process that is not endogenous.

math.PR↗

arXiv2018

A new characterization of endogeny

Aldous and Bandyopadhyay have shown that each solution to a recursive distributional equation (RDE) gives rise to recursive tree process (RTP), which is a sort of Markov chain in which time has a tree-like structure and in which the state of each vertex is a random function of its descendants. If the state at the root is measurable with respect to the sigma field generated by the random functions attached to all vertices, then the RTP is said to be endogenous. For RTPs defined by continuous maps, Aldous and Bandyopadhyay showed that endogeny is equivalent to bivariate uniqueness, and they asked if the continuity hypothesis can be removed. We introduce a higher-level RDE that through its $n$-th moment measures contains all $n$-variate RDEs. We show that this higher-level RDE has minimal and maximal fixed points with respect to the convex order, and that these coincide if and only if the corresponding RTP is endogenous. As a side result, this allows us to answer the question of Aldous and Bandyopadhyay positively.

math.PR↗