Longterm behaviour of interacting stochastic (partial) differential equations and combinatorial stochastic processes, with a focus on the method of duality

• Applicant: Professor Dr. Jochen Blath
• Subject Area: Mathematics
• Term: from 2011 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 209674139

• Final Report Year: 2016

Final Report Abstract

Duality is a mathematical tool that relates two stochastic processes in such a way that one can deduce properties of one process by analysing the other one. Our project focussed on three aspects of the theory of duality, in particular in relation to stochastic processes arising in modelling evolution. In the first part, we looked at interacting particle systems and interacting stochastic (P)DEs modelling the evolution of populations in a spatial domain. Of particular interest to us is the continuous-space symbiotic branching model, which describes the interaction of a population consisting of two types that can only branch in the presence of each other. By varying a correlation parameter, this model generalizes several well-known population models such as the stepping stone model. Our main result shows that for negative correlations the diffusively rescaled system converges to a limit that satisfies a separation-of-types property. This shows that the diffusive regime is the right scaling to understand the interface, i.e. the region where both types are present. Moreover, we give a new duality for the limiting system, which is essential in identifying the limit in a special case as a system of annihilating Brownian motions with drift. Finally, we establish the link to a well-understood discrete-space version of the model, which allows us to show a geometric characterization of the separation-of-types property. In the second part of the project, we looked at certain discrete combinatorial dual processes. Here our main work concentrated on the effect of long dormancy in populations. For the first (non-Markovian) model, we identify three parameter regimes with very different behaviour. In a second paper, we present a Markovian seed bank model, which allows us to identify a new combinatorial dual process, the seedbank coalescent. Unlike the classical Kingman coalescent, this new model does not ‘come down from infinity’, and we identify the growth rate of the time to the most recent common ancestor as the sample size increases. Another line of research is to understand the effect of the pedigree on the gene genealogy in a diploid, bi-parental population. Here, we look at the geometry induced by the cyclical model introduced by Wakeley et al. Finally, in the third part of the project, we investigated general properties of duality. In particular, we wrote a survey paper bringing together well-known results, but also unveiling new aspects of the general theory. We focussed on a functional analytic formulation, transfer of properties from dual properties (such as monotonicity), and several notions of pathwise duality.

Publications

• Graphical representation of certain moment dualities and application to population models with balancing selection. Electron. Comm. Probab. 18 (2013), paper no. 14
  S. Jansen and N. Kurt
  
• On the notion(s) of duality for Markov processes. Probab. Surv. 11:59–120, 2014
  S. Jansen and N. Kurt
  
• A new look at duality for the symbiotic branching model. Preprint
  M. Hammer, M. Ortgiese and F. Völlering
  
• Genetic Variability Under the Seedbank Coalescent. Genetics 200(3):921–934, 2015
  J. Blath, B. Eldon, A. González Casanova, N. Kurt and M. Wilke Berenguer
  
• A new coalescent for seed-bank models. Ann. Appl. Probab. 26(2):851–897, 2016
  J. Blath, A. González Casanova, N. Kurt and M. Wilke Berenguer
  
• The scaling limit of the interface of the continuous-space symbiotic branching model. Ann. Probab. 44(2):807–866, 2016
  J. Blath, M. Hammer and M. Ortgiese