• Corpus ID: 15617254

SPDE approximation for random trees

@article{Bakhtin2009SPDEAF,
  title={SPDE approximation for random trees},
  author={Yuri Bakhtin},
  journal={Markov Processes and Related Fields},
  year={2009},
  volume={17},
  pages={1-36}
}
  • Yuri Bakhtin
  • Published 14 September 2009
  • Mathematics
  • Markov Processes and Related Fields
We consider the genealogy tree for a critical branching process conditioned on non-extinction. We enumerate vertices in each generation of the tree so that for each two generations one can define a monotone map describing the ancestor--descendant relation between their vertices. We show that under appropriate rescaling this family of monotone maps converges in distribution in a special topology to a limiting flow of discontinuous monotone maps which can be seen as a continuum tree. This flow is… 

Figures from this paper

Geometry of Large Random Trees: SPDE Approximation
In this chapter we present a point of view at large random trees. We study the geometry of large random rooted plane trees under Gibbs distributions with nearest neighbour interaction. In the first
Limit theorems for flows of branching processes
We construct two kinds of stochastic flows of discrete Galton-Watson branching processes. Some scaling limit theorems for the flows are proved, which lead to local and nonlocal branching
Some limit theorems for flows of branching processes 1
Continuous-state branching processes (CB-processes) arose as weak limits of rescaled discrete Galton-Watson branching processes; see, e.g., Jǐrina (1958) and Lamperti (1967). Continuous-state
Some limit theorems for flows of branching processes
We construct two kinds of stochastic flows of discrete Galton-Watson branching processes. Some scaling limit theorems for the flows are proved, which lead to local and nonlocal branching
Limit Theorems for Continuous-Time Branching Flows
TLDR
Some scaling limit theorems for the flow are proved, which lead to the path-valued branching processes and nonlocal branching superprocesses over the positive half line studied in Li (2012).
LIMIT THEOREMS FOR CONTINUOUS-TIME BRANCHING FLOWS
We construct a flow of continuous-time and discrete-state branching processes. Some scaling limit theorems for the flow are proved, which lead to the path-valued branching processes and nonlocal
LIMIT THEOREMS FOR CONTINUOUS-TIME BRANCHING FLOWS
We construct a flow of continuous-time and discrete-state branching processes. Some scaling limit theorems for the flow are proved, which lead to the path-valued branching processes and nonlocal
Limit theorems for continuous-time branching flows
We construct a flow of continuous time and discrete state branching processes. Some scaling limit theorems for the flow are proved, which lead to the path-valued branching processes and nonlocal

References

SHOWING 1-10 OF 20 REFERENCES
Thermodynamic limit for large random trees
TLDR
An infinite random tree consistent with Gibbs distributions on finite random plane trees with bounded branching is introduced and it is shown that it satisfies a certain form of the Markov property.
Measure-valued Markov branching processes conditioned on non-extinction
We consider a particular class of measure-valued Markov branching processes that are constructed as “superprocesses” over some underlying Markov process. Such a processX dies out almost surely, so we
Large Deviations for Random Trees
TLDR
A Large Deviation Principle (LDP) is proved for the distribution of degrees of vertices of the tree in a large random tree under Gibbs distributions from the analysis of RNA secondary structures.
Genealogical processes for Fleming-Viot models with selection and recombination
Infinite population genetic models with general type space incorporating mutation, selection and recombination are considered. The Fleming– Viot measure-valued diffusion is represented in terms of a
Two representations of a conditioned superprocess
We consider a class of measure-valued Markov processes constructed by taking a superprocess over some underlying Markov process and conditioning it to stay alive forever. We obtain two
Subdiffusive behavior of random walk on a random cluster
On considere deux cas de marche aleatoire {X n } n≥0 sur un graphe aleatoire #7B-G. Dans le cas ou #7B-G est l'arbre d'un processus de branchement critique, conditionne par la non-extinction, si h(x)
Markov Processes: Characterization and Convergence
Introduction. 1. Operator Semigroups. 2. Stochastic Processes and Martingales. 3. Convergence of Probability Measures. 4. Generators and Markov Processes. 5. Stochastic Integral Equations. 6. Random
Spatial Branching Processes, Random Snakes, and Partial Differential Equations
I An Overview.- I.1 Galton-Watson processes and continuous-state branching processes.- I.2 Spatial branching processes and superprocesses.- I.3 Quadratic branching and the Brownian snake.- I.4 Some
The Galton-Watson process conditioned on the total progeny
Let Zk denote the number in the kth generation of a Galton-Watson process initiated by one individual and let N be the total progeny, i.e., N = E',, Z,. As n x the limiting behaviour of the process
...
...