• Corpus ID: 119321658

Stochastic averaging for multiscale Markov processes with an application to a Wright-Fisher model with fluctuating selection

@article{Hutzenthaler2015StochasticAF,
  title={Stochastic averaging for multiscale Markov processes with an application to a Wright-Fisher model with fluctuating selection},
  author={Martin Hutzenthaler and Peter Pfaffelhuber and C. H. Printz},
  journal={arXiv: Probability},
  year={2015}
}
Let $Z = (Z_t)_{t\in[0,\infty)}$ be an ergodic Markov process and, for every $n\in\mathbb{N}$, let $Z^n = (Z_{n^2 t})_{t\in[0,\infty)}$ drive a process $X^n$. Classical results show under suitable conditions that the sequence of non-Markovian processes $(X^n)_{n\in\mathbb{N}}$ converges to a Markov process and give its infinitesimal characteristics. Here, we consider a general sequence $(Z^n)_{n\in\mathbb{N}}$. Using a general result on stochastic averaging from [Kur92], we derive conditions… 
A unified framework for limit results in Chemical Reaction Networks on multiple time-scales
If (XN )N=1,2,... is a sequence of Markov processes which solve the martingale problems for some operators (GN )N=1,2,..., it is a classical task to derive a limit result as N → ∞, in particular a
The spatial Lambda-Fleming-Viot process with fluctuating selection
We are interested in populations in which the fitness of different genetic types fluctuates in time and space, driven by temporal and spatial fluctuations in the environment. For simplicity, our
The effective strength of selection in random environment
We analyse a family of Wright Fisher models with selection in a random environment and skewed offspring distribution. We provide a calculable criterion to quantify the strength of different shapes of

References

SHOWING 1-10 OF 41 REFERENCES
Diffusion approximations of Markov chains with two time scales and applications to population genetics, II
For N = 1, 2, …, let {(XN (k), YN (k)), k = 0, 1, …} be a time-homogeneous Markov chain in . Suppose that, asymptotically as N → ∞, the ‘infinitesimal' covariances and means of XN ([·/ε N ]) are aij
On the extinction of Continuous State Branching Processes with catastrophes
We consider continuous state branching processes (CSBP's) with additional multiplicative jumps modeling dramatic  events in a random environment. These jumps  are described by a Levy process with
Infinite dimensional stochastic differential equations and their applications
The central aim o f this paper is to construct som e classes o f infinite dimensional stochastic processes related to population genetics or statistical m echanics by m aking u s e o f infinite
Infinite rate mutually catalytic branching in infinitely many colonies: construction, characterization and convergence
We construct a mutually catalytic branching process on a countable site space with infinite “branching rate”. The finite rate mutually catalytic model, in which the rate of branching of one
On the scaling limits of Galton-Watson processes in varying environments
We establish a general sufficient condition for a sequence of Galton–Watson branching processes in varying environments to converge weakly. This condition extends previ- ous results by allowing
A Liapounov bound for solutions of the Poisson equation
In this paper we consider ψ-irreducible Markov processes evolving in discrete or continuous time on a general state space. We develop a Liapounov function criterion that permits one to obtain
Central limit theorems and diffusion approximations for multiscale Markov chain models
Ordinary differential equations obtained as limits of Markov processes appear in many settings. They may arise by scaling large systems, or by averaging rapidly fluctuating systems, or in systems
Branching diffusions in random environment
We consider the diffusion approximation of branching processes in random environment (BPREs). This diffusion approximation is similar to and mathematically more tractable than BPREs. We obtain the
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
Population Genetics of Polymorphism and Divergence Under Fluctuating Selection
TLDR
This work uses classical diffusion approximations to model temporal fluctuations in the selection coefficients to find the expected distribution of mutation frequencies in the population and finds that fluctuating selection will lead to an increase in the ratio of divergence to polymorphism similar to that observed under positive directional selection.
...
...