## The Annals of Applied Probability

### Wright–Fisher diffusions in stochastic spatial evolutionary games with death–birth updating

Yu-Ting Chen

#### Abstract

We investigate stochastic spatial evolutionary games with death–birth updating in large finite populations. Within growing spatial structures subject to appropriate conditions, the density processes of a fixed type are proven to converge to the one-dimensional Wright–Fisher diffusions. Convergence in the Wasserstein distance of the laws of the occupation measures also holds. The proofs study the convergences under certain voter models by an equivalence between their laws and the laws of the evolutionary games. In particular, the additional growing dimensions in minimal systems that close the dynamics of the game density processes are cut off in the limit.

As another application of this equivalence of laws, we consider a first-derivative test among the major methods for these evolutionary games in a large population of size $N$. Requiring only the assumption that the stationary probabilities of the corresponding voting kernel are comparable to uniform probabilities, we prove that the test is applicable at least up to weak selection strengths in the usual biological sense [i.e., selection strengths of the order $\mathcal{O}(1/N)$].

#### Article information

Source
Ann. Appl. Probab., Volume 28, Number 6 (2018), 3418-3490.

Dates
Revised: February 2018
First available in Project Euclid: 8 October 2018

https://projecteuclid.org/euclid.aoap/1538985626

Digital Object Identifier
doi:10.1214/18-AAP1390

Mathematical Reviews number (MathSciNet)
MR3861817

Zentralblatt MATH identifier
06994397

#### Citation

Chen, Yu-Ting. Wright–Fisher diffusions in stochastic spatial evolutionary games with death–birth updating. Ann. Appl. Probab. 28 (2018), no. 6, 3418--3490. doi:10.1214/18-AAP1390. https://projecteuclid.org/euclid.aoap/1538985626

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