The Annals of Applied Probability

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

Yu-Ting Chen

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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

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

Received: May 2017
Revised: February 2018
First available in Project Euclid: 8 October 2018

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C22: Interacting particle systems [See also 60K35] 60F05: Central limit and other weak theorems 60J60: Diffusion processes [See also 58J65]

Voter model Wright–Fisher diffusion evolutionary game


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.

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