scholarly journals Gaussian estimates for spatially inhomogeneous random walks on Zd

2006 ◽  
Vol 34 (1) ◽  
pp. 264-283 ◽  
Author(s):  
Sami Mustapha
Keyword(s):  
Random Walks ◽  
Spatially Inhomogeneous ◽  
Inhomogeneous Random Walks ◽  
Gaussian Estimates

Minima of independent time-inhomogeneous random walks

2020 ◽  
Vol 23 (03) ◽  
pp. 2050021
Author(s):  
Wanting Hou ◽  
Wenming Hong
Keyword(s):  
Random Walks ◽  
Second Order ◽  
Inhomogeneous Random Walks

In this paper, we will consider the minima of an exponentially growing number of independent time-inhomogeneous random walks, where the first- and second-order limit behaviors for the minima have been obtained.

scholarly journals Spatially Inhomogeneous Populations with Seed-Banks: I. Duality, Existence and Clustering

2021 ◽  
Author(s):  
Frank den Hollander ◽  
Shubhamoy Nandan
Keyword(s):  
Random Walks ◽  
Seed Bank ◽  
Seed Banks ◽  
Model Parameters ◽  
Geographic Space ◽  
Spatially Inhomogeneous ◽  
And Migration ◽  
Coalescing Random Walks ◽  
Colony Migration

AbstractWe consider a system of interacting Moran models with seed-banks. Individuals live in colonies and are subject to resampling and migration as long as they are active. Each colony has a seed-bank into which individuals can retreat to become dormant, suspending their resampling and migration until they become active again. The colonies are labelled by $${\mathbb {Z}}^d$$ Z d , $$d \ge 1$$ d ≥ 1 , playing the role of a geographic space. The sizes of the active and the dormant population are finite and depend on the location of the colony. Migration is driven by a random walk transition kernel. Our goal is to study the equilibrium behaviour of the system as a function of the underlying model parameters. In the present paper, under a mild condition on the sizes of the active populations, the system is well defined and has a dual. The dual consists of a system of interacting coalescing random walks in an inhomogeneous environment that switch between an active state and a dormant state. We analyse the dichotomy of coexistence (= multi-type equilibria) versus clustering (= mono-type equilibria) and show that clustering occurs if and only if two random walks in the dual starting from arbitrary states eventually coalesce with probability one. The presence of the seed-bank enhances genetic diversity. In the dual this is reflected by the presence of time lapses during which the random walks are dormant and do not move.

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