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The spatial Λ-coalescent [article]

Vlada Limic, Anja Sturm
2005 arXiv   pre-print
This paper extends the notion of the -coalescent of Pitman (1999) to the spatial setting. The partition elements of the spatial Λ-coalescent migrate in a (finite) geographical space and may only coalesce if located at the same site of the space. We characterize the Λ-coalescents that come down from infinity, in an analogous way to Schweinsberg (2000). Surprisingly, all spatial coalescents that come down from infinity, also come down from infinity in a uniform way. This enables us to study
more » ... time asymptotics of spatial Λ-coalescents on large tori in d> 3 dimensions. Our results generalize and strengthen those of Greven et al. (2005), who studied the spatial Kingman coalescent in this context.
arXiv:math/0511536v1 fatcat:eys62h4gdnd5vgfjdocif5dqaa

Efficient Random Walks on Riemannian Manifolds [article]

Michael Herrmann, Simon Schwarz, Anja Sturm, Max Wardetzky
2022 arXiv   pre-print
According to a version of Donsker's theorem, geodesic random walks on Riemannian manifolds converge to the respective Brownian motion. From a computational perspective, however, evaluating geodesics can be quite costly. We therefore introduce approximate geodesic random walks based on the concept of retractions. We show that these approximate walks converge to the correct Brownian motion in the Skorokhod topology as long as the geodesic equation is approximated up to second order. As a result
more » ... obtain an efficient algorithm for sampling Brownian motion on compact Riemannian manifolds.
arXiv:2202.00959v1 fatcat:gkk5bz3chnetxkmzbwhfn3xqcu

The Algebraic Approach to Duality: An Introduction [article]

Anja Sturm, Jan M. Swart, Florian Völlering
2018 arXiv   pre-print
This survey article gives an elementary introduction to the algebraic approach to Markov process duality, as opposed to the pathwise approach. In the algebraic approach, a Markov generator is written as the sum of products of simpler operators, which each have a dual with respect to some duality function. We discuss at length the recent suggestion by Giardin\'a, Redig, and others, that it may be a good idea to choose these simpler operators in such a way that they form an irreducible
more » ... ion of some known Lie algebra. In particular, we collect the necessary background on representations of Lie algebras that is crucial for this approach. We also discuss older work by Lloyd and Sudbury on duality functions of product form and the relation between intertwining and duality.
arXiv:1802.07150v1 fatcat:xcewlvtlwbeuzmro7ta3a5qouy

On spatial coalescents with multiple mergers in two dimensions [article]

Benjamin Heuer, Anja Sturm
2012 arXiv   pre-print
This extends in particular the work of Limic and Sturm (2006) , which dealt with the case d ≥ 3. The two dimensional case considered here is biologically the most relevant.  ...  This is in contrast to results in d ≥ 3 as in Limic and Sturm (2006) , where there is a nontrivial distribution of ancestral lines (far apart from each other) for all large L.  ... 
arXiv:1209.5437v1 fatcat:dflbjlr3jnfpxjpj5itzwjdjky

Tightness of voter model interfaces [article]

Anja Sturm, Jan M. Swart
2007 arXiv   pre-print
Consider a long-range, one-dimensional voter model started with all zeroes on the negative integers and all ones on the positive integers. If the process obtained by identifying states that are translations of each other is positively recurrent, then it is said that the voter model exhibits interface tightness. In 1995, Cox and Durrett proved that one-dimensional voter models exhibit interface tightness if their infection rates have a finite third moment. Recently, Belhaouari, Mountford, and
more » ... le have improved this by showing that a finite second moment suffices. The present paper gives a new short proof of this fact. We also prove interface tightness for a long range swapping voter model, which has a mixture of long range voter model and exclusion process dynamics.
arXiv:0706.4405v2 fatcat:uenlq7mzmnczti6ganlj77dvcu

Mechanisms to synchronize neuronal activity

Anja K. Sturm, Peter König
2001 Biological cybernetics  
Temporal aspects of neuronal activity have received increasing attention in recent years. Oscillatory dynamics and the synchronization of neuronal activity are hypothesized to be of functional relevance to information processing in the brain. Here we review theoretical studies of single neurons at dierent levels of abstraction, with an emphasis on the implications for properties of networks composed of such units. We then discuss the in¯uence of dierent types of couplings and choices of
more » ... rs to the existence of a stable state of synchronous or oscillatory activity. Finally we relate these theoretical studies to the available experimental data, and suggest future lines of research.
doi:10.1007/s004220000209 pmid:11252634 fatcat:dbtf57mw6vazlgxt5ydefkamga

The spatial $\Lambda$-coalescent

Vlada Limic, Anja Sturm
2006 Electronic Journal of Probability  
This paper extends the notion of the Λ-coalescent of Pitman (1999) to the spatial setting. The partition elements of the spatial Λ-coalescent migrate in a (finite) geographical space and may only coalesce if located at the same site of the space. We characterize the Λ-coalescents that come down from infinity, in an analogous way to Schweinsberg (2000) . Surprisingly, all spatial coalescents that come down from infinity, also come down from infinity in a uniform way. This enables us to study
more » ... e-time asymptotics of spatial Λ-coalescents on large tori in d ≥ 3 dimensions. Some of our results generalize and strengthen the corresponding results in Greven et al. (2005) concerning the spatial Kingman coalescent.
doi:10.1214/ejp.v11-319 fatcat:gi42klbggzdfxj3yjyxcksn76i

Tightness of voter model interfaces

Anja Sturm, Jan Swart
2008 Electronic Communications in Probability  
Consider a long-range, one-dimensional voter model started with all zeroes on the negative integers and all ones on the positive integers. If the process obtained by identifying states that are translations of each other is positively recurrent, then it is said that the voter model exhibits interface tightness. In 1995, Cox and Durrett proved that one-dimensional voter models exhibit interface tightness if their infection rates have a finite third moment. Recently, Belhaouari, Mountford, and
more » ... le have improved this by showing that a finite second moment suffices. The present paper gives a new short proof of this fact. We also prove interface tightness for a long range swapping voter model, which has a mixture of long range voter model and exclusion process dynamics.
doi:10.1214/ecp.v13-1360 fatcat:k26eok6r4ffrdgxbttodj4463m

On pathwise uniqueness for stochastic heat equations with non-Lipschitz coefficients [article]

Leonid Mytnik, Edwin Perkins, Anja Sturm
2005 arXiv   pre-print
For the details of the proof we refer to Sturm [Stu02] Proposition 3.2.3. There, the setting is a bit different as it works in the setting of Remark 1.3 with bounded k.  ...  (b) In the case where the correlation kernel is bounded, existence has been shown for more general initial conditions and solution spaces in Sturm [Stu03] : Define L p λ (R d ) := L p (R d , e −λ|x| dx  ... 
arXiv:math/0507545v1 fatcat:aaf56jdfojavpa3cubth5jqlce

New results on pathwise uniqueness for the heat equation with colored noise [article]

Thomas Rippl, Anja Sturm
2012 arXiv   pre-print
We discuss improvements of the sufficient conditions obtained in Mytnik, Perkins and Sturm (2006) that relate the Hölder coefficient with the singularity of the correlation kernel of the noise.  ... 
arXiv:1212.4352v1 fatcat:o74fxnv53vakbptemqyqootody

Coalescent results for diploid exchangeable population models [article]

Matthias Birkner, Huili Liu, Anja Sturm
2018 arXiv   pre-print
We consider diploid bi-parental analogues of Cannings models: in a population of fixed size N the next generation is composed of V_i,j offspring from parents i and j, where V=(V_i,j)_1< i≠ j < N is a (jointly) exchangeable (symmetric) array. Every individual carries two chromosome copies, each of which is inherited from one of its parents. We obtain general conditions, formulated in terms of the vector of the total number of offspring to each individual, for the convergence of the properly
more » ... d ancestral process for an n-sample of genes towards a (Ξ-)coalescent. This complements Möhle and Sagitov's (2001) result for the haploid case and sharpens the profile of Möhle and Sagitov's (2003) study of the diploid case, which focused on fixed couples, where each row of V has at most one non-zero entry. We apply the convergence result to several examples, in particular to two diploid variations of Schweinsberg's (2003) model, leading to Beta-coalescents with two-fold and with four-fold mergers, respectively.
arXiv:1709.02563v2 fatcat:cipfx3qyjzfw7ioq3kwjzpk7am

Random Function Iterations for Consistent Stochastic Feasibility [article]

Neal Hermer, D. Russell Luke, Anja Sturm
2018 arXiv   pre-print
We study the convergence of stochastic fixed point iterations in the consistent case (in the sense of Butnariu and Flåm (1995)) in several different settings, under decreasingly restrictive regularity assumptions of the fixed point mappings. The iterations are Markov chains and, for the purposes of this study, convergence is understood in very restrictive terms. We show that sufficient conditions for geometric (linear) convergence in expectation of stochastic projection algorithms presented in
more » ... edić (2011), are in fact necessary for geometric (linear) convergence in expectation more generally of iterated random functions.
arXiv:1808.05426v2 fatcat:awmknstttzcdjlurokkwfw72zm

Nonexpansive Markov Operators and Random Function Iterations for Stochastic Fixed Point Problems [article]

Neal Hermer, D. Russell Luke, Anja Sturm
2022 arXiv   pre-print
We study the convergence of random function iterations for finding an invariant measure of the corresponding Markov operator. We call the problem of finding such an invariant measure the stochastic fixed point problem. This generalizes earlier work studying the stochastic feasibility problem, namely, to find points that are, with probability 1, fixed points of the random functions . When no such points exist, the stochastic feasibility problem is called inconsistent, but still under certain
more » ... mptions, the more general stochastic fixed point problem has a solution and the random function iterations converge to an invariant measure for the corresponding Markov operator. We show how common structures in deterministic fixed point theory can be exploited to establish existence of invariant measures and convergence in distribution of the Markov chain. This framework specializes to many applications of current interest including, for instance, stochastic algorithms for large-scale distributed computation, and deterministic iterative procedures with computational error. The theory developed in this study provides a solid basis for describing the convergence of simple computational methods without the assumption of infinite precision arithmetic or vanishing computational errors.
arXiv:2205.15897v1 fatcat:ygn24xz6tbh2fli2ycwurluewi

Subcritical contact processes seen from a typical infected site

Anja Sturm, Jan Swart
2014 Electronic Journal of Probability  
E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. Abstract What is the long-time behavior of the law of a contact process started with a single infected site, distributed according to counting measure on the lattice? This question is related to the configuration as seen from a typical infected site and gives rise to the definition of so-called eigenmeasures, which are possibly infinite measures on the set of nonempty configurations that are preserved under the dynamics up to
more » ... a time-dependent exponential factor. In this paper, we study eigenmeasures of contact processes on general countable groups in the subcritical regime. We prove that in this regime, the process has a unique spatially homogeneous eigenmeasure. As an application, we show that the law of the process as seen from a typical infected site, chosen according to a Campbell law, converges to a long-time limit. We also show that the exponential decay rate of the expected number of infected sites is continuously differentiable and strictly increasing as a function of the recovery rate, and we give a formula for the derivative in terms of the long time limit law of the process as seen from a typical infected site. Subcritical contact processes seen from a typical infected site a are symmetric, or more generally if one has a(i, j) > 0 iff a(j, i) > 0, then all three conditions are equivalent. In general, irreducibility implies (1.3) which implies weak irreducibility, but none of the converse implications hold. It is well-known that contact processes can be constructed by a graphical representation. Let ω = (ω r , ω i ) be a pair of independent, locally finite random subsets of Λ × R and Λ × Λ × R, respectively, produced by Poisson point processes with intensity δ and a(i, j), respectively. This is usually visualized by plotting Λ horizontally and R vertically, marking points (i, s) ∈ ω r with a recovery symbol (e.g., * ), and drawing an infection arrow from (i, t) to (j, t) for each (i, j, t) ∈ ω i . For any (i, s), (j, u) ∈ Λ × R with s ≤ u, by definition, an open path from (i, s) to (j, u) is a cadlag function π : [s, u] → Λ such that {(π(t), t) : t ∈ [s, u]} ∩ ω r = ∅ and (π(t−), π(t), t) ∈ ω i whenever π(t−) = π(t). Thus, open paths must avoid recovery symbols and may follow infection arrows. We write
doi:10.1214/ejp.v19-2904 fatcat:4c4fridbcjbsljcgr35xqcariq

Random Function Iterations for Stochastic Fixed Point Problems [article]

Neal Hermer, D. Russell Luke, Anja Sturm
2022 arXiv   pre-print
This generalizes earlier work studying the stochastic feasibility problem}, namely, to find points that are, with probability 1, fixed points of the random functions [Hermer, Luke, Sturm, 2019].  ... 
arXiv:2007.06479v2 fatcat:k747glaxsbfodexn74zp2ojb7q
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