Interacting Fisher–Wright diffusions in a catalytic medium

Greven, Andreas; Klenke, Achim; Wakolbinger, Anton

doi:10.1007/pl00008777

Cited by 11 publications

(5 citation statements)

References 29 publications

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“…(3) Interacting measure-valued diffusions, for example, Fleming-Viot process [17], mutually catalytic diffusions [18], catalytic interacting diffusions [30].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…(2) Interacting diffusions, for example, Fisher-Wright diffusion [9,10,13,14,24,25,32,33,40,41,44], critical Ornstein-Uhlenbeck process [19,20], Feller's branching diffusion [15,41], parabolic Anderson model with Brownian noise [7]. (3) Interacting measure-valued diffusions, for example, Fleming-Viot process [17], mutually catalytic diffusions [18], catalytic interacting diffusions [30].…”

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confidence: 99%

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Phase transitions for the long-time behavior of interacting diffusions

Greven¹,

Hollander²

2007

Ann. Probab.

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Let $(\{X_i(t)\}_{i\in \mathbb{Z}^d})_{t\geq 0}$ be the system of interacting diffusions on $[0,\infty)$ defined by the following collection of coupled stochastic differential equations: \begin{eqnarray}dX_i(t)=\sum\limits_{j\in \mathbb{Z}^d}a(i,j)[X_j(t)-X_i(t)] dt+\sqrt{bX_i(t)^2} dW_i(t), \eqntext{i\in \mathbb{Z}^d,t\geq 0.}\end{eqnarray} Here, $a(\cdot,\cdot)$ is an irreducible random walk transition kernel on $\mathbb{Z}^d\times \mathbb{Z}^d$, $b\in (0,\infty)$ is a diffusion parameter, and $(\{W_i(t)\}_{i\in \mathbb{Z}^d})_{t\geq 0}$ is a collection of independent standard Brownian motions on $\mathbb{R}$. The initial condition is chosen such that $\{X_i(0)\}_{i\in \mathbb{Z}^d}$ is a shift-invariant and shift-ergodic random field on $[0,\infty)$ with mean $\Theta\in (0,\infty)$ (the evolution preserves the mean). We show that the long-time behavior of this system is the result of a delicate interplay between $a(\cdot,\cdot)$ and $b$, in contrast to systems where the diffusion function is subquadratic. In particular, let $\hat{a}(i,j)={1/2}[a(i,j)+a(j,i)]$, $i,j\in \mathbb{Z}^d$, denote the symmetrized transition kernel. We show that: (A) If $\hat{a}(\cdot,\cdot)$ is recurrent, then for any $b>0$ the system locally dies out. (B) If $\hat{a}(\cdot,\cdot)$ is transient, then there exist $b_*\geq b_2>0$ such that: (B1)d The system converges to an equilibrium $\nu_{\Theta}$ (with mean $\Theta$) if $0b_*$. (B3) $\nu_{\Theta}$ has a finite 2nd moment if and only if $0b_2$. The equilibrium $\nu_{\Theta}$ is shown to be associated and mixing for all $0b_2$. We further conjecture that the system locally dies out at $b=b_*$. For the case where $a(\cdot,\cdot)$ is symmetric and transient we further show that: (C) There exists a sequence $b_2\geq b_3\geq b_4\geq ... >0$ such that: (C1) $\nu_{\Theta}$ has a finite $m$th moment if and only if $0b_m$. (C3) $b_2\leq (m-1)b_m<2$. uad(C4) $\lim_{m\to\infty}(m-1)b_m=c=\sup_{m\geq 2}(m-1)b_m$. The proof of these results is based on self-duality and on a representation formula through which the moments of the components are related to exponential moments of the collision local time of random walks. Via large deviation theory, the latter lead to variational expressions for $b_*$ and the $b_m$'s, from which sharp bounds are deduced. The critical value $b_*$ arises from a stochastic representation of the Palm distribution of the system. The special case where $a(\cdot,\cdot)$ is simple random walk is commonly referred to as the parabolic Anderson model with Brownian noise. This case was studied in the memoir by Carmona and Molchanov [Parabolic Anderson Problem and Intermittency (1994) Amer. Math. Soc., Providence, RI], where part of our results were already established.Comment: Published at http://dx.doi.org/10.1214/00911790600000...

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“…(3) Interacting measure-valued diffusions, for example, Fleming-Viot process [17], mutually catalytic diffusions [18], catalytic interacting diffusions [30].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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confidence: 99%

Phase transitions for the long-time behavior of interacting diffusions

Greven¹,

Hollander²

2007

Ann. Probab.

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“…However, the Wright-Fisher diffusion on domain D = (0, 1) is a diffusion process whose diffusion matrix a(x) = x(1 − x) is not uniformly elliptic. The process has attracted a wide interest in the literature (see for example [32,30,4]). Fleischmann and Swart [30] studied the large-time behaviour of the corresponding superprocess with spatially independent, quadratic branching mechanism on [0, 1].…”

Section: Bounded Domainsmentioning

confidence: 99%

Spines, skeletons and the strong law of large numbers for superdiffusions

Eckhoff¹,

Kyprianou²,

Winkel³

2015

Ann. Probab.

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Consider a supercritical superdiffusion (Xt) t≥0 on a domain D ⊆ R d with branching mechanismThe skeleton decomposition provides a pathwise description of the process in terms of immigration along a branching particle diffusion. We use this decomposition to derive the Strong Law of Large Numbers (SLLN) for a wide class of superdiffusions from the corresponding result for branching particle diffusions. That is, we show that for suitable test functions f and starting measures µ,where W∞ is a finite, non-deterministic random variable characterised as a martingale limit. Our method is based on skeleton and spine techniques and offers structural insights into the driving force behind the SLLN for superdiffusions. The result covers many of the key examples of interest and, in particular, proves a conjecture by Fleischmann and Swart [30] for the super-Wright-Fisher diffusion.

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“…w does not have invariant harmonics, in our terminology. It seems that at present nobody can treat the long-time behavior of (3.12), but Greven, Klenke and Wakolbinger [16] treat a similar case where X 1 is replaced by the voter model. They show that the model clusters 6 when a is nearest neighbor on = ‫ޚ‬ d in the recurrent dimensions d = 1, 2.…”

Section: )mentioning

confidence: 99%

Clustering of linearly interacting diffusions and universality of their long-time limit distribution

Swart¹

2000

Probab Theory Relat Fields

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Let K ⊂ ‫ޒ‬ d (d ≥ 1) be a compact convex set and a countable Abelian group. We study a stochastic process X in K , equipped with the product topology, where each coordinate solves a SDE of the form dXHere a(·) is the kernel of a continuous-time random walk on and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel a S (i) = a(i) + a(−i) is recurrent, then each component X i (∞) is concentrated on {x ∈ K : σ (x) = 0} and the coordinates agree, i.e., the system clusters. Both these statements fail if a S is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components. This distribution turns out to be universal in all recurrent kernels a S on Abelian groups .

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Interacting Fisher–Wright diffusions in a catalytic medium

Cited by 11 publications

References 29 publications

Phase transitions for the long-time behavior of interacting diffusions

Phase transitions for the long-time behavior of interacting diffusions

Spines, skeletons and the strong law of large numbers for superdiffusions

Clustering of linearly interacting diffusions and universality of their long-time limit distribution

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