Renormalization of Hierarchically Interacting Isotropic Diffusions

Hollander, W.Th.F. den; Swart, Jan M.

doi:10.1023/b:joss.0000026734.93723.b9

Cited by 9 publications

(8 citation statements)

References 19 publications

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“…Thus, we may say that Theorem 2 describes a situation in which the systems exhibits 'large time-scale universality', where we borrow a term from renormalization theory. For a renormalization analysis of models of the type in (1.1) we refer to [1,2,[9][10][11]17]. We note that the 'universal object' found in this work coincides with our w * in (2.15).…”

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confidence: 76%

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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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Section: )supporting

confidence: 76%

“…We note that the 'universal object' found in this work coincides with our w * in (2.15). In particular, the function g * in (3.9) occurred first in [17].…”

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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“…In terms of R n we may write K n (ξ) = E τ 1 ,...,τn exp n Ω (π 1 R n )(dy) log P |y| (0, Σy) P 2|y| (0, 0) , (4. 33) where π 1 R n is the projection of R n onto the first coordinate. Step 4.…”

Section: 11mentioning

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%

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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“…The latter has been the object of intense study. A sample of relevant papers and overviews is [Shi80], [Daw93], [DG93], [DGV95], [DG96], [EF96], [HS98], [Hol06], [DGdH + 08], [GdHKK14]. We expect the presence of the seed-bank to affect the long-time behaviour of the system not only quantitatively but also qualitatively.…”

Section: Background and Outline 1background And Goalsmentioning

confidence: 99%