Infinite rate mutually catalytic branching

Klenke, Achim; Mytnik, Leonid

doi:10.1214/09-aop520

Cited by 10 publications

(20 citation statements)

References 18 publications

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“…It is clear that (Λ t ) t≥0 is increasing with Λ 0 = 0. We check condition (14): By [19], Theorem 5 (see also [18], Theorem 1.1(b)), we can find a sequence γ k ↑ ∞ and a set I ⊆ (0, ∞) of full Lebesgue measure such that the finite dimensional distributions of (Λ [γ k ] t ) t∈I converge weakly to those of (Λ t ) t∈I as k → ∞. Fix t ∈ I.…”

Section: Properties Of Limit Pointsmentioning

confidence: 99%

“…Proof of Proposition 5.1. By Definition 1.8, there exist increasing processes (Λ t ) t≥0 ∈ D [0,∞) (M tem ) and (Λ t ) t≥0 ∈ D [0,∞) (M rap ), with Λ 0 = Λ 0 = 0 and satisfying (14), such that for all test functions, expression (15) is a martingale. For the purposes of the proof, we may assume that (µ, ν, Λ) and (μ,ν,Λ) are defined on a common sample space Ω and are independent of each other.…”

mentioning

confidence: 99%

“…By assumption (54), the integrand in the above display converges to 0 for all x ∈ R and almost all r ∈ [0, T ] as ε ↓ 0. Hence using conditions (14) and (55) together with dominated convergence, we are done. The argument for the second term in the difference is completely analogous.…”

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confidence: 99%

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The scaling limit of the interface of the continuous-space symbiotic branching model

2016

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The continuous-space symbiotic branching model describes the evolution of two interacting populations that can reproduce locally only in the simultaneous presence of each other. If started with complementary Heaviside initial conditions, the interface where both populations coexist remains compact. Together with a diffusive scaling property, this suggests the presence of an interesting scaling limit. Indeed, in the present paper, we show weak convergence of the diffusively rescaled populations as measure-valued processes in the Skorokhod, respectively the Meyer-Zheng, topology (for suitable parameter ranges). The limit can be characterized as the unique solution to a martingale problem and satisfies a "separation of types" property. This provides an important step toward an understanding of the scaling limit for the interface. As a corollary, we obtain an estimate on the moments of the width of an approximate interface. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Probability, 2016, Vol. 44, No. 2, 807-866. This reprint differs from the original in pagination and typographic detail. 1 4 J. BLATH, M. HAMMER AND M. ORTGIESE before Corollary 1.2. Recently, analogous results have been derived by Döring and Mytnik in the case ̺ ∈ (−1, 1) in [9, 10]. Returning to the continuous-space set-up, for ̺ = −1 (the stepping stone model) Tribe [27] proves a "functional limit theorem": For a pair of (continuous) functions (u, v), define R(u, v) := sup{x : u(x) > 0}, L(u, v) = inf{x : v(x) > 0}. (5)Note that for a solution (u t , v t ) t≥0 of the symbiotic branching model, the interface at time t is contained in the interval [L(u t , v t ), R(u t , v t )]. It is proved in [27] for ̺ = −1 and for continuous initial conditions u 0 = 1 − v 0 which satisfy −∞ < L(u 0 , v 0 ) ≤ R(u 0 , v 0 ) < ∞ that under Brownian rescaling, the motion of the position of the right endpoint of the interface t → 1 n R(u n 2 t , 1 − u n 2 t ), t ≥ 0, converges to a Brownian motion as n → ∞.The above results suggest the existence of an interesting diffusive scaling limit for the continuous-space symbiotic branching model (and its interface) for ̺ > −1. This is the starting point of our investigation. However, compared to the case ̺ = −1, the situation is more involved here: For example, the total mass of the solution is not necessarily bounded, and in particular, moments of the solution may diverge as t → ∞, depending on ̺. For instance, second moments diverge for ̺ ≥ 0. In order to state this result, which was obtained in [3]

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Section: Properties Of Limit Pointsmentioning

confidence: 99%

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confidence: 99%

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The scaling limit of the interface of the continuous-space symbiotic branching model

2016

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“…For ̺ ≥ 0, it is still an open problem how to construct the infinite rate limit of this model. If we replace the real line as site space by some discrete space and replace 1 2 ∂ 2 x by the generator of some Markov chain on this site space, then for ̺ = 0 the infinite rate process was studied in great detail in [KM10], [KM12a], [KM12b] and [KO10]. The main tool for showing (weak) uniqueness for the solutions of (1.1) is a self-duality relation that goes back to Mytnik [Myt98] for the case ̺ = 0 and Etheridge and Fleischmann [EF04] in the case ̺ = 0.…”

Section: Introduction 1motivation and First Resultsmentioning

confidence: 99%

Infinite rate symbiotic branching on the real line: The tired frogs model

Klenke¹,

Mytnik²

2020

Ann. Inst. H. Poincaré Probab. Statist.

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Consider a population of infinitesimally small frogs on the real line. Initially the frogs on the positive half-line are dormant while those on the negative half-line are awake and move according to the heat flow. At the interface, the incoming wake frogs try to wake up the dormant frogs and succeed with a probability proportional to their amount among the total amount of involved frogs at the specific site. Otherwise, the incoming frogs also fall asleep. This frog model is a special case of the infinite rate symbiotic branching process on the real line with different motion speeds for the two types.We construct this frog model as the limit of approximating processes and compute the structure of jumps. We show that our frog model can be described by a stochastic partial differential equation on the real line with a jump type noise. * f, g = f, Ag for all suitable f, g and where f, g = i∈S f (i)g(i). Let E := [0, ∞) 2 \ (0, ∞) 2 andBy a solution of the martingale problem MP S we understand an E-valued Markov process (X 1 , X 2 ) with càdlàg paths such thatfor some orthogonal zero mean martingales M (k), k ∈ S. As usual, uniqueness of the solution to a martingale problem means uniqueness in law.

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“…The analogous statement holds for x = (0, x 2 ) ∈ E. The measure ν x can be thought of as the prototypic measure for jumps away from x when there is an immigration of the respective other type. Due to symmetry and a scaling relation, all the measures ν x are simple transformations (described below implicitly, see also [KM10], discussion before (5.5)) of the measure ν := ν (1,0) . This measure ν on E can be explicitly described in terms of its Lebesgue densities…”

Section: Letmentioning

confidence: 99%

Finite system scheme for mutually catalytic branching with infinite branching rate

Döring¹,

Klenke²,

Mytnik³

2017

Ann. Appl. Probab.

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For many stochastic diffusion processes with mean field interaction, convergence of the rescaled total mass processes towards a diffusion process is known.Here we show convergence of the so-called finite system scheme for interacting jump-type processes known as mutually catalytic branching processes with infinite branching rate. Due to the lack of second moments the rescaling of time is different from the finite rate mutually catalytic case. The limit of rescaled total mass processes is identified as the finite rate mutually catalytic branching diffusion. The convergence of rescaled processes holds jointly with convergence of coordinate processes, where the latter converge at a different time scale.

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Infinite rate mutually catalytic branching

Cited by 10 publications

References 18 publications

The scaling limit of the interface of the continuous-space symbiotic branching model

The scaling limit of the interface of the continuous-space symbiotic branching model

Infinite rate symbiotic branching on the real line: The tired frogs model

Finite system scheme for mutually catalytic branching with infinite branching rate

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