Infinite rate mutually catalytic branching in infinitely many colonies: construction, characterization and convergence

Klenke, Achim; Mytnik, Leonid

doi:10.1007/s00440-011-0376-1

Cited by 13 publications

(36 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The claimed universality of MCB(∞) was also established in Theorem 1.5 of [KM12]: In fact, the diffusion function is not necessarily g(u, v) = uv as for mutually catalytic branching. The diffusion function only needs to vanish on E and be positive on (0, ∞) 2 .…”

Section: Letmentioning

confidence: 75%

“…Recall from (1.3) the transition semigroup e tA N of A N . By Lemma 3.7 of [KM12], we get the mixed second moment bound…”

Section: Proof Of Theoremmentioning

confidence: 93%

“…The idea is that each coordinate X N t (k) experiences a drift towards the mean of all coordinates. In addition, it is a (non-trivial) consequence of the particular form of the jump function J that solutions are forced to remain only at the boundary E of [0, ∞) 2 : Jumps go from E to E and the compensator cancels with the drift (compare Section 2 of [KM12]). Also note that the dr-contribution does not play a role for the jump target but instead only determines the jump rate which is proportional to I N .…”

Section: Letmentioning

confidence: 99%

See 2 more Smart Citations

Finite system scheme for mutually catalytic branching with infinite branching rate

Döring¹,

Klenke²,

Mytnik³

2017

Ann. Appl. Probab.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Letmentioning

confidence: 75%

“…Recall from (1.3) the transition semigroup e tA N of A N . By Lemma 3.7 of [KM12], we get the mixed second moment bound…”

Section: Proof Of Theoremmentioning

confidence: 93%

Section: Letmentioning

confidence: 99%

See 1 more Smart Citation

Finite system scheme for mutually catalytic branching with infinite branching rate

Döring¹,

Klenke²,

Mytnik³

2017

Ann. Appl. Probab.

Self Cite

View full text Add to dashboard Cite

show abstract

“…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.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…In two forthcoming papers, we construct the infinite rate process on a countable site space S via a stochastic differential equation with jump-type noise and give a characterization via a martingale problem [9]. Furthermore, we will investigate the long-time behaviour and give conditions for segregation and for coexistence of types [10].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Infinite rate mutually catalytic branching

Klenke¹,

Mytnik²

2010

Ann. Probab.

Self Cite

View full text Add to dashboard Cite

Consider the mutually catalytic branching process with finite branching rate $\gamma$. We show that as $\gamma\to\infty$, this process converges in finite-dimensional distributions (in time) to a certain discontinuous process. We give descriptions of this process in terms of its semigroup in terms of the infinitesimal generator and as the solution of a martingale problem. We also give a strong construction in terms of a planar Brownian motion from which we infer a path property of the process. This is the first paper in a series or three, wherein we also construct an interacting version of this process and study its long-time behavior.Comment: Published in at http://dx.doi.org/10.1214/09-AOP520 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

show abstract

Infinite rate mutually catalytic branching in infinitely many colonies: construction, characterization and convergence

Cited by 13 publications

References 12 publications

Finite system scheme for mutually catalytic branching with infinite branching rate

Finite system scheme for mutually catalytic branching with infinite branching rate

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

Infinite rate mutually catalytic branching

Contact Info

Product

Resources

About