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Mutually Catalytic Super Branching Random Walks: Large Finite Systems and Renormalization Analysis

Large Finite Systems and Renormalization Analysis

We study features of the longtime behavior and the spatial continuum limit for the diffusion limit of the following particle model. Consider populations consisting of two types of particles located on sites labeled by a countable group. The populations of each of the types evolve as follows: Each particle performs a random walk and dies or splits in two with probability $\frac{1}{2}$ and the branching rates of a particle of each type at a site $x$ at time $t$ is proportional to the size of the population at $x$ at time $t$ of the other type. The diffusion limit of ``small mass, large number of initial particles'' is a pair of two coupled countable collections of interacting diffusions, the mutually catalytic super branching random walk. Consider now increasing sequences of finite subsets of sites and define the corresponding finite versions of the process. We study the evolution of these large finite spatial systems in size-dependent time scales and compare them with the behavior of the infinite systems, which amounts to establishing the so-called finite system scheme. A dichotomy is known between transient and recurrent symmetrized migrations for the infinite system, namely, between convergence to equilibria allowing for coexistence in the first case and concentration on monotype configurations in the second case. Correspondingly we show (i) in the recurrent case both large finite and infinite systems behave similar in all time scales, (ii) in the transient case we see for small time scales a behavior resembling the one of the infinite system, whereas for large time scales the system behaves as in the finite case with fixed size and finally in intermediate scales interesting behavior is exhibited, the system diffuses through the equilibria of the infinite system which are indexed by the pair of intensities and this diffusion process can be described as mutually catalytic diffusion on $(\mathbb{R}^+)^2$. At the same time, the above finite system asymptotics can be applied to mean-field systems of $N$ exchangeable mutually catalytic diffusions. This is the building block for a renormalization analysis of the spatially infinite hierarchical model and leads to an association of this system with the so-called interaction chain, which reflects the behavior of the process on large space-time scales. Similarly we introduce the concept of a continuum limit in the hierarchical mean field limit and show that this limit always exists and that the small-scale properties are described by another Markov chain called small scale characteristics. Both chains are analyzed in detail and exhibit the following interesting effects. The small scale properties of the continuum limit exhibit the dichotomy, overlap or segregation of densities of the two populations, as a function of the underlying random walk kernel. A corresponding concept to study hot spots is presented. Next we look in the transient regime for global equilibria and their equilibrium fluctuations and in the recurrent regime on the formation of monotype regions. For particular migration kernels in the recurrent regime we exhibit diffusive clustering, which means that the sizes (suitably defined) of monotype regions have a random order of magnitude as time proceeds and its distribution is explicitly identifiable. On the other hand in the regime of very large clusters we identify the deterministic order of magnitude of monotype regions and determine the law of the random size. These two regimes occur for different migration kernels than for the cases of ordinary branching or Fisher-Wright diffusion. Finally we find a third regime of very rapid deterministic spatial cluster growth which is not present in other models just mentioned. A further consequence of the analysis is that mutually catalytic branching has a fixed point property under renormalization and gives a natural example different from the trivial case of multitype models consisting of two independent versions of the fixed points for the one type case.

[DF1] [DF2 [DG1] [DG2 [DG3) [DG4] [DG5] (DGV [DGW] [DM) [FG [FK] D.A.
Dawson, K. Fleischmann, A continuous super-Brownian motion in a super-
Brownian medium, J. Theoret. Probab., 10(1), 213-276 (1997). D.A. Dawson, K.
Fleischmann, ...

Child Development , An Active Learning Approach

Psychology, Human development

Facts101 is your complete guide to Child Development , An Active Learning Approach. In this book, you will learn topics such as as those in your book plus much more. With key features such as key terms, people and places, Facts101 gives you all the information you need to prepare for your next exam. Our practice tests are specific to the textbook and we have designed tools to make the most of your limited study time.

Ide Penataan Interior hunian tipe Studio

Hunian tipe studio adalah sebuah hunian berbentuk ruang kosong tanpa penyekat sebagai bagian unit dari hunian bertingkat. Ruangan tersebut dapat digunakan untuk berbagai aktivitas. Rumah kompak untuk ditinggali sendiri ataupun keluarga baru tersebut kini semakin marak dan menjadi incaran konsumen. GRIYA KREASI

Selain aktif menulis beberapa jurnal ilmiah untuk majalah di lingkungan institusi
pendidikan, narasumber talkshow di beberapa media massa dan jejaring sosial,
penulis juga turut berpartisipasi dalam program yang diadakan oleh pemerintah
 ...

Matematika sebagai alat bantu - Persiapan Olimpiade Fisika

Matematika sebagai alat bantu

Buku ‘Matematika Sebagai Alat Bantu’ adalah salah satu dari Seri Bahan Persiapan Olimpiade Fisika SMA yang dibahas oleh pakar pembimbing Olimpiade Nasional Prof. Yohanes Surya, Ph.D. Materi Matematika yang dibahas terfokus pada materi pendukung konsep Fisika. Di setiap akhir bab terdapat soal-soal yang dibahas secara sistematis dan detail. Beberapa pembahasan soal disertai alternatif pemecahan soal. Hal ini dapat memberikan wawasan lebih luas dan pemahaman yang lebih kuat pada siswa. Buku ini tidak hanya baik digunakan oleh para pelajar yang akan mengikuti Olimpiade Fisika, tapi baik digunakan oleh siapa saja yang ingin memahami Fisika dengan cara yang sistematis dan menyenangkan.

Buku ‘Matematika Sebagai Alat Bantu’ adalah salah satu dari Seri Bahan Persiapan Olimpiade Fisika SMA yang dibahas oleh pakar pembimbing Olimpiade Nasional Prof.

Strategi pemimpin pasar

Strategy for marketing various business in Indonesia; collection of articles.

Produk membutuhkan strategi pemasaran, finansial dan manufakturing yang
berbeda untuk setiap tahapan. Siklus hidup produk untuk produk industri
biasanya lebih panjang dibandingkan dengan produk konsumen, karena
tingginya ...