Marked Branching Diffusions

2013 ◽  
pp. 417-464
Keyword(s):  
Branching Diffusions

Bisexual Branching Diffusions

1993 ◽  
Author(s):  
Leonid Mytnik ◽  
Robert J. Adler
Keyword(s):  
Branching Diffusions

Bisexual branching diffusions

1995 ◽  
Vol 27 (04) ◽  
pp. 980-1018
Author(s):  
Leonid Mytnik ◽  
Robert J. Adler
Keyword(s):  
Martingale Problem ◽  
Dimensional System ◽  
Brownian Particles ◽  
Branching Law ◽  
Limiting Behaviour ◽  
General State Space ◽  
New Methodologies ◽  
The Individual ◽  
Branching Diffusions ◽  
Two Populations

We study the limiting behaviour of large systems of two types of Brownian particles undergoing bisexual branching. Particles of each type generate individuals of both types, and the respective branching law is asymptotically critical for the two-dimensional system, while being subcritical for each individual population.The main result of the paper is that the limiting behaviour of suitably scaled sums and differences of the two populations is given by a pair of measure and distribution valued processes which, together, determine the limit behaviours of the individual populations.Our proofs are based on the martingale problem approach to general state space processes. The fact that our limit involves both measure and distribution valued processes requires the development of some new methodologies of independent interest.

scholarly journals Hybrid PDE solver for data-driven problems and modern branching

2017 ◽  
Vol 28 (6) ◽  
pp. 949-972 ◽  
Author(s):  
FRANCISCO BERNAL ◽  
GONÇALO DOS REIS ◽  
GREIG SMITH
Keyword(s):  
High Performance ◽  
Large Scale ◽  
Data Science ◽  
Data Driven ◽  
Stochastic Representation ◽  
Non Linear ◽  
Research Questions ◽  
Stochastic Representations ◽  
Branching Diffusions ◽  
Pde Solvers

The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgement in the modelling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for non-linear PDEs using branching diffusions, which have significantly broadened the scope of PDD. We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods. We close this work with an invitation to the fully non-linear case and open research questions.

scholarly journals Boundedness of one-dimensional branching Markov processes

1997 ◽  
Vol 10 (4) ◽  
pp. 307-332 ◽  
Author(s):  
F. I. Karpelevich ◽  
Yu. M. Suhov
Keyword(s):  
Markov Processes ◽  
Necessary Conditions ◽  
Random Variable ◽  
Expected Number ◽  
One Dimensional ◽  
Sufficient And Necessary Conditions ◽  
Type Boundary ◽  
The One ◽  
Branching Diffusions ◽  
Jump Markov Processes

A general model of a branching Markov process on ℝ is considered. Sufficient and necessary conditions are given for the random variable M=supt≥0max1≤k≤N(t)Ξk(t) to be finite. Here Ξk(t) is the position of the kth particle, and N(t) is the size of the population at time t. For some classes of processes (smooth branching diffusions with Feller-type boundary points), this results in a criterion stated in terms of the linear ODEσ2(x)2f″(x)+a(x)f′(x)=λ(x)(1−k(x))f(x). Here σ(x) and a(x) are the diffusion coefficient and the drift of the one-particle diffusion, respectively, and λ(x) and k(x) the intensity of branching and the expected number of offspring at point x, respectively. Similarly, for branching jump Markov processes the conditions are expressed in terms of the relations between the integral μ(x)∫π(x,dy)(f(y)−f(x)) and the product λ(x)(1−k(x))f(x), where λ(x) and k(x) are as before, μ(x) is the intensity of jumping at point x, and π(x,dy) is the distribution of the jump from x to y.

Singular Spacetime Itô's Integral and a Class of Singular Interacting Branching Particle Systems

2003 ◽  
Vol 06 (02) ◽  
pp. 321-335 ◽  
Author(s):  
Hao Wang
Keyword(s):  
Brownian Motion ◽  
Unique Solution ◽  
Martingale Problem ◽  
Particle Systems ◽  
Dual Process ◽  
Singular Function ◽  
Singular Coefficient ◽  
Duality Method ◽  
Super Brownian Motion ◽  
Branching Diffusions

In Wang,8 a class of interacting measure-valued branching diffusions [Formula: see text] with singular coefficient were constructed and characterized as a unique solution to ℒε-martingale problem by a limiting duality method since in this case the dual process does not exist. In this paper, we prove that for any ε ≠ 0 the superprocess with singular motion coefficient is just the super-Brownian motion. The singular motion coefficient is handled as a sequential limit motivated by Antosik et al.1 Thus, the limiting superprocess is investigated and identified as the motion coefficient converges to a singular function. The representation of the singular spacetime Itô's integral is derived.

Sign in / Sign up

Export Citation Format

Share Document