"Hao Wang"

Outline

Classical Model for Super-Brownian Motion

Brownian branching particle systems

Spatial motion assumption

Branching mechanism

Proportional rescaling limit

Empirical measure-valued processes

Multiplicative property and infinite divisibility

MuP is equivalent to infinite divisibility

Log-Laplace functional and evolution equation

Structural properties of Super-Brownian motion

SPDE for Super-Brownian motion

New Model, spatial motion assumption

New Model, assumptions

New Model, branching mechanism

Dependency of the particles in the new model

Loss of the multiplicative property

Construction : pregenerator

Construction : existence of solution of MP

Construction : uniqueness of solution of MP

Change the form of the generator

Change the form of the generator

Dual generator

Duality

Duality for singular coefficient

Existence of the density process

Purely-atomic measure state

Degenerate and coalescence

Fellerís criterion of accessibility

Inaccessibility of zero

Commutativity of semigroups

Derivations of SPDEs for density processes

Degenerate SPDE for purely-atomic superprocess

Simple facts

One dimensional case

Multidimensional case

A generalized new model and SDSM

New Model, spatial motion assumption

New Model, branching mechanism

Dependency of the particles in the new model

Construction : pregenerator

Construction : existence of solution of MP

State classification for this location dependent branching model

Generator decomposition technique

Always purely-atomic measures

Variable clock speed

Number of atoms dominated

This is all for this presentation