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# 数学代写|随机过程代写Stochastic Porcess代考|MATH544

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## 数学代写|随机过程代写Stochastic Porcess代考|Branching Processes with a Continuum of States

If a branching process consists of a large number of particles of various types then relative indices are sometimes introduced to describe the size of the population such as the ratio of the number of particles of a given type and at a given time to a certain parameter which characterizes the order of magnitude of the number of particles in the population. In other analogous situations the size of the population is characterized by the mass of particles of a given type or by the geometrical volume occupied by these particles. In these cases the state of the system can be described by a vector whose dimension is equal to the number of types of particles and whose components are equal to the total volume (mass) of particles of the given type. Unlike the previous case the state of the process is now characterized by an arbitrary non-negative vector. The basic property of a branching process in this case can be stated generally in the following manner: each separate part of the population is transformed in the course of time independently of the evolution of the remaining part of the population.

In certain cases populations with an arbitrary set of particle types are of interest. For example, along with the inner and sharp differences among the particles, i.e. the differences in the characteristics which previously (in the discrete case) were referred to as the type of a particle, the type of the particle is also characterized by some other parameter (such as energy, or geometrical position in the space) which takes on infinitely many values.

In this section populations are considered which consist of $m$ types of continuous components (masses), while in the next section, processes with a finite number of particles belonging to a finite number of various types are studied; however, it is also assumed that each particle in the population moves in a phase space in accordance with a certain Markov process.

Let $E$ be an arbitrary set of types of particles and let $\mathscr{E}$ be the $\sigma$-algebra of subsets of $E$ containing singletons ${e}$ where $e$ is a point belonging to $E$.

We introduce the space $\mathrm{B}^*(\mathscr{E})$ of all bounded complex-valued $\mathscr{E}$-measurable functions on $E$ and the space $\mathscr{M}(\mathscr{E})\left(\mathscr{M}_{+}(\mathscr{E})\right)$ of all finite charges (measures) on $\mathscr{E}$.

## 数学代写|随机过程代写Stochastic Porcess代考|General Markov Processes with Branching

Up until now we have studied branching processes in which the state of the process was completely determined by the number of particles of each type and (in certain cases) by the “age” of the particles. In many cases, however, one must also take into account the position of the particle in a certain phase space to describe the evolution of the system and moreover, this position varies continuously in time, while the duration of life of a particle and the probabilities of its changes (transformations) depend on its trajectories in the phase space. The basic property of a branching process, the independence of the evolution of the individual particles, is however retained. The present section is devoted to processes of this kind.
The constructive description of a process. Let a certain measurable space ${\mathscr{X}, \mathfrak{B}}$, called the phase space of the process, be given. In this phase space particles of $m$ types $\left(T_1, T_2, \ldots, T_m\right)$ are in motion. The number of particles of each type may vary from 0 to $\infty$. If at a certain instant of time the total number of particles becomes $\infty$, then the process is cut-off at that instant. If at the initial time there is exactly one particle in the phase space and its type is $T_k$, then this particle moves along the trajectory of a certain generally cut-off homogeneous Markov process $\left{\mathscr{F}^{(k)}, \mathscr{N}^{(k)}, \mathrm{P}x^{(k)}\right}$. Denote by $\zeta_k$ the cut-off moment of the process. Then at the moment $\zeta_k$, in place of one particle of type $T_k, n_1, n_2, \ldots, n_m$ particles of the types $T_1, \ldots, T_m$ respectively appear in the phase space and, moreover, the positions of the particles of type $T_i$ are the points $x_1^{(i)}, \ldots, x{n_i}^{(i)}$ respectively. These may be characterized by the random measure $\mu^i$ on ${\mathscr{X}, \mathfrak{B}}$ defined by
$$\mu^i(B)=\sum_{k=1}^{n_i} \chi_B\left(x_k^{(i)}\right)$$

# 随机过程代写

## 数学代写|随机过程代写Stochastic Porcess代考|General Markov Processes with Branching

$$\mu^i(B)=\sum_{k=1}^{n_i} \chi_B\left(x_k^{(i)}\right)$$

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