Fourteenth Conference on Stochastic Processes and their Applications

<span title="">1985</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/oeoum6frfvactev64cas33x6ta" style="color: black;">Stochastic Processes and their Applications</a> </i> &nbsp;
We consider a model of a disordered ferromagnet. At each site i of the square lattice hZ there is a bounded real-valued spin (T{. These spins interact as follows. A typical example of stochastic differential equations is of the form dX, = a(X,) dB, + a(X,) dt, where {X,} is an R'-valued sample continuous process and {B,} is an Rd-valued Wiener process. Our purpose is to discuss an infinite dimensional extension of this equation where the state spaces of X, and B, are Schwartz spaces of
more &raquo; ... ions, because practically all problems appearing in applications can be discussed in this frame-work. In this note we will discuss basic concepts such as random variables, stochastic processes, Wiener processes, stochastic integrals and stochastic differential equations Four/eenth Conference on Stochastic Processes in infinite dimensions and deal with concrete stochastic differential equations from our view-point. A simple coupling argument is used to construct a sequence of general stochastic epidemics, indexed by the initial number of susceptibles, from a time-homogeneous birth-and-death process. This construction provides us with a simple proof of the threshold behaviour of the general stochastic epidemic. The method is quite general and extensions to other epidemic models, such as chain-binomial and host-vector models, are briefly described.
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