Dimensional Markov Research Articles

Branching Processes with Immigration and Related Limit Theorems

Branching Processes with Immigration and Related Limit Theorems

On Gaussian noise envelopes

The first-passage time problem for a continuous one-dimensional Markov process is reviewed, and upper bounds are obtained for both the probability of failure (or passage and the moments of the time to failure, in terms of the mean time to failure. In addition, stationary Gaussian variables arising from systems with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> degrees of freedom that have autocorrelation functions of the form \begin{equation} R(r) = e^{-b \mid \tau \mid} \sum_{k=1}^{N} d_k^2 \cos \omega_k \tau \end{equation} are shown to be derivable from a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</tex> -dimensional (or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</tex> - 1, if one of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega_k</tex> is zero) Markov process that possesses a "pseudoenvelope," which is itself the result of a one-dimensional Markov process. This pseudo-envelope can be used as a bound on the magnitude of the Gaussian variable, and its first-passage time problem can be solved explicitly or utilized to obtain convenient bounds for the probability of failure of the Gaussian process.

On Two Dimensional Markov Processes with Branching Property

On Two Dimensional Markov Processes with Branching Property

On two dimensional Markov processes with branching property

Introduction. A continuous state branching Markov process (C.B.P.) was introduced by Jirina [8] and recently Lamperti [10] determined all such processes on the half line. (A quite similar result was obtained independently by the author.) This class of Markov processes contains as a special case the diffusion processes (which we shall call Feller's diffusions) studied by Feller [2]. The main objective of the present paper is to extend Lamperti's result to multi-dimensional case. For simplicity we shall consider the case of 2-dimensions though many arguments can be carried over to the case of higher dimensions(2). In Theorem 2 below we shall characterize all C.B.P.'s in the first quadrant of a plane and construct them. Our construction is in an analytic way, by a similar construction given in Ikeda, Nagasawa and Watanabe [5], through backward equations (or in the terminology of [5] through S-equations) for a simpler case and then in the general case by a limiting procedure. A special attention will be paid to the case of diffusions. We shall show that these diffusions can be obtained as a unique solution of a stochastic equation of Ito (Theorem 3). This fact may be of some interest since the solutions of a stochastic equation with coefficients H65lder continuous of exponent 1/2 (which is our case) are not known to be unique in general. Next we shall examine the behavior of sample functions near the boundaries (x1-axis or x2-axis). We shall explain, for instance, the case of x1-axis. There are two completely different types of behaviors. In the first case x1-axis acts as a pure exit boundary: when a sample function reaches the x1-axis then it remains on it moving as a one-dimensional Feller diffusion up to the time when it hits the origin and then it is stopped. In the second case, there is a point x0 on x1-axis such thaty'2 = (0, x0) acts as a reflecting boundary and L1= (xo, oo) acts as a pure entrance boundary (Theorem 4 and Corollaries).

On the probability density of the output of a low-pass system when the input is a Markov step process

Forward equations are derived for the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(N + 1)</tex> - dimensional Markov process generated when a Markov step signal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{s(t)\}</tex> is the input to an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N^{th}</tex> -order system of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dX/dt = U(X; s)</tex> . As examples, the joint probability densities of input and output are found for a symmetric three-level signal smoothed by an RC low-pass filter, and partial results are obtained for a doubly integrated Rice telegraph signal.

Infinitesimal Operators of Markov Processes

In 1931 A. Kolmogorov showed [9] that a wide class of one dimensional Markov processes can be described by the differential equation \[(1)\qquad \frac{{\partial u}}{{\partial t}} + a\frac{{\partial ^2 u}}{{\partial x^2 }} + b\frac{{\partial u}}{{\partial x}}.\]Are there one-dimensional Markov processes governed by equations of the type \[ (2)\qquad \frac{{\partial u}}{{\partial t}}\mathfrak{A}u, \] where $\mathfrak{A}$ is a differential operator of an order higher than 2? This question remained unsettled until 1954–1955 when it was completely solved by W. Feller. Feller showed that every one-dimensional Markov process with a continuous path function is described by (2), where $\mathfrak{A}$ is the generalized second derivative.The purely analytical method of Feller is essentially connected with the one-dimensional character of the problem. It is very difficult to extend this method to the case of two (and more) dimensions.In this paper, a method is developed which can be applied to n dimensions as well as...