On the optimal search for a target whose motion is a Markov process

We will consider the optimal search for a target whose motion is a Markov process. The classical detection law leads to the use of multiplicative functionals and the search is equivalent to the termination of the Markov process with a termination density. A general condition for the optimality is derived and for Markov processes in n-dimensional Euclidean space with continuous transition functions we derive a simple necessary condition which generalizes the result of Hellman (1972).

Download Full-text

On the Asymptotic Behaviour of a Diffusion Process with Singular Drift

Nagoya Mathematical Journal ◽

10.1017/s0027763000016573 ◽

1975 ◽

Vol 57 ◽

pp. 87-106

Author(s):

Hitoshi Kaneta

Keyword(s):

Brownian Motion ◽

Markov Process ◽

Euclidean Space ◽

Diffusion Process ◽

Asymptotic Behaviour ◽

Markov Processes ◽

Smooth Boundary ◽

Dimensional Euclidean Space ◽

Singular Drift ◽

Strong Markov

We discuss some peculiar features of the diffusion process whose characterization is given below. Let D be a bounded domain in the d-dimensional Euclidean space Ed with a smooth boundary ∂D. The domain D contains open balls (i = 1, · · ·, n) which are mutually disjoint. Our process is a diffusion process on the state space D ∪ ∂D which is locally equivalent to the Brownian motion except on the spheres ∂ and the boundary ∂D. By a diffusion process we mean a continuous strong Markov process. As to the terminology about Markov processes we refer to [2].

Download Full-text

Proportional intensities and strong ergodicity for Markov processes

Journal of Applied Probability ◽

10.1017/s0021900200097060 ◽

1983 ◽

Vol 20 (01) ◽

pp. 185-190 ◽

Cited By ~ 3

Author(s):

Mark Scott ◽

Dean L. Isaacson

Keyword(s):

Markov Process ◽

Markov Processes ◽

Discrete Time ◽

Continuous Time ◽

Homogeneous Markov Process ◽

Strong Ergodicity ◽

Transition Functions ◽

The Matrix ◽

Intensity Functions ◽

Time Point

By assuming the proportionality of the intensity functions at each time point for a continuous-time non-homogeneous Markov process, strong ergodicity for the process is determined through strong ergodicity of a related discrete-time Markov process. For processes having proportional intensities, strong ergodicity implies having the limiting matrix L satisfy L · P(s, t) = L, where P(s, t) is the matrix of transition functions.

Download Full-text

Multiple points for symmetric Lévy processes

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100054311 ◽

1978 ◽

Vol 83 (1) ◽

pp. 83-90 ◽

Cited By ~ 15

Author(s):

John Hawkes

Keyword(s):

Markov Process ◽

Euclidean Space ◽

Lévy Process ◽

Lévy Processes ◽

Transition Function ◽

Levy Processes ◽

Levy Process ◽

Dimensional Euclidean Space ◽

Multiple Points ◽

Image Position

Let Xt be a Lévy process in Rd, d-dimensional euclidean space. That is X is a Markov process whose transition function satisfies

Download Full-text

Unisolvence on Multidimensional Spaces

Canadian Mathematical Bulletin ◽

10.4153/cmb-1968-056-6 ◽

1968 ◽

Vol 11 (3) ◽

pp. 469-474 ◽

Cited By ~ 1

Author(s):

Charles B. Dunham

Keyword(s):

Continuous Function ◽

Euclidean Space ◽

Compact Subset ◽

Compact Space ◽

General Condition ◽

Chebyshev Approximation ◽

Continuous Functions ◽

Dimensional Euclidean Space ◽

Variable Degree ◽

Convexity Conditions

In this note we consider the possibility of unisolvence of a family of real continuous functions on a compact subset X of m-dimensional Euclidean space. Such a study is of interest for two reasons. First, an elegant theory of Chebyshev approximation has been constructed for the case where the approximating family is unisolvent of degree n on an interval [α, β]. We study what sort of theory results from unisolvence of degree n on a more general space. Secondly, uniqueness of best Chebyshev approximation on a general compact space to any continuous function on X can be shown if the approximating family is unisolvent of degree n and satisfies certain convexity conditions. It is therefore of importance to Chebyshev approximation to consider the domains X on which unisolvence can occur. We will also study a more general condition on involving a variable degree.

Download Full-text

Immersions of Metric Spaces into Euclidean Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1965-095-6 ◽

1965 ◽

Vol 17 ◽

pp. 1015-1018

Author(s):

Takeo Akasaki

Keyword(s):

Euclidean Space ◽

Topological Space ◽

Metric Space ◽

Metric Spaces ◽

Necessary Condition ◽

Dimensional Euclidean Space ◽

Compact Metric Space ◽

Singular Cohomology ◽

Smith Theory ◽

Combinatorial Methods

In a recent paper on isotopy invariants (1), S. T. Hu denned the enveloping space Em(X) of any given topological space X for each integer m > 1. By an application of the Smith theory to the singular cohomology of the enveloping space Em(X), he obtained his immersion classes for every n = 1, 2, 3, . . . and proved (3) the main theorem that a necessary condition for a compact metric space X to be immersible into the ^-dimensional Euclidean space Rn is . This theorem was proved earlier by W. T. Wu (4) for finitely triangulable spaces X using purely combinatorial methods.

Download Full-text

Proportional intensities and strong ergodicity for Markov processes

Journal of Applied Probability ◽

10.2307/3213734 ◽

1983 ◽

Vol 20 (1) ◽

pp. 185-190 ◽

Cited By ~ 10

Author(s):

Mark Scott ◽

Dean L. Isaacson

Keyword(s):

Markov Process ◽

Markov Processes ◽

Discrete Time ◽

Continuous Time ◽

Homogeneous Markov Process ◽

Strong Ergodicity ◽

Transition Functions ◽

The Matrix ◽

Intensity Functions ◽

Time Point

By assuming the proportionality of the intensity functions at each time point for a continuous-time non-homogeneous Markov process, strong ergodicity for the process is determined through strong ergodicity of a related discrete-time Markov process. For processes having proportional intensities, strong ergodicity implies having the limiting matrix L satisfy L · P(s, t) = L, where P(s, t) is the matrix of transition functions.

Download Full-text

The Emergence and Formation of the Theory of Optimal Set Partitioning for Sets of the n Dimensional Euclidean Space. Theory and Application

Journal of Automation and Information Sciences ◽

10.1615/jautomatinfscien.v50.i9.10 ◽

2018 ◽

Vol 50 (9) ◽

pp. 1-24 ◽

Cited By ~ 4

Author(s):

Elena M. Kiseleva

Keyword(s):

Euclidean Space ◽

Set Partitioning ◽

Dimensional Euclidean Space ◽

Space Theory ◽

Optimal Set

Download Full-text

CLASSICAL MARKOV PROCESSES FROM QUANTUM LÉVY PROCESSES

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025799000060 ◽

1999 ◽

Vol 02 (01) ◽

pp. 105-129 ◽

Cited By ~ 1

Author(s):

UWE FRANZ

Keyword(s):

Markov Process ◽

Hopf Algebra ◽

Markov Processes ◽

Lévy Processes ◽

Sufficient Conditions ◽

Markov Property ◽

Vacuum State ◽

Levy Processes ◽

Quantum Process ◽

Quantum Markov Processes

We show how classical Markov processes can be obtained from quantum Lévy processes. It is shown that quantum Lévy processes are quantum Markov processes, and sufficient conditions for restrictions to subalgebras to remain quantum Markov processes are given. A classical Markov process (which has the same time-ordered moments as the quantum process in the vacuum state) exists whenever we can restrict to a commutative subalgebra without losing the quantum Markov property.8 Several examples, including the Azéma martingale, with explicit calculations are presented. In particular, the action of the generator of the classical Markov processes on polynomials or their moments are calculated using Hopf algebra duality.

Download Full-text

On Vitali's Theorem for Groups of Motions of Euclidean Space

Georgian Mathematical Journal ◽

10.1515/gmj.1999.323 ◽

1999 ◽

Vol 6 (4) ◽

pp. 323-334

Author(s):

A. Kharazishvili

Keyword(s):

Euclidean Space ◽

Dimensional Euclidean Space ◽

Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

Download Full-text