Differential equations in which the Poisson process plays a role

Kaplan, S.

doi:10.1090/s0002-9904-1964-11112-5

Cited by 46 publications

(26 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…After showing that such a motion is governed by a pair of partial differential equations, Kac comments, "The amazing thing is that these two equations can be combined into a hyperbolic equation"-namely, the telegraph equation, 1W = <P£_2a8F v8t2 ~~ V 8x2 v dt ' Kac refers to an earlier discussion of this model by Sidney Goldstein [3], and attributes its origin to G. I. Taylor. A few later papers [8], [13], follow up certain aspects of Kac's analysis.…”

mentioning

confidence: 99%

Theory of Random Evolutions with Applications to Partial Differential Equations

Griego¹,

Hersh²

1971

Transactions of the American Mathematical Society

View full text Add to dashboard Cite

Abstract. The selection from a finite number of strongly continuous semigroups by means of a finite-state Markov chain leads to the new notion of a random evolution. Random evolutions are used to obtain probabilistic solutions to abstract systems of differential equations. Applications include one-dimensional first order hyperbolic systems. An important special case leads to consideration of abstract telegraph equations and a generalization of a result of Kac on the classical 7¡-dimensional telegraph equation is obtained and put in a more natural setting. In this connection a singular perturbation theorem for an abstract telegraph equation is proved by means of a novel application of the classical central limit theorem and a representation of the solution for the limiting equation is found in terms of a transformation formula involving the Gaussian distribution.In a little-known section of an out-of-print book [7], Mark Kac has considered a particle which moves on a line at speed v, and reverses direction according to a Poisson process with intensity a. After showing that such a motion is governed by a pair of partial differential equations, Kac comments, "The amazing thing is that these two equations can be combined into a hyperbolic equation"-namely, the telegraph equation, 1W =

show abstract

mentioning

confidence: 99%

Theory of Random Evolutions with Applications to Partial Differential Equations

Griego¹,

Hersh²

1971

Transactions of the American Mathematical Society

View full text Add to dashboard Cite

show abstract

“…The main purpose of this paper is to obtain explicitly the distributions of the random walks, by extending and then applying an important result of Kaplan,3) so that the solutions to the wave equations with a damping term can be given, up to an integral, when the solutions to the corresponding wave equation without the damping term are known.…”

Section: )~6)mentioning

confidence: 99%

“…Then the s.olution F(x, t) can be obtained by2),5b) (3) where the average is with respect to the randomized time Set) which is given by (4) where N(r) is a random variable Poisson distributed with intensity aCt), with N(O) =0, and for r'2. r'…”

Section: )~6)mentioning

confidence: 99%

Poisson Random Walk for Solving Wave Equations

Foong¹,

Kolck²

1992

Progress of Theoretical Physics

View full text Add to dashboard Cite

We reconsider the connection between random walks of Poisson type and wave equations, following M. Kac's approach which was based on the work by S. Goldstein.' A scheme for calculating the distribution of the displacement from the origin at an arbitrary time for the random walk with a time dependent intensity is given, and it is applied to three cases. A simple example showing the use of the distribution is given. § 1. IntroductionThe connections between Brownian motion, which is a random walk of Gaussian type, and diffusion equations are well known. However, the connections between random walks of Poisson type and wave equations are less explored. Aconnection between the walk and the telegrapher equation, which is a wave equation with dissipation, in 1 space dimension was probably first realized many years ago by Goldstein. The main purpose of this paper is to obtain explicitly the distributions of the random walks, by extending and then applying an important result of Kaplan,3) so that the solutions to the wave equations with a damping term can be given, up to an integral, when the solutions to the corresponding wave equation without the damping term are known.In § 2, we give the path integral approach for solving the wave equations with dissipation. In § 3, a scheme for calculating the distributions of the displacement for the random walk is given. Section 4 focuses on the actual computations of the distributions, although a simple model solution of wave equations is included. The conclusion is in § 5. § 2. The path integral approach to wave equations Consider the following wave equation with dissipation with general boundary conditions: t) Fellow of CNPq, Brazil.

show abstract

“…The telegraph equation was first considered from a probabilistic point of view by GOLDSTEIlV [3] who showed that it can be obtained by considering a sequence of step-dependent random walks. KAc [6] [7] simplified and extended KAC'S proof to cover the temporally inhomogenous process related to the equation utt + a (t) ut = uxx.…”

Section: --/T U]mentioning

confidence: 99%

Differential equations with a small parameter and the central limit theorem for functions defined on a finite Markov chain

Pinsky

1968

Z. Wahrscheinlichkeitstheorie verw Gebiete

View full text Add to dashboard Cite

Differential equations in which the Poisson process plays a role

Cited by 46 publications

References 1 publication

Theory of Random Evolutions with Applications to Partial Differential Equations

Theory of Random Evolutions with Applications to Partial Differential Equations

Poisson Random Walk for Solving Wave Equations

Differential equations with a small parameter and the central limit theorem for functions defined on a finite Markov chain

Contact Info

Product

Resources

About