One-dimensional semi-Markov evolutions with general Erlang sojourn times

It is a well-known result that almost all sample paths of Brownian motion or Wiener process {W (t)} have infinitely many zero-crossings in the interval (0, δ) for δ > 0. Under the Kac condition, the telegraph process weakly converges to the Wiener process. We estimate the number of intersections of a level or the number of level-crossings for the telegraph process. Passing to the limit under the Kac condition, we also obtain an estimate for the level-crossings for the Wiener process. Вiдомо, що майже всi вибiрковi траєкторiї броунiвського руху чи вiнерiвського процесу {W (t)} мають нескiнченно багато нульових перетинiв в iнтервалi (0, δ) при δ > 0. За умови Каца телеграфний процес слабко збiгається до вiнерiвського процесу. В роботi оцiнюється число перетинiв рiвня для телеграфного процесу. Переходячи до границi за умови Каца, ми також отримуємо оцiнку перетинiв рiвня для вiнерiвського процесу.

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“…It is well-known, Pogorui and Rodríguez-Dagnino [1], that this set of equations relating ϕ 0 (s, u) and ϕ 1 (s, u) can be represented as…”

Section: Proof Consider the Laplace Transform Ofmentioning

confidence: 99%

The First Passage Time and Estimation of the Number of Level-Crossings for a Telegraph Process

2015

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“…The biwave equation is given by

{((), \frac{\partial^{2}}{\partial x^{2}} MathClass-bin- \frac{\partial^{2}}{\partial y^{2}})}^{2} u (x MathClass-punc, y) MathClass-rel= 0 MathClass-punc,

and it has been obtained in in the context of random evolutions and studied in . In this case, we have

P (ξ_{0} MathClass-punc, ξ_{1}) MathClass-rel= {((), ξ_{0}^{2} MathClass-bin- ξ_{1}^{2})}^{2}

.…”

Section: Examplesmentioning

confidence: 99%

“…This equation is called the bi‐telegraph equation, and it was obtained in after finding the distribution of the position of a particle that moves on the line driven by a point process with two‐Erlang sojourn times.…”

Section: The Bi‐telegraph Equationmentioning

confidence: 99%

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Solutions for PDEs with constant coefficients and derivability of functions ranged in commutative algebras

Pogorui

Rodríguez‐Dagnino

Shapiro

2013

Math Methods in App Sciences

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“…This model describes the evolution of a particle on a line. The telegraph process has been extended in many ways since then (see [2][3][4] and references therein). In this paper, we study the fading evolution, which was introduced in [5].…”

Section: Introductionmentioning

confidence: 99%