We study the asymmetric one-dimensional telegraph process in the bounded domain. Lower boundary is absorbing and upper boundary is reflecting with delay. Point stays in the upper boundary until switch of regime occurs. We obtain the distribution of this process in terms of Laplace trasforms.
We consider a telegraph process with elastic boundary at the origin studied recently in the literature. It is a particular random motion with finite velocity which starts at $xgeq 0$, and its dynamics is determined by upward and downward switching ra
tes $lambda$ and $mu$, with $lambda>mu$, and an absorption probability (at the origin) $alphain(0,1]$. Our aim is to study the asymptotic behavior of the absorption time at the origin with respect to two different scalings: $xtoinfty$ in the first case; $mutoinfty$, with $lambda=betamu$ for some $beta>1$ and $x>0$, in the second case. We prove several large and moderate deviation results. We also present numerical estimates of $beta$ based on an asymptotic Normality result for the case of the second scaling.
Suppose that a $d$-dimensional domain is filled with a gas of (in general, interacting) diffusive particles with density $n_0$. A particle is absorbed whenever it reaches the domain boundary. Employing macroscopic fluctuation theory, we evaluate the
probability ${mathcal P}$ that no particles are absorbed during a long time $T$. We argue that the most likely gas density profile, conditional on this event, is stationary throughout most of the time $T$. As a result, ${mathcal P}$ decays exponentially with $T$ for a whole class of interacting diffusive gases in any dimension. For $d=1$ the stationary gas density profile and ${mathcal P}$ can be found analytically. In higher dimensions we focus on the simple symmetric exclusion process (SSEP) and show that $-ln {mathcal P}simeq D_0TL^{d-2} ,s(n_0)$, where $D_0$ is the gas diffusivity, and $L$ is the linear size of the system. We calculate the rescaled action $s(n_0)$ for $d=1$, for rectangular domains in $d=2$, and for spherical domains. Near close packing of the SSEP $s(n_0)$ can be found analytically for domains of any shape and in any dimension.
A fractional Ficks law and fractional hydrostatics for the one dimensional exclusion process with long jumps in contact with infinite reservoirs at different densities on the left and on the right are derived.
Consider a non-relativistic quantum particle with wave function inside a region $Omegasubset mathbb{R}^3$, and suppose that detectors are placed along the boundary $partial Omega$. The question how to compute the probability distribution of the time
at which the detector surface registers the particle boils down to finding a reasonable mathematical definition of an ideal detecting surface; a particularly convincing definition, called the absorbing boundary rule, involves a time evolution for the particles wave function $psi$ expressed by a Schrodinger equation in $Omega$ together with an absorbing boundary condition on $partial Omega$ first considered by Werner in 1987, viz., $partial psi/partial n=ikappapsi$ with $kappa>0$ and $partial/partial n$ the normal derivative. We provide here a discussion of the rigorous mathematical foundation of this rule. First, for the viability of the rule it plays a crucial role that these two equations together uniquely define the time evolution of $psi$; we point out here how the Hille-Yosida theorem implies that the time evolution is well defined and given by a contraction semigroup. Second, we show that the collapse required for the $N$-particle version of the problem is well defined. Finally, we also prove analogous results for the Dirac equation.
Telegraph equation describing the compression of electromagnetic waves in a waveguide (resonator) with moving boundary are derived. It is shown that the character of oscillations of the compressed electromagnetic field depends on the parameters of th
e resonator, and under certain conditions, the oscillations of voltage (current) yield the exponential growth, leading to a noticeable change in the radiation losses.
Igor G. Pospelov
,Stanislav A. Radionov
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(2015)
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"Telegraph process in the bounded domain with absorbing lower boundary and reflecting with delay upper boundary"
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Stanislav Radionov
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