The paper is devoted to spaces of generalized smoothness on so-called h-sets. First we find quarkonial representations of isotropic spaces of generalized smoothness on ℝⁿ and on an h-set. Then we investigate representations of such spaces via differences, which are very helpful when we want to find an explicit representation of the domain of a Dirichlet form on h-sets. We prove that both representations are equivalent, and also find the domain of some time-changed Dirichlet form on an h-set.

We study one-dimensional Lévy processes with Lévy-Khintchine exponent ψ(ξ²), where ψ is a complete Bernstein function. These processes are subordinate Brownian motions corresponding to subordinators whose Lévy measure has completely monotone density; or, equivalently, symmetric Lévy processes whose Lévy measure has completely monotone density on (0,∞). Examples include symmetric stable processes and relativistic processes. The main result is a formula for the generalized eigenfunctions of transition...

Let $X$ be a $\mu$-symmetric Hunt process on a LCCB space $𝙴$. For an open set $𝙶\subseteq 𝙴$, let ${\tau }_{𝙶}$ be the exit time of $X$ from $𝙶$ and $A^{𝙶}$ be the generator of the process killed when it leaves $𝙶$. Let $r:[0,\infty [\to [0,\infty [$ and $R\left(t\right)={\int }_0^{t}r\left(s\right)\mathrm{d}s$. We give necessary and sufficient conditions for ${𝔼}_{\mu }R\left({\tau }_{𝙶}\right)lt;\infty$ in terms of the behavior near the origin of the spectral measure of $-A^{𝙶}$. When $r\left(t\right)=t^l$, $l\ge 0$, by means of this condition we derive the Nash inequality for the killed process. In the diffusion case this permits to show that the existence of moments of order $l+1$ for ${\tau }_{𝙶}$ implies the...

In this paper, we consider a birth–death process with generator $ℒ$ and reversible invariant probabilityπ. Given an increasing function ρ and the associated Lipschitz norm ‖⋅‖Lip(ρ), we find an explicit formula for ${\parallel (-ℒ)}^{-1}{\parallel }_{Lip\left(\rho \right)}$. As a typical application, with spectral theory, we revisit one variational formula of M. F. Chen for the spectral gap of $ℒ$ inL2(π). Moreover, by Lyons–Zheng’s forward-backward martingale decomposition theorem, we get convex concentration inequalities for additive functionals of birth–death...

Let X be a regular continuous positively recurrent Markov process with state space ℝ, scale function S and speed measure m. For a∈ℝ denote Ba+=supx≥am(]x, +∞[)(S(x)−S(a)), Ba−=supx≤am(]−∞; x[)(S(a)−S(x)). It is well known that the finiteness of Ba± is equivalent to the existence of spectral gaps of generators associated with X. We show how these quantities appear independently in the study of the exponential moments of hitting times of X. Then we establish a very direct relation between exponential...

This paper examines appropriate protocols for high speed multiple access communication systems where the bandwidth is divided into two separate asymmetric channels. Both channels operate using slotted non-persistent CSMA or CSMA/CD techniques. Free stations access the first channel while all retransmissions occur in the second channel. We define the stability regions and the rules for optimal bandwidth allocation among the two channels for improvement of the system performance in case of infinite...

We consider the optimal stopping problem for a discrete-time Markov process on a Borel state space $X$. It is supposed that an unknown transition probability $p(·|x)$, $x\in X$, is approximated by the transition probability $\tilde{p}(·|x)$, $x\in X$, and the stopping rule ${\tilde{\tau }}_{*}$, optimal for $\tilde{p}$, is applied to the process governed by $p$. We found an upper bound for the difference between the total expected cost, resulting when applying ${\tilde{\tau }}_{*}$, and the minimal total expected cost. The bound given is a constant times ${\displaystyle {sup}_{x\in X}\parallel p(·|x)-\tilde{p}(·|x)\parallel }$, where $\parallel ·\parallel$ is the total variation...

The paper is concerned with stability analysis for a class of impulsive Hopfield neural networks with Markovian jumping parameters and time-varying delays. The jumping parameters considered here are generated from a continuous-time discrete-state homogenous Markov process. By employing a Lyapunov functional approach, new delay-dependent stochastic stability criteria are obtained in terms of linear matrix inequalities (LMIs). The proposed criteria can be easily checked by using some standard numerical...

We study the asymptotic behaviour of discrete time processes which are products of time dependent transformations defined on a complete metric space. Our sufficient condition is applied to products of Markov operators corresponding to stochastically perturbed dynamical systems and fractals.