Let $D$ be a domain of finite Lebesgue measure in $\bR^d$ and let $X^D_t$ be the symmetric $α$-stable process killed upon exiting $D$. Each element of the set $\{λ_i^α\}_{i=1}^\infty$ of eigenvalues associated to $X^D_t$, regarded as a function of $α\in(0,2)$, is right continuous. In addition, if $D$ is Lipschitz and bounded, then each $ λ_i^α$ is continuous in $α$ and the set of associated eigenfunctions is precompact. We also prove that if $D$ is a domain of finite Lebesgue measure, then for all $0<α<β\leq 2$ and $i\geq 1$, \[λ_i^α\leq [ λ^β_i]^{α/β}.\] Previously, this bound had been known only for $β=2$ and $α$ rational.
We deduce bilateral interrelations between Fourier coefficients for lacunar trigonometric series and modulus of their continuity. We obtain also as an application some conditions for continuity and discontinuity for Gaussian periodic stationary random centered processes.
We introduce the notion of linear multifractional stable sheets in the broad sense (LMSS) with $α\in(0,2]$, to include both linear multifractional Brownian sheets ($α=2$) and linear multifractional stable sheets ($α<2$). The purpose of the present paper is to study the existence and joint continuity of the local times of LMSS, and also the local Hölder condition of the local times in the set variable. Among the main results of this paper, Theorem 2.4 provides a sufficient and necessary condition for the existence of local times of LMSS; Theorem 3.1 shows a sufficient condition for the joint co…
该论文将在JOTP上发表。本文证明了几类有界增量过程的Holder连续性。本文推广了Kwapien和Rosinski关于随机过程的样本H{<$}lder连续性和优化测度的结果. \textit{Seminar on Stochastic Analysis, Random Fields and Applications IV, Progr. in Probab.} { \bf 58}，155- 163. Birkh{ä}user，Basel.
We consider piecewise deterministic Markov processes with degenerate transition kernels of the "house-of-cards"-type. We use a splitting scheme based on jump times to prove the absolute continuity, as well as some regularity, of the invariant measure of the process. Finally, we obtain finer results on the regularity of the one-dimensional marginals of the invariant measure, using integration by parts with respect to the jump times.
In this work, we solve an open problem mentioned in Xiao (2010) and provide sharp bounds on the rate of growth of maximal moments for stationary symmetric stable random fields using structure theorem of finitely generated abelian groups and ergodic theory of quasi-invariant group actions. We also investigate the relationship between this rate of growth and the path regularity properties of self-similar stable random fields with stationary increments, and establish uniform modulus of continuity of such fields. In the process, a new notion of weak effective dimension is introduced for stable ran…
In this paper we study time-inhomogeneous affine processes beyond the common assumption of stochastic continuity. In this setting times of jumps can be both inaccessible and predictable. To this end we develop a general theory of finite dimensional affine semimartingales under very weak assumptions. We show that the corresponding semimartingale characteristics have affine form and that the conditional characteristic function can be represented with solutions to measure differential equations of Riccati type. We prove existence of affine Markov processes and affine semimartingales under mild co…
We consider a family of Bessel Processes that depend on the starting point $x$ and dimension $δ$, but are driven by the same Brownian motion. Our main result is that almost surely the first time a process hits $0$ is jointly continuous in $x$ and $δ$, provided $δ\le 0$. As an application, we show that the SLE($κ$) welding homeomorphism is continuous in $κ$ for $κ\in [0,4]$. Our motivation behind this is to study the well known problem of the continuity of SLE$_κ$ in $κ$. The main tool in our proofs is random walks with increments distributed as infinite mean Inverse-Gamma laws.
We consider the joint lower expectation of a finite-state imprecise stochastic process, defined using either the Ville-Vovk-Shafer natural extension or the Williams natural extension. In both cases, we show that it is continuous with respect to the pointwise convergence of non-decreasing sequences of real-valued functions $f_n$, $n\in\mathbb{N}_0$, where each $f_n$ is $n$-measurable. For the Ville-Vovk-Shafer natural extension, a similar result is shown to hold for non-increasing sequences, provided that they converge to a bounded function.
We prove that if we are given a generator of a cadlag Markov process and an open domain $G$ in the state space, on which the generator has the local property expressed in a suitable way on a class $\mathcal{C}$ of test functions that is sufficiently rich, then the Markov process has continuous paths when it passes through $G$. The result holds for any Markov process which is associated with the generator merely on $\mathcal{C}$. This points out that the path continuity of the process is an a priori property encrypted by the generator acting on enough test functions, and this property can be ea…
Let $ \overline B=\{ \overline B_{t},t\in R^{1} \}$ be Brownian motion killed after an independent exponential time with mean $2/λ^{2}$. The process $\overline B$ has potential densities, \[ u(x,y) ={e^{-λ|y-x|}\over λ},\qquad x,y\in R^{ 1}, \] which is also the covariance of an Ornstein--Uhlenbeck process. Let $f$ be an excessive function for $\overline B$. Then, \[ {e^{-λ|y-x|}\over λ}+f(y),\qquad x,y\in R^{ 1}, \] is the kernel of an $α$-permanental process $ X_α=\{ X_α(t), t\in R^{ 1}\}$ for all $α>0$. It is shown that for all $k\ge 1$ and intervals $Δ\subseteq [0,1] $, \[ \limsup_{h\to 0}…
Let $μ$ denote a Borel probability measure and let $\{ μ_{t} \}_{t\geq 1}$ denote the free additive convolution semigroup of Nica and Speicher. We show that the support of these measures varies continuously in the Hausdorff metric for $t >1$. We utilize complex analytic methods and, in particular, a characterization of the absolutely continuous portion of these supports due to Huang.
In this article, we define the new concept of local coupling property for Markov processes and study its relationship with distributional properties of the transition probability. In the special case of Lévy processes we show that this property is equivalent to the absolute continuity of the transition probability and also provide a sufficient condition for it in terms of the Lévy measure. Our result is stronger than existing results for absolute continuity of Lévy distributions.
We consider a supercritical branching process $Z_n$ in a stationary and ergodic random environment $ξ=(ξ_n)_{n\ge0}$. Due to the martingale convergence theorem, it is known that the normalized population size $W_n=Z_n/ (\mathbb E (Z_n|ξ))$ converges almost surely to a random variable $W$. We prove that if $W$ is not concentrated at $0$ or $1$ then for almost every environment $ξ$ the law of $W$ conditioned on the environment $ξ$ is absolutely continuous with a possible atom at $0$. The result generalizes considerably the main result of \cite{kaplan:1974}, and of course it covers the well-known…
A CLT for the $L^{2}$ moduli of continuity of local times of Levy processes
Let $X=\{X_{t},t\in R_{+}\}$ be a symmetric Lévy process with local time $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ . When the Lévy exponent $ψ(\la)$ is regularly varying at infinity with index $1<β\leq 2$ and satisfies some additional regularity conditions && \sqrt{hψ^{2}(1/h)} \lc \int (L^{x+h}_{1}- L^{x}_{1})^{2} dx- E( \int (L^{x+h}_{1}- L^{x}_{1})^{2} dx) \rc \nn && {1 in} \stackrel{\mathcal{L}}{\Longrightarrow} (8 c_{β,1} )^{1/2} η( \int ( L_{1} ^{x})^{2} dx)^{1/2} \nn , as $h\rar 0$ , where $η$ is a normal random variable with mean zero and variance one that is independent of $L^{…
We consider discrete-time uncertain processes with finite state space and study the properties of game-theoretic upper expectations developed by Shafer and Vovk. We start by proving some basic properties, e.g. monotonicity, law of iterated upper expectations,... Afterwards, we show that this operator satisfies several, more involved continuity properties, including continuity with respect to non-decreasing sequences and continuity with respect to specific sequences of so-called `finitary measurable' functions, which are functions that depend on a finite number of states.
For two independent Lévy processes $ξ$ and $η$ and an exponentially distributed random variable $τ$ with parameter $q>0$, independent of $ξ$ and $η$, the killed exponential functional is given by $V_{q,ξ,η} := \int_0^τ\mathrm{e}^{-ξ_{s-}} \, \mathrm{d} η_s$. Interpreting the case $q=0$ as $τ=\infty$, the random variable $V_{q,ξ,η}$ is a natural generalization of the exponential functional $\int_0^\infty \mathrm{e}^{-ξ_{s-}} \, \mathrm{d} η_s$, the law of which is well-studied in the literature as it is the stationary distribution of a generalised Ornstein-Uhlenbeck process. In this paper we sh…
