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.
英文摘要：
The paper will be published in JOTP.
In the paper we prove Holder Continuity for ceratian classes of processes with bounded increments. The paper generalizes results obtained by Kwapien and Rosinski in Sample H{ö}lder continuity of stochastic processes and majorizing measures. \textit{Seminar on Stochastic Analysis, Random Fields and Applications IV, Progr. in Probab.} {\bf 58}, 155--163. Birkh{ä}user, Basel.
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.
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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}…
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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^{…
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There are numerous applications of the classical (deterministic) Gronwall inequality. Recently, Michael Scheutzow discovered a stochastic Gronwall inequality which provides upper bounds for $p$-th moments, $p\in(0,1)$, of the supremum of nonnegative scalar continuous processes which satisfy a linear integral inequality. In this article we complement this with upper bounds for $p$-th moments, $p\in[2,\infty)$, of the supremum of general Itô processes which satisfy a suitable one-sided affine-linear growth condition. As example applications, we improve known results on strong local Lipschitz con…
We study the absolute continuity with respect to the Lebesgue measure of the distribution of the nodal volume associated with a smooth, non-degenerated and stationary Gaussian field $(f(x), {x \in \mathbb R^d})$. Under mild conditions, we prove that in dimension $d\geq 3$, the distribution of the nodal volume has an absolutely continuous component plus a possible singular part. This singular part is actually unavoidable baring in mind that some Gaussian processes have a positive probability to keep a constant sign on some compact domain. Our strategy mainly consists in proving closed Kac--Rice…
