Group theoretic dimension of stationary symmetric \alpha-stable random fields

The growth rate of the partial maximum of a stationary stable process was first studied in the works of Samorodnitsky (2004a,b), where it was established, based on the seminal works of Rosi\'nski (1995,2000), that the growth rate is connected to the ergodic theoretic properties of the flow that generates the process. The results were generalized to the case of stable random fields indexed by Z^d in Roy and Samorodnitsky (2008), where properties of the group of nonsingular transformations generating the stable process were studied as an attempt to understand the growth rate of the partial maximum process. This work generalizes this connection between stable random fields and group theory to the continuous parameter case, that is, to the fields indexed by R^d.Comment: To appear in Journal of Theoretical Probability. Affiliation of the authors are update

Schramm-Loewner Equations Driven by Symmetric Stable Processes

We consider shape, size and regularity of the hulls of the chordal Schramm-Loewner evolution driven by a symmetric alpha-stable process. We obtain derivative estimates, show that the complements of the hulls are Hoelder domains, prove that the hulls have Hausdorff dimension 1, and show that the trace is right-continuous with left limits almost surely.Comment: 22 pages, 4 figure

Inversions of Levy Measures and the Relation Between Long and Short Time Behavior of Levy Processes

The inversion of a Levy measure was first introduced (under a different name) in Sato 2007. We generalize the definition and give some properties. We then use inversions to derive a relationship between weak convergence of a Levy process to an infinite variance stable distribution when time approaches zero and weak convergence of a different Levy process as time approaches infinity. This allows us to get self contained conditions for a Levy process to converge to an infinite variance stable distribution as time approaches zero. We formulate our results both for general Levy processes and for the important class of tempered stable Levy processes. For this latter class, we give detailed results in terms of their Rosinski measures

Maximal LpL^p-regularity for stochastic evolution equations

We prove maximal $L^p$-regularity for the stochastic evolution equation \{{aligned} dU(t) + A U(t)\, dt& = F(t,U(t))\,dt + B(t,U(t))\,dW_H(t), \qquad t\in [0,T], U(0) & = u_0, {aligned}. under the assumption that $A$ is a sectorial operator with a bounded $H^\infty$-calculus of angle less than $\frac12\pi$ on a space $L^q(\mathcal{O},\mu)$. The driving process $W_H$ is a cylindrical Brownian motion in an abstract Hilbert space $H$. For $p\in (2,\infty)$ and $q\in [2,\infty)$ and initial conditions $u_0$ in the real interpolation space \XAp we prove existence of unique strong solution with trajectories in L^p(0,T;\Dom(A))\cap C([0,T];\XAp), provided the non-linearities F:[0,T]\times \Dom(A)\to L^q(\mathcal{O},\mu) and B:[0,T]\times \Dom(A) \to \g(H,\Dom(A^{\frac12})) are of linear growth and Lipschitz continuous in their second variables with small enough Lipschitz constants. Extensions to the case where $A$ is an adapted operator-valued process are considered as well. Various applications to stochastic partial differential equations are worked out in detail. These include higher-order and time-dependent parabolic equations and the Navier-Stokes equation on a smooth bounded domain \OO\subseteq \R^d with $d\ge 2$. For the latter, the existence of a unique strong local solution with values in (H^{1,q}(\OO))^d is shown.Comment: Accepted for publication in SIAM Journal on Mathematical Analysi

Convolution-type derivatives, hitting-times of subordinators and time-changed C0C_0-semigroups

In this paper we will take under consideration subordinators and their inverse processes (hitting-times). We will present in general the governing equations of such processes by means of convolution-type integro-differential operators similar to the fractional derivatives. Furthermore we will discuss the concept of time-changed $C_0$-semigroup in case the time-change is performed by means of the hitting-time of a subordinator. We will show that such time-change give rise to bounded linear operators not preserving the semigroup property and we will present their governing equations by using again integro-differential operators. Such operators are non-local and therefore we will investigate the presence of long-range dependence.Comment: Final version, Potential analysis, 201

Measurement of ion emission from plasmas obtained with a 600 fs KrF laser

Ion emission from plasmas obtained by the use of a 600 fs, 36 mJ KrF laser operating at 248 nm was measured and analysed for a variety of targets at different laser intensities. The intensity was set by changing the distance between the focusing lens and the target. It was found that the ions emitted originate from impurities and ions from the bulk of the target can be produced only in the subsequent shots. Proton emission was identified from some targets, but the energy of the protons was low (less than 12 keV). A new silicon carbide semiconductor detector proved to be applicable for the collection of the ions