Dispersion of volume under the action of isotropic Brownian flows

We study transport properties of isotropic Brownian flows. Under a transience condition for the two-point motion, we show asymptotic normality of the image of a finite measure under the flow and -- under slightly stronger assumptions -- asymptotic normality of the distribution of the volume of the image of a set under the flow. Finally, we show that for a class of isotropic flows, the volume of the image of a nonempty open set (which is a martingale) converges to a random variable which is almost surely strictly positive.Comment: To appear in "Stochastic processes and their applications

Isotropic Ornstein-Uhlenbeck flows

Isotropic Brownian flows (IBFs) are a fairly natural class of stochastic flows which has been studied extensively by various authors. Their rich structure allows for explicit calculations in several situations and makes them a natural object to start with if one wants to study more general stochastic flows. Often the intuition gained by understanding the problem in the context of IBFs transfers to more general situations. However, the obvious link between stochastic flows, random dynamical systems and ergodic theory cannot be exploited in its full strength as the IBF does not have an invariant probability measure but rather an infinite one. Isotropic Ornstein-Uhlenbeck flows are in a sense localized IBFs and do have an invariant probability measure. The imposed linear drift destroys the translation invariance of the IBF, but many other important structure properties like the Markov property of the distance process remain valid and allow for explicit calculations in certain situations. The fact that isotropic Ornstein-Uhlenbeck flows have invariant probability measures allows one to apply techniques from random dynamical systems theory. We demonstrate this by applying the results of Ledrappier and Young to calculate the Hausdorff dimension of the statistical equilibrium of an isotropic Ornstein-Uhlenbeck flow

The War and the Working Class

Political pamphlet in support of the Communist Party. 24 pages. Student Publications: The Campus Newspaper Collectio

Futures Cross-Hedging with a Stationary Basis

This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.When managing risk, frequently only imperfect hedging instruments are at hand. We show how to optimally cross-hedge risk when the spread between the hedging instrument and the risk is stationary. For linear risk positions we derive explicit formulas for the hedge error, and for nonlinear positions we show how to obtain numerically efficient estimates. Finally, we demonstrate that even in cases with no clear-cut decision concerning the stationarity of the spread, it is better to allow for mean reversion of the spread rather than to neglect it.Peer Reviewe

Uniform shrinking and expansion under isotropic Brownian flows

We study some finite time transport properties of isotropic Brownian flows. Under a certain nondegeneracy condition on the potential spectral measure, we prove that uniform shrinking or expansion of balls under the flow over some bounded time interval can happen with positive probability. We also provide a control theorem for isotropic Brownian flows with drift. Finally, we apply the above results to show that under the nondegeneracy condition the length of a rectifiable curve evolving in an isotropic Brownian flow with strictly negative top Lyapunov exponent converges to zero as $t\to \infty$ with positive probability

Criteria for strong and weak random attractors

The theory of random attractors has different notions of attraction, amongst them pullback attraction and weak attraction. We investigate necessary and sufficient conditions for the existence of pullback attractors as well as of weak attractors

Eigenschaften isotroper Brownscher und Ornstein-Uhlenbeckscher Flüsse

Die vorliegende Arbeit behandelt gewisse makroskopische Eigenschaften isotroper Brownscher Flüsse (IBF) und isotroper Ornstein-Uhlenbeckscher Flüsse auf $\R^d$ mit $d\ge2$. \\ Die IBF, untersucht von Le Jan und Baxendale/Harris, sind eine Klasse von Flüssen mit unabhängigen und stationären Inkrementen und einer Verteilung, die invariant bezüglich starrer Bewegungen des $\R^d$ ist. Zunächst befassen wir uns mit der zeitlichen Asymptotik der Länge einer rektifizierbaren Kurve unter dem Fluss. Wir liefern fast sichere exponentielle untere und obere Abschätzungen für diese Länge und zeigen, dass im Falle von strikt negativen Lyapunov Exponenten die untere Abschätzung mit positiver Wahrscheinlichkeit erreicht wird. Im Fall von positiven Lyapunov Exponenten wissen wir im allgemeinen nicht, ob diese Abschätzungen erreicht werden, allerdings geben wir eine hinreichende Bedingung über die Gleichmäßigkeit der räumlichen Verteilung des Bildes der Kurve, unter welcher f.s. die obere Abschätzung angenommen wird. Weiter befassen wir uns mit der Evolution des Volumens einer Borelmenge unter einem IBF. Diese Problematik wurde von Le Jan, Darling und Kunita behandelt. Le Jan und Darling haben unter anderem gezeigt, dass im Falle von strikt negativen Lyapunov Exponenten das Volumen asymptotisch für große Zeiten verschwindet, sogar wenn der Durchmesser der Menge wächst. Le Jan hat auch bewiesen, dass im Fall von strikt positiven Lyapunov Exponenten das Volumen mit positiver Wahrscheinlichkeit nicht verschwindet. Wir haben gezeigt, dass für IBF nahe dem volumenerhaltenden Fall, das Volumen einer offenen Menge f.s. überlebt. Wir behandeln die räumliche Verteilung der Menge und zeigen, dass sich asymptotisch eine Normalverteilung ergibt. Ein weiterer Schwerpunkt dieser Arbeit ist die Frage, ob mit positiver Wahrscheinlichkeit ein nichtvolumenerhaltender IBF eine Kugel strikt in sich selbst abbildet. Unter einer Nichtdegeneriertheits-Bedingung an das zugehörige Potentialmaß zeigen wir, dass in der Tat mit strikt positiver Wahrscheinlichkeit das Bild der Kugel zu einem festen Zeitpunkt uniform geschrumpft ist. Ein wesentlicher Nachteil der IBF ist die Tatsache, dass sie über kein invariantes Wahrscheinlichkeitsmaß verfügen und deshalb wichtige Resultate der Ergodentheorie nicht anwendbar sind. Dies motiviert die Betrachtung der IBF mit einem Drift, der die Existenz eines invarianten Wahrscheinlichkeitsmasses sichert. Allerdings geht dann ein Teil der reichen Struktur der IBF verloren. Dennoch, falls der zusätzliche Drift linear ist, bleiben die meisten Eigenschaften, z.B. dass der Abstandprozess eine Diffusion ist, erhalten. Diese Klasse nennen wir isotrope Ornstein-Uhlenbeck Flüsse (IOUF). Wir zeigen, dass jeder IOUF einen globalen schwachen Mengenattraktor besitzt. Ferner untersuchen wir das statistische Gleichgewicht eines IOUF. Da die IOUF invariante Wahrscheinlichkeitsmaße haben, können wir die Ergebnisse von Ledrappier und Young benutzen, um zu zeigen, dass das statistische Equilibrium eine fast sicher konstante punktweise Dimension fast überall besitzt und diese auch bestimmen.In this thesis we study some macroscopic properties of isotropic Brownian flows (IBF) and isotropic Ornstein-Uhlenbeck flows (IOUF) in $\R^d$ with $d\ge2$. \\ IBFs were studied extensively by Baxendale and Harris and independently by Le Jan. They are a very natural family of stochastic flows with independent and stationary increments and with a distribution, which is invariant with respect to spatial rigid motions on $\R^d$. First we consider the asymptotics of the evolution of a smooth rectifiable curve under the action of an IBF. We provide almost sure lower and upper exponential bounds for the length of the curve and show that in case the top Lyapunov exponent is strictly negative the lower bound is attained with positive probability. In the case of a positive Lyapunov exponent we do not know if these bounds are attained, however we give a sufficient condition concerning the uniformity of the spatial spreading'' of the curve, under which the upper bound is attained a.s.. This might help the further research on the problem. \\ Further we consider the evolution of the volume of a set under the action of an IBF. This problem has also been studied by Le Jan, Darling and Kunita. Le Jan and Darling proved that in the case of a strictly negative top Lyapunov exponent the volume of a bounded set vanishes almost surely with time (even if its diameter grows). Le Jan has also shown that in the case of a positive Lyapunov exponent the volume does not vanish asymptotically with strictly positive probability. We show that if the IBF is near to the volume preserving case then the volume of an open set does persist (i.e.~does not converge to zero) almost surely. We also study the spatial distribution of the volume of an open set moved by the flow and prove that on a large scale there is a certain kind of asymptotic normality. Another question we consider is if an IBF with strictly positive probability uniformly shrinks a ball. Clearly this is not possible in the volume preserving case. We give an affirmative answer to this question under a certain nondegeneracy'' condition on the potential measure associated to the flow. \\ Unfortunately IBFs viewed as random dynamical systems do not have finite invariant measure and therefore many results and tools from ergodic theory do not apply to IBFs. This motivates the study of IBFs with drift ensuring the existence of an invariant probability measure, however some of the rich structure of the IBF is being lost. Nevertheless, in case this drift is linear, the flows we obtain inherit most of the properties from the IBFs without drift, e.g. the distance process is a diffusion. %and has the advantage of an invariant probability measure. We call them isotropic Ornstein-Uhlenbeck flows (IOUF). One of the central notions in the random dynamical systems theory is the notion of an attractor. We show that the isotropic Ornstein-Uhlenbeck flows have a global weak set attractor. Further we consider the statistical equilibrium of an IOUF. Exploiting the fact that IOUFs do have an invariant measure, we apply the results of Ledrappier and Young to show that the statistical equilibrium has a.s.~constant pointwise dimension almost everywhere and give a formula for this dimension