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

Bee colony assessments with the Liebefeld method: How do individual beekeepers influence results and are photo assessments an option to reduce variability?

Colony strength, food storage and brood development are a fundamental part of each honeybee field study. Colony assessments are used to compare and assess those for beehive over time. At present, most colony assessments are made by experienced beekeepers according to Liebefeld method. This method is based on an estimation of areas covered by honeybees, food and brood stages on each side of a comb. Areas are counted from a grid separating the comb side into 8 sections which are protocolled with an accuracy of 0.5 sections. An assessment for a hive takes up to 20 min and even with two field locations, it is necessary to split assessments between beekeepers. So, it is important to make estimates as comparable as possible. For this purpose, beekeepers practice the assessments on pre-determined photographs to “calibrate themselves”. The advantage of the Liebefeld assessment is that the condition of bee hive is estimated with minimum disturbance of the bees. Digital photography is under discussion to gain data with high precision and accuracy with one major disadvantage. To be able to see food and brood stages in photographs, bees have to be removed from combs. This, however, results in a disturbance of the colony – especially if the assessments take place in short time intervals of 7 ± 1 days. An experiment was performed to evaluate the variation between individual beekeepers and to compare the results to data generated with photographs. For the experiment, five colonies were assessed each by four beekeepers independently according to Liebefeld method. Each comb side of the five colonies was photographed with and without honeybees sitting on it for precise analysis at the computer for a number of bees, nectar cells, pollen cells, eggs, open brood and capped brood. The number of bees and cells with the different contents were generated by an area-based assessment in ImageJ as well as a detailed counting with help of HiveAnalyzer® Software. Data from beekeeper estimations were then compared with assessments based on digital photography. With the results of the experiment, we tried to answer several questions. With the study, we wanted to determine the level of variation between the beekeepers for the live stages and food stores estimated.Colony strength, food storage and brood development are a fundamental part of each honeybee field study. Colony assessments are used to compare and assess those for beehive over time. At present, most colony assessments are made by experienced beekeepers according to Liebefeld method. This method is based on an estimation of areas covered by honeybees, food and brood stages on each side of a comb. Areas are counted from a grid separating the comb side into 8 sections which are protocolled with an accuracy of 0.5 sections. An assessment for a hive takes up to 20 min and even with two field locations, it is necessary to split assessments between beekeepers. So, it is important to make estimates as comparable as possible. For this purpose, beekeepers practice the assessments on pre-determined photographs to “calibrate themselves”. The advantage of the Liebefeld assessment is that the condition of bee hive is estimated with minimum disturbance of the bees. Digital photography is under discussion to gain data with high precision and accuracy with one major disadvantage. To be able to see food and brood stages in photographs, bees have to be removed from combs. This, however, results in a disturbance of the colony – especially if the assessments take place in short time intervals of 7 ± 1 days. An experiment was performed to evaluate the variation between individual beekeepers and to compare the results to data generated with photographs. For the experiment, five colonies were assessed each by four beekeepers independently according to Liebefeld method. Each comb side of the five colonies was photographed with and without honeybees sitting on it for precise analysis at the computer for a number of bees, nectar cells, pollen cells, eggs, open brood and capped brood. The number of bees and cells with the different contents were generated by an area-based assessment in ImageJ as well as a detailed counting with help of HiveAnalyzer® Software. Data from beekeeper estimations were then compared with assessments based on digital photography. With the results of the experiment, we tried to answer several questions. With the study, we wanted to determine the level of variation between the beekeepers for the live stages and food stores estimated

Einige asymptotische Eigenschaften stochastischer Flüsse

Die vorliegende Arbeit behandelt einige asymptotische Eigenschaften gewisser stochastischer Fluesse auf dem Euklidischen Raum, deren Verteilungen meistens invariant gegenueber Rotationen des Zustandsraumes sind. Im Detail sind dies zuerst die isotropen Brownschen Fluesse, deren Verteilungen zusaetzlich invariant gegenueber Translationen des Zustandsraumes sind und deren Studium bis auf Yaglom (1957) oder Baxendale, Harris und Le Jan (1980er) zurueck geht. Desweiteren werden isotrope Ornstein-Uhlenbeck-Fluesse behandelt, die seit Dimitroff (2006) untersucht werden sowie repulsive isotrope Fluesse, die erst in dieser Arbeit eingefuehrt werden. Alle diese stochastischen Fluesse werden durch eine gemeinsame stochastische Differentialgleichung verbunden, die durch Spezifikationen eines reellen Parameters in die Gleichungen fuer die oben genannten Klassen stochastischer Fluesse uebergeht. Zunaechst wird in einem ersten Kurzkapitel zum Aufwaermen ein Lemma ueber das raeumliche asymptotische Verhalten isotroper Fluesse bewiesen, das spaeter dazu dienen wird, den Nachweis dafuer zu erbringen, dass die durch stereographische Projektion gewonnene Kugelversion eines Ornstein-Uhlenbeck Flusses nicht stetig differenzierbar ist und sich daher der Anwendung der bekannten Ergebnisse von Ledrappier und Young (1980er) entzieht. Ein weiteres Kurzkapitel behandelt rudimentaer die Frage, wann die n-Punkt Bewegung eines stochastischen Flusses eine stetige oder sogar differenzierbare Dichte besitzt, die zudem abseits der verallgemeinerten Diagonalen strikt positiv ist. Die Anwendbarkeit des gefundenen Kriteriums auf eine grosse Teilklasse der hier behandelten Fluesse wird nachgewiesen. Im darauf folgenden ersten Hauptkapitel wird die asymptotische Form der Menge aller von einer kompakten Menge besuchten Punkte untersucht und nachgewiesen, dass diese Form im Falle eines planaren isotropen Brownschen Flussess in Wahrscheinlichkeit in einem gewissen Sinn deterministisch ist. Die Beweisfuehrung folgt dabei teilweise Dolgopyat, Kaloshin und Koralov, die ein aehnliches Resultat in einem anders gelagerten Fall untersucht haben (2004). Der wesentliche Schritt ist jedoch neu, weil die dort angewandte Argumentationsweise im betrachteten Fall nicht anwendbar ist. Die in den hier betrachteten Faellen nicht gegebene raeumliche Periodizitaet wird dabei durch die Invarianz-Eigenschaften isotroper Brownscher Fluesse gegenueber Zeitumkehr ersetzt. Die sogenannte Margulis-Ruelle Ungleichung besagt, das die Entropie eines zufaelligen dynamischen Systems nach oben durch die Summe seiner positiven Lyapunov-Exponenten abgeschaetzt werden kann. Diese Ungleichung wird auf den Fall des zufaelligen dynamischen Systems, das auf kanonische Weise aus einem isotropen Ornstein-Uhlenbeck Fluss erhalten werden kann, erweitert. In den letzten beiden Hauptkapiteln wird die Frage nach dem asymptotischen Verhalten des Supremums der raeumlichen Ableitungen eines stochastischen Flusses behandelt. Es wird nachgewiesen, dass dieses Supremum (genommen ueber eine kompakte Startmenge) mit der Zeit hoechstens exponentiell schnell waechst und eine Schranke fuer die Rate wird angegeben. Dieses Resultat ist eine Vorstufe, die benoetigt wird, um aus der Margulis-Ruelle Ungleichung die Pesinsche Formel machen zu koennen, d.h. in dieser Gleichheit zu erreichen. Zunaechst werden die ersten Ableitungen eines isotropen Flusses betrachtet und danach wird das Resultat - allerdings mit i.A. schlechteren Konstanten - auf eine sehr viel allgemeinere Klasse stochastischer Fluesse sowie Ableitungen beliebiger Ordnung verallgemeinert. Zuletzt werden Moeglichkeiten fuer weitere Untersuchungen und offene Fragen aufgezeigt.The present work treats several asymptotic properties of stochastic flows on the Euclidean space, whose distributions are frequently assumed to be invariant under rotations of the state space. These stochastic flows include the isotropic Brownian flows, which have been studied since Yaglom (1957) or Baxendale, Harris and Le Jan (1980s). Furthermore isotropic Ornstein-Uhlenbeck Flows are treated, which are considered since Dimitroff (2006) as well as repulsive isotropic flows, which are about to be introduced in this work. All these classes of stochastic flows are linked by a single stochastic differential equation which passes to one of the named cases by the specification of one real parameter. First we define the classes of models and cite important facts from the literature. Afterwards the spatial asymptotic behaviour of isotropic stochastic flows is treated in a short warm up chapter. A lemma is proved that serves to show that the unit ball based random dynamical system coming from an isotropic Ornstein-Uhlenbeck flow is not sufficiently smooth to apply well known results concerning Pesin's formula from Ledrappier and Young (1980s) and hence to motivate the self contained study of this subject. Another short chapter is concerned with the following question. When does the finite-point motion of a given stochastic flow admit a continuous (or even smooth) density which is strictly positive apart from the generalized diagonal? It is also shown that a large subclass of the isotropic flows belongs to the scope of the obtained results. The following first main chapter treats the asymptotic behaviour of the shape of the set of points in the plane that has been visited up to some time. It is shown in the case of a planar isotropic Brownian flow that this shape is deteministic in probability. Dolgopyat, Kaloshin and Koralov give a similar result in a different setting (2004). But since the core of their proof - the spatial periodicity of their model - fails to hold for all isotropic flows it has to be replaced by a different feature of the isotropic Brownian flows namely their invariance properties w.r.t. time reversion. The so-called Margulis-Ruelle inequality asserts that the entropy of a random dynamical system can be estimated from above through the sum of its positive Lyapunov exponents. This inequality is extended to the case of a random dynamical system coming from an isotropic Ornstein-Uhlenbeck flow. The last two main chapters are devoted to the asymptotic expansion of the spatial derivative of a stochastic flow taking the supremum over a compact set of initial points in space. It is shown that this expansion is at most exponentially fast in time and a deterministic bound on the expansion speed is obtained. This result can be seen as a first step towards Pesin's formula for isotropic Ornstein-Uhlenbeck flows. First the case of first order derivatives of an isotropic flow is treated and afterwards the result is generalized - with worse constants - to a much more general class of stochastic flows and derivatives of arbitrary order. Finally some open questions are listed and possible directions of further research are discussed