170 research outputs found
Harnack-Ungleichungen und Anwendungen auf stochastische Gleichungen
No full textOuyang S-X. Harnack inequalities and applications for stochastic equations. Bielefeld (Germany): Bielefeld University; 2009.In dieser Dissertation studieren wir hauptsächlich Wangs Harnack-Ungleichungen und ihre Anwendungen auf Übergangshalbgruppen, die zu stochastischen Gleichungen gehören.
Wir betrachten endlich-dimensionale stochastische (gewöhnliche) Differentialgleichungen mit irregulärem Drift, unendlich-dimensionale (semi-)lineare stochastische partielle Differentialgleichungen mit Gauß'schem oder Lévy-Rauschen, mehrwertige stochastische Differentialgleichungen in endlich-dimensionalen Räumen und mehrwertige stochastische Evolutionsgleichungen in Banach-Räumen.
Die Anwendungen der Harnack-Ungleichungen beinhalten das Studium von Regularisierungs-Eigenschaften wie der starken Feller-Eigenschaft, Wärmeleitungskern-Abschätzungen und Hyper-Beschränktheit usw. für die Übergangshalbgruppen.
Die wichtigste benutzte Methode, um Harnack-Ungleichungen zu erhalten, ist die Transformation von Maßen, genauer Bildmaß-Transformationen und die Girsanov-Transformation (zusammen mit einem Kopplungsargument). Die letztere Methode wurde von Arnaudon et al. in 2006 eingeführt.
Mithilfe einfacher Bildmaß-Transformationen zeigen wir eine optimale Harnack-Ungleichung für die Gauß'sche Ornstein-Uhlenbeck-Halbgruppe. Dies ist eine Verbesserung der von Röckner und Wang in 2003 erhaltenen Harnack-Ungleichung. Darüber hinaus zeigen wir, dass diese Ungleichung äquivalent zur starken Feller-Eigenschaft der Gauß'schen Ornstein-Uhlenbeck-Halbgruppe ist.
Durch Koppeln und die Girsanov-Transformation erhalten wir eine Harnack-Ungleichung für Ornstein-Uhlenbeck-Prozesse mit Lévy-Rauschen. Der Drift in der Girsanov-Transformation ist eine Null-Kontrolle eines linearen Systems. Durch Optimierung über alle Null-Kontrollen erhalten wir eine Harnack-Ungleichung, die die gleiche Form hat wie die für Ornstein-Uhlenbeck-Prozesse mit Gauß'schem Rauschen. Diese Harnack-Ungleichung verallgemeinert und verbessert die von Röckner und Wang in 2003 erhaltene.
Mit der gleichen Methode und der Wahl geeigneter Drifts erhalten wir auch Harnack-Ungleichungen für andere stochastische Gleichungen.
Wichtige Aspekte der Methode von Arnaudon et al. sind absolute Stetigkeit und das erfolgreiche Koppeln von Prozessen. Wir behandeln diese Themen in zwei Kapiteln der Arbeit.
Wir zeigen ein Girsanov-Theorem für Lévy-Prozesse in unendlich-dimensionalen Räumen. Des weiteren untersuchen wir die absolute Stetigkeit von Lévy-Prozessen. Dieser Teil ist eine unendlich-dimensionale Version der Vorlesungen von Sato über Dichte-Transformationen von Lévy-Prozessen in 2000.
Wir zeigen ein Klebe-Lemma, dass das Martingalproblem für die Summe zweier durch Stoppzeiten getrennter Operatoren löst. Es ist eine Verallgemeinerung eines Lemmas von Chen und Li, 1989. Wir wenden dieses Resultat an, um die Existenz einer schwachen Lösung der gekoppelten Gleichung zu erhalten.
Für mehrwertige stochastische Evolutionsgleichungen untersuchen wir auch die Konzentrationseigenschaft der invarianten Maße. Diese Eigenschaft verallgemeinert die von Zhang in 2007 untersuchte.
Des weiteren untersuchen wir Entropie-Kosten- und HWI-Ungleichungen für Ornstein-Uhlenbeck-Prozesse mit Gauß'schem Rauschen.In this thesis, we mainly study Wang's Harnack inequalities and their applications for the transition semigroups associated with stochastic equations.
Among the stochastic equations, we consider finite dimensional stochastic differential equations with irregular drifts; infinite dimensional (semi-)linear stochastic partial differential equations with Gaussian or Lévy noise; multivalued stochastic differential equations in finite dimension and multivalued stochastic evolution equations in Banach spaces.
The applications of Harnack inequalities include the study of regularizing properties like strong Feller property, heat kernel estimates and hyperboundedness etc. for the transition semigroups.
The main method we used to establish Harnack inequalities is measure transformation. There are two aspects: image measure transformation and Girsanov transformation (combined with a coupling argument). The latter method was introduced by Arnaudon et al. in 2006.
By simple image measure transformation (e.g. use the Cameron-Martin formula in Gaussian case), we prove an optimal Harnack inequality for the Gaussian Ornstein-Uhlenbeck semigroup. This is an improvement of the Harnack inequality obtained by Röckner and Wang in 2003. Moreover, we prove that this inequality is equivalent with the strong Feller property of the Gaussian Ornstein-Uhlenbeck semigroup.
By coupling and Girsanov transformation we obtain a Harnack inequality for Lévy driven Ornstein-Uhlenbeck processes. The drift in the Girsanov transformation is a null control of some linear system. By optimizing over all null controls, we obtain a Harnack inequality which is of the same form as the Harnack inequality for Ornstein-Uhlenbeck processes driven by Wiener processes. This Harnack inequality also generalizes and improves the one obtained by Röckner and Wang in 2003.
By using the same procedure and taking proper drifts, we also establish Harnack inequalities for other stochastic equations.
Two crucial ingredients of the method of Arnaudon et al. are the absolute continuity and successful coupling of processes. We also devote two chapters on these topics.
We prove a Girsanov theorem for Lévy processes in infinite dimensional spaces. Moreover, we studied the absolute continuity of Lévy processes. This part is an infinite dimensional version of the results in the lecture notes of Sato on density transformation of Lévy processes in 2000.
We observed a gluing lemma which solves the martingale problem for the sum of two operators separated by some stopping time. It is a generalization of a lemma stated by Chen and Li in 1989. We apply this result for the existence of the weak solution for coupled equations. This makes it possible for us to establish Harnack inequality for stochastic differential equations with weak solutions.
For multivalued stochastic evolution equations, we also studied the concentration property of the invariant measures. This property generalizes the one studied by Zhang in 2007.
As a complement, we also study entropy cost and HWI inequalities for Ornstein-Uhlenbeck processes driven by Wiener processes
Elliptic and parabolic Harnack inequalitys: an abstract approach
No full textFil: Toledano, Ricardo Daniel. Universidad Nacional del Litoral. Facultad de Ingeniería Química; Argentina.Necessary and sufficient conditions are determined in order to obtain Holder regularity for a family of functions defined in a space of homogeneous type. Such conditions are obtained from the study of the validity of the Harnack inequality in this abstract context. The Harnack inequality is a useful tool in the theory of Holder regularity of solutions of partial differential equations of both elliptic and parabolic type.Se determinan condiciones necesarias y suficientes para que una familia de funciones definidas en un espacio de tipo homogéneo tenga regularidad Holder. Tales condiciones se determinan a partir de la validez, en este contexto abstracto, de la desigualdad de Harnack la cual constituye una herramienta fundamental en la teoría clásica del estudio de regularidad de soluciones de ecuaciones diferenciales en derivadas parciales elípticas y parabólicas.Consejo Nacional de Investigaciones Científicas y TécnicasUniversidad Nacional del Litora
Elliptic and parabolic Harnack inequalitys: an abstract approach
No full textFil: Toledano, Ricardo Daniel. Universidad Nacional del Litoral. Facultad de Ingeniería Química; Argentina.Necessary and sufficient conditions are determined in order to obtain Holder regularity for a family of functions defined in a space of homogeneous type. Such conditions are obtained from the study of the validity of the Harnack inequality in this abstract context. The Harnack inequality is a useful tool in the theory of Holder regularity of solutions of partial differential equations of both elliptic and parabolic type.Se determinan condiciones necesarias y suficientes para que una familia de funciones definidas en un espacio de tipo homogéneo tenga regularidad Holder. Tales condiciones se determinan a partir de la validez, en este contexto abstracto, de la desigualdad de Harnack la cual constituye una herramienta fundamental en la teoría clásica del estudio de regularidad de soluciones de ecuaciones diferenciales en derivadas parciales elípticas y parabólicas.Consejo Nacional de Investigaciones Científicas y TécnicasUniversidad Nacional del Litora
Hölder continuity and Harnack estimate for non-homogeneous parabolic equations
No full textIn this paper we continue the study on intrinsic Harnack inequality for non-homogeneous parabolic equations in non-divergence form initiated by the first author in Arya (Calc Var Partial Differ Equ 61:30–31, 2022). We establish a forward-in-time intrinsic Harnack inequality, which in particular implies the Hölder continuity of the solutions. We also provide a Harnack type estimate on global scale which quantifies the strong minimum principle. In the time-independent setting, this together with Arya (2022) provides an alternative proof of the generalized Harnack inequality proven by the second author in Julin (Arch Ration Mech Anal 216:673–702, 2015).peerReviewe
Multi-Harnack smoothings of real plane branches
No full textThis paper is motivated by the results of G. Mikhalkin about a certain class of real algebraic curves, called Harnack curves, in toric surfaces. Mikhalkin has proved the existence of such curves as well as topological uniqueness of their real locus.
The authors are concerned about an analogous statement in the case of a smoothing of a real plane branch (C, 0) _ (C2, 0) (an analytically irreducible germ of a real curve). They introduce the class of multi-Harnack smoothings of (C, 0) and prove its existence along with its topological uniqueness.
Theorem 9.3. Any real plane branch (C, 0) has a multi-Harnack smoothing.
Theorem 9.4. Let (C, 0) be a real branch. The topological type of multi-Harnack smoothings of (C, 0) is unique. There are at most two signed topological types of multi-Harnack smoothings of (C, 0). These types depend only on the sequence {(nj ,mj)}, which determines and is determined by the embedded topological type of (C, 0) _ (C2, 0).
In terms of the parameters, multi-Harnack smoothings are multi-semi-quasi-homogeneous, which lets the authors analyze also the asymptotic multi-scales of the ovals.Ministerio de Educación y Ciencia (MEC)Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu
The Harnack Distance on Bounded Domains in ℂ
No full textIn [1], the author has constructed the Harnack metric on the space and studied the conformal invariant as well as relations among the Harnack metric, the Bergman metric and the Carathéodory metric. In this paper, the authors obtain the Harnack distance on the domain in . Then we construct the Harnack metric when is a bounded domain. The main results of the paper show that, the Harnack metric on the bounded domain is complete and the topology induced by that metric is equivalent to the topology that is induced by the normal metric on . Moreover, by applying the conformal mapping theory and the conformal invariant of the Harnack distance, the authors obtain some formulas of the Harnack distance between two arbitrary points in some specific domains in the complex plane
H\"older Continuity and Harnack estimate for non-homogeneous parabolic equations
No full textIn this paper we continue the study on intrinsic Harnack inequality for non-
homogeneous parabolic equations in non-divergence form initiated by the first
author in [1]. We establish a forward-in-time intrinsic Harnack inequality,
which in particular implies the H\"older continuity of the solutions. We also
provide a Harnack type estimate on global scale which quantifies the strong
minimum principle. In the time-independent setting, this together with [1]
provides an alternative proof of the generalized Harnack inequality proven by
the second author in [9]
Interpolating between constrained Li–Yau and Chow–Hamilton Harnack inequalities for a nonlinear parabolic equation
No full textAbstractWe establish a one-parameter family of Harnack inequalities connecting the constrained trace Li–Yau differential Harnack inequality for a nonlinear parabolic equation to the constrained trace Chow–Hamilton Harnack inequality for this nonlinear equation with respect to evolving metrics related to the Ricci flow on a 2-dimensional closed manifold. This result can be regarded as a nonlinear version of the previous work of Y. Zheng and the author [J.-Y. Wu, Y. Zheng, Interpolating between constrained Li–Yau and Chow–Hamilton Harnack inequalities on a surface, Arch. Math., 94 (2010) 591–600]
Adolf von Harnack, Mission et expansion du christianisme aux trois premiers siècles
No full textAprès la traduction du Marcion de Harnack en 2003, cette nouvelle traduction d'une œuvre majeure du grand historien du christianisme servira toute perspective historique sur les premiers siècles du christianisme. Cette traduction repose sur la quatrième édition (1924) d'un ouvrage constamment revu et remanié qui servit de manuel tout au long du xxe siècle. Composé comme une grande fresque historique d'au moins quatre siècles, Harnack relève le défi de présenter ce que l'historien Th. Mommsen ..
Adolf von Harnack, Mission et expansion du christianisme aux trois premiers siècles
No full textAprès la traduction du Marcion de Harnack en 2003, cette nouvelle traduction d'une œuvre majeure du grand historien du christianisme servira toute perspective historique sur les premiers siècles du christianisme. Cette traduction repose sur la quatrième édition (1924) d'un ouvrage constamment revu et remanié qui servit de manuel tout au long du xxe siècle. Composé comme une grande fresque historique d'au moins quatre siècles, Harnack relève le défi de présenter ce que l'historien Th. Mommsen ..
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