Academic literature on the topic 'Stochastic Differential Geometry'
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Journal articles on the topic "Stochastic Differential Geometry"
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Kendall, Wilfrid S. "Stochastic differential geometry: An introduction." Acta Applicandae Mathematicae 9, no. 1-2 (1987): 29–60. http://dx.doi.org/10.1007/bf00580820.
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Catuogno, Pedro, and Paulo Ruffino. "Geometry of Stochastic Delay Differential Equations." Electronic Communications in Probability 10 (2005): 190–95. http://dx.doi.org/10.1214/ecp.v10-1151.
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Thiruthummal, Abhiram Anand, and Eun-jin Kim. "Monte Carlo Simulation of Stochastic Differential Equation to Study Information Geometry." Entropy 24, no. 8 (August 12, 2022): 1113. http://dx.doi.org/10.3390/e24081113.
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Information Geometry is a useful tool to study and compare the solutions of a Stochastic Differential Equations (SDEs) for non-equilibrium systems. As an alternative method to solving the Fokker–Planck equation, we propose a new method to calculate time-dependent probability density functions (PDFs) and to study Information Geometry using Monte Carlo (MC) simulation of SDEs. Specifically, we develop a new MC SDE method to overcome the challenges in calculating a time-dependent PDF and information geometric diagnostics and to speed up simulations by utilizing GPU computing. Using MC SDE simulations, we reproduce Information Geometric scaling relations found from the Fokker–Planck method for the case of a stochastic process with linear and cubic damping terms. We showcase the advantage of MC SDE simulation over FPE solvers by calculating unequal time joint PDFs. For the linear process with a linear damping force, joint PDF is found to be a Gaussian. In contrast, for the cubic process with a cubic damping force, joint PDF exhibits a bimodal structure, even in a stationary state. This suggests a finite memory time induced by a nonlinear force. Furthermore, several power-law scalings in the characteristics of bimodal PDFs are identified and investigated.
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Kendall, Wilfrid S. "Stochastic Differential Geometry, a Coupling Property, and Harmonic Maps." Journal of the London Mathematical Society s2-33, no. 3 (June 1986): 554–66. http://dx.doi.org/10.1112/jlms/s2-33.3.554.
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Dimakis, Aristophanes, and Folkert M�ller-Hoissen. "Stochastic differential calculus, the Moyal *-product, and noncommutative geometry." Letters in Mathematical Physics 28, no. 2 (June 1993): 123–37. http://dx.doi.org/10.1007/bf00750305.
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Manton, Jonathan H. "A Primer on Stochastic Differential Geometry for Signal Processing." IEEE Journal of Selected Topics in Signal Processing 7, no. 4 (August 2013): 681–99. http://dx.doi.org/10.1109/jstsp.2013.2264798.
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Kühnel, Line, Stefan Sommer, and Alexis Arnaudon. "Differential geometry and stochastic dynamics with deep learning numerics." Applied Mathematics and Computation 356 (September 2019): 411–37. http://dx.doi.org/10.1016/j.amc.2019.03.044.
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Agrachev, Andrei, Ugo Boscain, Robert Neel, and Luca Rizzi. "Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling." ESAIM: Control, Optimisation and Calculus of Variations 24, no. 3 (2018): 1075–105. http://dx.doi.org/10.1051/cocv/2017037.
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We relate some constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular attention to walks where the choice of step is influenced by a volume on the manifold. A primary motivation is to explore how one can pass, in the parabolic scaling limit, from geodesics, orthonormal frames, and/or volumes to diffusions, and hence their infinitesimal generators, on sub-Riemannian manifolds, which is interesting in light of the fact that there is no completely canonical notion of sub-Laplacian on a general sub-Riemannian manifold. However, even in the Riemannian case, this random walk approach illuminates the geometric significance of Ito and Stratonovich stochastic differential equations as well as the role played by the volume.
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Ashyralyev, Allaberen, and Ülker Okur. "Stability of Stochastic Partial Differential Equations." Axioms 12, no. 7 (July 24, 2023): 718. http://dx.doi.org/10.3390/axioms12070718.
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In this paper, we study the stability of the stochastic parabolic differential equation with dependent coefficients. We consider the stability of an abstract Cauchy problem for the solution of certain stochastic parabolic differential equations in a Hilbert space. For the solution of the initial-boundary value problems (IBVPs), we obtain the stability estimates for stochastic parabolic equations with dependent coefficients in specific applications.
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Andrianantenainarinoro, T. R. H., R. A. Randrianomenjanahary, and T. J. Rabeherimanana. "AMPLITUDE ADJUSTMENT WITH FIWASVJ MODEL." Advances in Mathematics: Scientific Journal 11, no. 4 (April 28, 2022): 383–413. http://dx.doi.org/10.37418/amsj.11.4.7.
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Andrianantenainarinoro\cite{maReference101} remarked that the price amplitudes of financial models may not correspond to the reality and we propose here a model in continuous time Fractionally Integrated WASC Stochastic Volatility Jump. To do this, we introduce a fractal index in the WASC Stochastic Volatility Jump model and we have two others characteristics: amplitude adjustment and memory of process. We present also several theories in stochastic calculus, algebraic, differential geometry, numerical method and estimating method which can use to financial such us: sense of a fractional integral, relationship between trace and determinant operator, Euler's approximation for an unresolved differential equation and convergence speed.
Dissertations / Theses on the topic "Stochastic Differential Geometry"
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Yang, Weiye. "Stochastic analysis and stochastic PDEs on fractals." Thesis, University of Oxford, 2018. http://ora.ox.ac.uk/objects/uuid:43a7af74-c531-424a-9f3d-4277138affbb.
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Stochastic analysis on fractals is, as one might expect, a subfield of analysis on fractals. An intuitive starting point is to observe that on many fractals, one can define diffusion processes whose law is in some sense invariant with respect to the symmetries and self-similarities of the fractal. These can be interpreted as fractal-valued counterparts of standard Brownian motion on Rd. One can study these diffusions directly, for example by computing heat kernel and hitting time estimates. On the other hand, by associating the infinitesimal generator of the fractal-valued diffusion with the Laplacian on Rd, it is possible to pose stochastic partial differential equations on the fractal such as the stochastic heat equation and stochastic wave equation. In this thesis we investigate a variety of questions concerning the properties of diffusions on fractals and the parabolic and hyperbolic SPDEs associated with them. Key results include an extension of Kolmogorov's continuity theorem to stochastic processes indexed by fractals, and existence and uniqueness of solutions to parabolic SPDEs on fractals with Lipschitz data.
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Morgado, Leandro Batista 1977. "Dinâmica de semimartingales com saltos : decomposição e retardo." [s.n.], 2015. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306280.
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Orientador: Paulo Regis Caron RuffinoTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: Este trabalho aborda alguns aspectos da teoria de equações diferenciais estocásticas em relação a semimartingales com saltos, suas aplicações na decomposição de fluxos estocásticos em variedades, bem como algumas implicações de natureza geométrica. Inicialmente, em uma variedade munida de distribuições complementares, discutimos o problema da decomposição de fluxos estocásticos contínuos, isto é, gerados por EDE em relação ao movimento Browniano. Resultados anteriores garantem a existência de uma decomposição em difeomorfismos que preservam as distribuições até um tempo de parada. Usando a assim denominada equação de Marcus, bem como uma técnica que denominamos equação 'stop and go', vamos construir um fluxo estocástico próximo ao original, com a propriedade adicional que o fluxo construído pode ser decomposto além do tempo de parada inicial. Em seguida, trataremos da decomposição de fluxos estocásticos no caso descontínuo, isto é, para processos gerados por uma EDE em relação a um semimartingale com saltos. Após uma discussão sobre a existência da decomposição, obtemos as EDEs para as componentes respectivas, a partir de uma extensão que propomos da fórmula de Itô-Ventzel-Kunita. Finalmente, propomos um modelo de equações diferenciais estocásticas com retardo incluindo saltos. A ideia é modelar certos fenômenos em que a informação pode chegar ao receptor por diferentes canais: de forma contínua, mas com retardo, e em tempos discretos, de forma instantânea. Vamos abordar aspectos geométricos relacionados ao tema: transporte paralelo em curvas diferenciáveis com saltos, bem como possibilidade de levantamento de uma solução do nosso modelo de equação para o fibrado de bases de uma variedade diferenciável
Abstract: The main subject of this thesis is the theory of stochastic differential equations driven by semimartingales with jumps. We consider applications in the decomposition of stochastic flows in differentiable manifolds, and geometrical aspects about these equations. Initially, in a differentiable manifold endowed with a pair of complementary distributions, we discuss the decomposition of continuous stochastic flows, that is, flows generated by SDEs driven by Brownian motion. Previous results guarantee that, under some assumptions, there exists a decomposition in diffeomorphisms that preserves the distributions up to a stopping time. Using the so called Marcus equation, and a technique that we call 'stop and go' equation, we construct a stochastic flow close to the original one, with the property that the constructed flow can be decomposed further on the stopping time. After, we deal with the decomposition of stochastic flows in the discontinuous case, that is, processes generated by SDEs driven by semimartingales with jumps. We discuss the existence of this decomposition, and obtain the SDEs for the respective components, using an extension of the Itô-Ventzel-Kunita formula. Finally, we propose a model of stochastic differential equations including delay and jumps. The idea is to describe some phenomena such that the information comes to the receptor by different channels: continuously, with some delay, and in discrete times, instantaneously. We deal with geometrical aspects related with this subject: parallel transport in càdlàg curves, and lifting of solutions of these equations to the linear frame bundle of a differentiable manifold
Doutorado
Matematica
Doutor em Matemática
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Silva, Júnior Rinaldo Vieira da 1981. "Calculo estocastico em variedades Finsler." [s.n.], 2005. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306287.
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Orientador: Paulo Regis Caron RuffinoDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Nesta dissertação fizemos um estudo da teoria de difusão em variedades Finsler, onde abor-damos o transporte paralelo estocástico, desenvolvimento estocástico de Cartan e Movimento Browniano. O objetivo principal é obter uma descrição mais geométrica dos objetos citados acima ainda que por enquanto em coordenadas locais e assim termos um paralelo entre o cálculo estocástico em variedades Riemannianas e variedades Finsler
Abstract: In this work we study diffusion theory in Finsler manifolds. It includes the stochastic par-allel transport, stochastic Cartan development and Brownian motion. The main objective is to provide a geometric description of the objects mentioned and 50 to draw a compari-50n between stochastic calculus in Riemannian manifolds and stochastic calculus in Finsler manifolds
Mestrado
Matematica
Mestre em Matemática
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Björnberg, Jakob Erik. "Graphical representations of Ising and Potts models : Stochastic geometry of the quantum Ising model and the space-time Potts model." Doctoral thesis, KTH, Matematik (Inst.), 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-11267.
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HTML clipboard Statistical physics seeks to explain macroscopic properties of matter in terms of microscopic interactions. Of particular interest is the phenomenon of phase transition: the sudden changes in macroscopic properties as external conditions are varied. Two models in particular are of great interest to mathematicians, namely the Ising model of a magnet and the percolation model of a porous solid. These models in turn are part of the unifying framework of the random-cluster representation, a model for random graphs which was first studied by Fortuin and Kasteleyn in the 1970’s. The random-cluster representation has proved extremely useful in proving important facts about the Ising model and similar models. In this work we study the corresponding graphical framework for two related models. The first model is the transverse field quantum Ising model, an extension of the original Ising model which was introduced by Lieb, Schultz and Mattis in the 1960’s. The second model is the space–time percolation process, which is closely related to the contact model for the spread of disease. In Chapter 2 we define the appropriate space–time random-cluster model and explore a range of useful probabilistic techniques for studying it. The space– time Potts model emerges as a natural generalization of the quantum Ising model. The basic properties of the phase transitions in these models are treated in this chapter, such as the fact that there is at most one unbounded fk-cluster, and the resulting lower bound on the critical value in 
. In Chapter 3 we develop an alternative graphical representation of the quantum Ising model, called the random-parity representation. This representation is based on the random-current representation of the classical Ising model, and allows us to study in much greater detail the phase transition and critical behaviour. A major aim of this chapter is to prove sharpness of the phase transition in the quantum Ising model—a central issue in the theory— and to establish bounds on some critical exponents. We address these issues by using the random-parity representation to establish certain differential inequalities, integration of which gives the results. In Chapter 4 we explore some consequences and possible extensions of the results established in Chapters 2 and 3. For example, we determine the critical point for the quantum Ising model in and in ‘star-like’ geometries.HTML clipboard Statistisk fysik syftar till att förklara ett materials makroskopiska egenskaper i termer av dess mikroskopiska struktur. En särskilt intressant egenskap är är fenomenet fasövergång, det vill säga en plötslig förändring i de makroskopiska egenskaperna när externa förutsättningar varieras. Två modeller är särskilt intressanta för en matematiker, nämligen Ising-modellen av en magnet och perkolationsmodellen av ett poröst material. Dessa två modeller sammanförs av den så-kallade fk-modellen, en slumpgrafsmodell som först studerades av Fortuin och Kasteleyn på 1970-talet. fk-modellen har sedermera visat sig vara extremt användbar för att bevisa viktiga resultat om Ising-modellen och liknande modeller. I den här avhandlingen studeras den motsvarande grafiska strukturen hos två näraliggande modeller. Den första av dessa är den kvantteoretiska Isingmodellen med transverst fält, vilken är en utveckling av den klassiska Isingmodellen och först studerades av Lieb, Schultz och Mattis på 1960-talet. Den andra modellen är rumtid-perkolation, som är nära besläktad med kontaktmodellen av infektionsspridning. I Kapitel 2 definieras rumtid-fk-modellen, och flera probabilistiska verktyg utforskas för att studera dess grundläggande egenskaper. Vi möter rumtid-Potts-modellen, som uppenbarar sig som en naturlig generalisering av den kvantteoretiska Ising-modellen. De viktigaste egenskaperna hos fasövergången i dessa modeller behandlas i detta kapitel, exempelvis det faktum att det i fk-modellen finns högst en obegränsad komponent, samt den undre gräns för det kritiska värdet som detta innebär. I Kapitel 3 utvecklas en alternativ grafisk framställning av den kvantteoretiska Ising-modellen, den så-kallade slumpparitetsframställningen. Denna är baserad på slumpflödesframställningen av den klassiska Ising-modellen, och är ett verktyg som låter oss studera fasövergången och gränsbeteendet mycket närmare. Huvudsyftet med detta kapitel är att bevisa att fasövergången är skarp—en central egenskap—samt att fastslå olikheter för vissa kritiska exponenter. Metoden består i att använda slumpparitetsframställningen för att härleda vissa differentialolikheter, vilka sedan kan integreras för att lägga fast att gränsen är skarp. I Kapitel 4 utforskas några konsekvenser, samt möjliga vidareutvecklingar, av resultaten i de tidigare kapitlen. Exempelvis bestäms det kritiska värdet hos den kvantteoretiska Ising-modellen på
, samt i ‘stjärnliknankde’ geometrier.QC 20100705
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Stelmastchuk, Simão Nicolau 1977. "Martingales no fibrado de bases e seções harmonicas via calculo estocastico." [s.n.], 2007. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306326.
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Orientador: Pedro Jose CatuognoTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Neste trabalho estudamos os martingales no fibrado de bases e suas relações com os martingales no fibrado tangente. Caracterizamos as aplicações harmônicas a valores no fibrado de bases e as relacionamos com as aplicações harmônicas a valores no fibrado tangente. Numa segunda parte estudamos a harmonicidade das seções de um fibrado via geometria estocástica. Seja P(M;G) um fibrado principal e E(M;N; G; P) um fibrado associado a P(M;G). Entre outros resultados obtemos que: uma seção s : M - E é harmônica se, e somente se, o seu levantamento eqüivariante Fs : P - N é horizontalmente harmônico; e se a ação à esquerda de G × N em N não fixa pontos então não existe seção s : M - E harmônica ou toda seção harmônica é nula
Abstract: Neste trabalho estudamos os martingales no fibrado de bases e suas relações com os martingales no fibrado tangente. Caracterizamos as aplicações harmônicas a valores no fibrado de bases e as relacionamos com as aplicações harmônicas a valores no fibrado tangente. Numa segunda parte estudamos a harmonicidade das seções de um fibrado via geometria estocástica. Seja P(M;G) um fibrado principal e E(M;N; G; P) um fibrado associado a P(M;G). Entre outros resultados obtemos que: uma seção s : M - E é harmônica se, e somente se, o seu levantamento eqüivariante Fs : P - N é horizontalmente harmônico; e se a ação à esquerda de G × N em N não fixa pontos então não existe seção s : M - E harmônica ou toda seção harmônica é nula
Doutorado
Geometria Estocastica
Doutor em Matemática
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Mora, Marianne. "Sur la geometrie differentielle en statistique : sur la convergence des suites de fonctions variance des familles exponentielles naturelles." Toulouse 3, 1988. http://www.theses.fr/1988TOU30044.
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La these comporte deux parties independantes. La premiere partie traite de l'utilisation de la geometrie differentielle en statistique. Dans les chapitres i et ii sont rappelees les notions fondamentales de geometrie differentielle et de statistique. Le chapitre iii est consacre a la theorie des "strings" dont de nombreux exemples apparaissent en statistique inferentielle. Ce nouveau concept a ete introduit et etudie par m. M. O. E. Barndorff nielsen et p. Blaesild. Nous en donnons ici une nouvelle definition, purement mathematique, basee sur un concept de differenciation d'ordre superieur, les objets differencies etant des fonctions, champs de vecteurs tangents, contangents ou jets. Enfin, dans le chapitre iv, a partir de structures geometriques specifiques definies sur des modeles statistiques parametriques reguliers et basees sur un point de vue de conditionnement pour une statistique ancillaire donnee, nous elaborons des developpements asymptotiques pour les lois du vecteur score et de l'estimateur du maximum de vraisemblance. La seconde partie concerne les familles exponentielles naturelles k-dimensionnelles et les fonctions-variance qui les caracterisent. Dans ce contexte nous etablissons dans le chapitre v un theoreme de convergence qui montre que l'ensemble des fonctions variances est ferme pour la convergence uniforme sur tout compact
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Herzog, David Paul. "Geometry's Fundamental Role in the Stability of Stochastic Differential Equations." Diss., The University of Arizona, 2011. http://hdl.handle.net/10150/145150.
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We study dynamical systems in the complex plane under the effect of constant noise. We show for a wide class of polynomial equations that the ergodic property is valid in the associated stochastic perturbation if and only if the noise added is in the direction transversal to all unstable trajectories of the deterministic system. This has the interpretation that noise in the "right" direction prevents the process from being unstable: a fundamental, but not well-understood, geometric principle which seems to underlie many other similar equations. The result is proven by using Lyapunov functions and geometric control theory.
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Lagrange, Jean-Michel. "Reconstruction tomographique à partir d'un petit nombre de vues." Cachan, Ecole normale supérieure, 1998. http://www.theses.fr/1998DENS0038.
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Cette thèse est consacrée à la reconstruction tomographique d'un objet 3d a partir d'un petit nombre de vues, sous des hypothèses de symétrie. Les inversions classiques étant très instables, j'ai aborde ce problème par une modélisation de l'objet : la reconstruction consiste à déterminer les paramètres du modèle en comparant les projections de ce dernier aux données. L'éclairement étant parallèle, j'ai tout d'abord étudié la reconstruction d'une coupe plane. Le modèle 1d est alors décrit par zones et comporte n paramètres. Leur recherche s'exprime sous la forme de la minimisation sous contraintes d'un critère quadratique. Ce dernier n'étant pas différentiable, j'ai propose trois techniques permettant de le régulariser. J'ai, ensuite, étudié la sensibilité du vecteur paramétré au bruit présent sur les données après avoir déterminé la matrice de covariance de ce vecteur. J'ai enfin propose une reconstruction de toutes les coupes de l'objet par itération de cette approche 1d en ajoutant un terme de lissage entre les coupes. Je me suis ensuite orienté vers la construction d'un modèle 3d des objets, caractérisé par six grandeurs : trois surfaces séparatrices (engendrées par la rotation de trois courbes planes) et les champs de densité sur ces interfaces. La première difficulté a résidé dans le remplissage d'un champ 3d a partir des champs de densité sur les interfaces. J'ai alors propose un operateur elliptique adaptatif assurant cette opération. A interfaces fixées, j'ai implémenté la recherche des densités sur chacune d'elles. J'ai ensuite formalise la déformation de ces surfaces, c'est-a-dire des génératrices planes. Elle est caractérisée par la recherche d'une base optimale des déformations obtenues par ACP sur un jeu d'exemples. Ce jeu est construit de manière aléatoire : les réalisations sont obtenues par intégration de la solution d'une équation différentielle stochastique linéaire.
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Gairing, Jan Martin. "Variational and Ergodic Methods for Stochastic Differential Equations Driven by Lévy Processes." Doctoral thesis, Humboldt-Universität zu Berlin, 2018. http://dx.doi.org/10.18452/18984.
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Diese Dissertation untersucht Aspekte des Zusammenspiels von ergodischem Langzeitver-
halten und der Glättungseigenschaft dynamischer Systeme, die von stochastischen Differen-
tialgleichungen (SDEs) mit Sprüngen erzeugt sind. Im Speziellen werden SDEs getrieben
von Lévy-Prozessen und der Marcusschen kanonischen Gleichung untersucht. Ein vari-
ationeller Ansatz für den Malliavin-Kalkül liefert eine partielle Integration, sodass eine
Variation im Raum in eine Variation im Wahrscheinlichkeitsmaß überführt werden kann.
Damit lässt sich die starke Feller-Eigenschaft und die Existenz glatter Dichten der zuge-
hörigen Markov-Halbgruppe aus einer nichtstandard Elliptizitätsbedingung an eine Kom-
bination aus Gaußscher und Sprung-Kovarianz ableiten. Resultate für Sprungdiffusionen
auf Untermannigfaltigkeiten werden aus dem umgebenden Euklidischen Raum hergeleitet.
Diese Resultate werden dann auf zufällige dynamische Systeme angewandt, die von lin-
earen stochastischen Differentialgleichungen erzeugt sind. Ruelles Integrierbarkeitsbedin-
gung entspricht einer Integrierbarkeitsbedingung an das Lévy-Maß und gewährleistet die
Gültigkeit von Oseledets multiplikativem Ergodentheorem. Damit folgt die Existenz eines
Lyapunov-Spektrums. Schließlich wird der top Lyapunov-Exponent über eine Formel der
Art von Furstenberg–Khasminsikii als ein ergodisches Mittel der infinitesimalen Wachs-
tumsrate über die Einheitssphäre dargestellt.The present thesis investigates certain aspects of the interplay between the ergodic long time behavior and the smoothing property of dynamical systems generated by stochastic differential equations (SDEs) with jumps, in particular SDEs driven by Lévy processes and the Marcus’ canonical equation. A variational approach to the Malliavin calculus generates an integration-by-parts formula that allows to transfer spatial variation to variation in the probability measure. The strong Feller property of the associated Markov semigroup and the existence of smooth transition densities are deduced from a non-standard ellipticity condition on a combination of the Gaussian and a jump covariance. Similar results on submanifolds are inferred from the ambient Euclidean space. These results are then applied to random dynamical systems generated by linear stochas- tic differential equations. Ruelle’s integrability condition translates into an integrability condition for the Lévy measure and ensures the validity of the multiplicative ergodic theo- rem (MET) of Oseledets. Hence the exponential growth rate is governed by the Lyapunov spectrum. Finally the top Lyapunov exponent is represented by a formula of Furstenberg– Khasminskii–type as an ergodic average of the infinitesimal growth rate over the unit sphere.
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Friedrich, Benjamin M. "Nonlinear dynamics and fluctuations in biological systems." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2018. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-234307.
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The present habilitation thesis in theoretical biological physics addresses two central dynamical processes in cells and organisms: (i) active motility and motility control and (ii) self-organized pattern formation. The unifying theme is the nonlinear dynamics of biological function and its robustness in the presence of strong fluctuations, structural variations, and external perturbations. We theoretically investigate motility control at the cellular scale, using cilia and flagella as ideal model system. Cilia and flagella are highly conserved slender cell appendages that exhibit spontaneous bending waves. This flagellar beat represents a prime example of a chemo-mechanical oscillator, which is driven by the collective dynamics of molecular motors inside the flagellar axoneme. We study the nonlinear dynamics of flagellar swimming, steering, and synchronization, which encompasses shape control of the flagellar beat by chemical signals and mechanical forces. Mechanical forces can synchronize collections of flagella to beat at a common frequency, despite active motor noise that tends to randomize flagellar synchrony. In Chapter 2, we present a new physical mechanism for flagellar synchronization by mechanical self-stabilization that applies to free-swimming flagellated cells. This new mechanism is independent of direct hydrodynamic interactions between flagella. Comparison with experimental data provided by experimental collaboration partners in the laboratory of J. Howard (Yale, New Haven) confirmed our new mechanism in the model organism of the unicellular green alga Chlamydomonas. Further, we characterize the beating flagellum as a noisy oscillator. Using a minimal model of collective motor dynamics, we argue that measured non-equilibrium fluctuations of the flagellar beat result from stochastic motor dynamics at the molecular scale. Noise and mechanical coupling are antagonists for flagellar synchronization. In addition to the control of the flagellar beat by mechanical forces, we study the control of the flagellar beat by chemical signals in the context of sperm chemotaxis. We characterize a fundamental paradigm for navigation in external concentration gradients that relies on active swimming along helical paths. In this helical chemotaxis, the direction of a spatial concentration gradient becomes encoded in the phase of an oscillatory chemical signal. Helical chemotaxis represents a distinct gradient-sensing strategy, which is different from bacterial chemotaxis. Helical chemotaxis is employed, for example, by sperm cells from marine invertebrates with external fertilization. We present a theory of sensorimotor control, which combines hydrodynamic simulations of chiral flagellar swimming with a dynamic regulation of flagellar beat shape in response to chemical signals perceived by the cell. Our theory is compared to three-dimensional tracking experiments of sperm chemotaxis performed by the laboratory of U. B. Kaupp (CAESAR, Bonn). In addition to motility control, we investigate in Chapter 3 self-organized pattern formation in two selected biological systems at the cell and organism scale, respectively. On the cellular scale, we present a minimal physical mechanism for the spontaneous self-assembly of periodic cytoskeletal patterns, as observed in myofibrils in striated muscle cells. This minimal mechanism relies on the interplay of a passive coarsening process of crosslinked actin clusters and active cytoskeletal forces. This mechanism of cytoskeletal pattern formation exemplifies how local interactions can generate large-scale spatial order in active systems. On the organism scale, we present an extension of Turing’s framework for self-organized pattern formation that is capable of a proportionate scaling of steady-state patterns with system size. This new mechanism does not require any pre-pattering clues and can restore proportional patterns in regeneration scenarios. We analytically derive the hierarchy of steady-state patterns and analyze their stability and basins of attraction. We demonstrate that this scaling mechanism is structurally robust. Applications to the growth and regeneration dynamics in flatworms are discussed (experiments by J. Rink, MPI CBG, Dresden)Das Thema der vorliegenden Habilitationsschrift in Theoretischer Biologischer Physik ist die nichtlineare Dynamik funktionaler biologischer Systeme und deren Robustheit gegenüber Fluktuationen und äußeren Störungen. Wir entwickeln hierzu theoretische Beschreibungen für zwei grundlegende biologische Prozesse: (i) die zell-autonome Kontrolle aktiver Bewegung, sowie (ii) selbstorganisierte Musterbildung in Zellen und Organismen. In Kapitel 2, untersuchen wir Bewegungskontrolle auf zellulärer Ebene am Modelsystem von Zilien und Geißeln. Spontane Biegewellen dieser dünnen Zellfortsätze ermöglichen es eukaryotischen Zellen, in einer Flüssigkeit zu schwimmen. Wir beschreiben einen neuen physikalischen Mechanismus für die Synchronisation zweier schlagender Geißeln, unabhängig von direkten hydrodynamischen Wechselwirkungen. Der Vergleich mit experimentellen Daten, zur Verfügung gestellt von unseren experimentellen Kooperationspartnern im Labor von J. Howard (Yale, New Haven), bestätigt diesen neuen Mechanismus im Modellorganismus der einzelligen Grünalge Chlamydomonas. Der Gegenspieler dieser Synchronisation durch mechanische Kopplung sind Fluktuationen. Wir bestimmen erstmals Nichtgleichgewichts-Fluktuationen des Geißel-Schlags direkt, wofür wir eine neue Analyse-Methode der Grenzzykel-Rekonstruktion entwickeln. Die von uns gemessenen Fluktuationen entstehen mutmaßlich durch die stochastische Dynamik molekularen Motoren im Innern der Geißeln, welche auch den Geißelschlag antreiben. Um die statistische Physik dieser Nichtgleichgewichts-Fluktuationen zu verstehen, entwickeln wir eine analytische Theorie der Fluktuationen in einem minimalen Modell kollektiver Motor-Dynamik. Zusätzlich zur Regulation des Geißelschlags durch mechanische Kräfte untersuchen wir dessen Regulation durch chemische Signale am Modell der Chemotaxis von Spermien-Zellen. Dabei charakterisieren wir einen grundlegenden Mechanismus für die Navigation in externen Konzentrationsgradienten. Dieser Mechanismus beruht auf dem aktiven Schwimmen entlang von Spiralbahnen, wodurch ein räumlicher Konzentrationsgradient in der Phase eines oszillierenden chemischen Signals kodiert wird. Dieser Chemotaxis-Mechanismus unterscheidet sich grundlegend vom bekannten Chemotaxis-Mechanismus von Bakterien. Wir entwickeln eine Theorie der senso-motorischen Steuerung des Geißelschlags während der Spermien-Chemotaxis. Vorhersagen dieser Theorie werden durch Experimente der Gruppe von U.B. Kaupp (CAESAR, Bonn) quantitativ bestätigt. In Kapitel 3, untersuchen wir selbstorganisierte Strukturbildung in zwei ausgewählten biologischen Systemen. Auf zellulärer Ebene schlagen wir einen einfachen physikalischen Mechanismus vor für die spontane Selbstorganisation von periodischen Zellskelett-Strukturen, wie sie sich z.B. in den Myofibrillen gestreifter Muskelzellen finden. Dieser Mechanismus zeigt exemplarisch auf, wie allein durch lokale Wechselwirkungen räumliche Ordnung auf größeren Längenskalen in einem Nichtgleichgewichtssystem entstehen kann. Auf der Ebene des Organismus stellen wir eine Erweiterung der Turingschen Theorie für selbstorganisierte Musterbildung vor. Wir beschreiben eine neue Klasse von Musterbildungssystemen, welche selbst-organisierte Muster erzeugt, die mit der Systemgröße skalieren. Dieser neue Mechanismus erfordert weder eine vorgegebene Kompartimentalisierung des Systems noch spezielle Randbedingungen. Insbesondere kann dieser Mechanismus proportionale Muster wiederherstellen, wenn Teile des Systems amputiert werden. Wir bestimmen analytisch die Hierarchie aller stationären Muster und analysieren deren Stabilität und Einzugsgebiete. Damit können wir zeigen, dass dieser Skalierungs-Mechanismus strukturell robust ist bezüglich Variationen von Parametern und sogar funktionalen Beziehungen zwischen dynamischen Variablen. Zusammen mit Kollaborationspartnern im Labor von J. Rink (MPI CBG, Dresden) diskutieren wir Anwendungen auf das Wachstum von Plattwürmern und deren Regeneration in Amputations-Experimenten
Books on the topic "Stochastic Differential Geometry"
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Belopolskaya, Ya I., and Yu L. Dalecky. Stochastic Equations and Differential Geometry. Dordrecht: Springer Netherlands, 1990. http://dx.doi.org/10.1007/978-94-009-2215-0.
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Belopolskaya, Ya I. Stochastic Equations and Differential Geometry. Dordrecht: Springer Netherlands, 1990.
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L, Dalet͡skiĭ I͡U, ed. Stochastic equations and differential geometry. Dordrecht, Netherlands: Kluwer Academic Publishers, 1990.
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Emery, Michel. Stochastic calculus in manifolds. Berlin: Springer-Verlag, 1989.
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An introduction to the geometry of stochastic flows. London: Imperial College Press, 2004.
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E, Gliklikh I͡U. Ordinary and stochastic differential geometry as a tool for mathematical physics. Dordrecht: Kluwer Academic Publishers, 1996.
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L, Dalet͡skiĭ I͡U. Stokhasticheskie uravnenii͡a i different͡sialʹnai͡a geometrii͡a. Kiev: Gol. izd-vo izdatelʹskogo obʺedinenii͡a "Vyshcha shkola", 1989.
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Debashish, Goswami, ed. Quantum stochastic processes and noncommutative geometry. Cambridge: Cambridge University Press, 2007.
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E, Gliklikh I͡U. Global analysis in mathematical physics: Geometric and stochastic methods. New York: Springer, 1997.
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Stochastic calculus in manifolds. Berlin: Springer-Verlag, 1989.
Find full textBook chapters on the topic "Stochastic Differential Geometry"
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Kallenberg, Olav. "Stochastic Differential Geometry." In Foundations of Modern Probability, 801–26. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-61871-1_36.
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Hsu, Elton. "Basic stochastic differential geometry." In Graduate Studies in Mathematics, 35–69. Providence, Rhode Island: American Mathematical Society, 2002. http://dx.doi.org/10.1090/gsm/038/03.
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Kendall, Wilfrid S. "Stochastic Differential Geometry: An Introduction." In Stochastic and Integral Geometry, 29–60. Dordrecht: Springer Netherlands, 1987. http://dx.doi.org/10.1007/978-94-009-3921-9_3.
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Emery, Michel. "Some vocabulary from differential geometry." In Stochastic Calculus in Manifolds, 9–20. Berlin, Heidelberg: Springer Berlin Heidelberg, 1989. http://dx.doi.org/10.1007/978-3-642-75051-9_2.
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Belopol’skaya, Ya I., and Yu L. Daletskiǐ. "Stochastic equations and differential geometry." In Lecture Notes in Mathematics, 131–58. Berlin, Heidelberg: Springer Berlin Heidelberg, 1986. http://dx.doi.org/10.1007/bfb0075963.
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Hsu, Pei. "Probabilistic Methods in Differential Geometry." In Seminar on Stochastic Processes, 1989, 123–34. Boston, MA: Birkhäuser Boston, 1990. http://dx.doi.org/10.1007/978-1-4612-3458-6_7.
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Applebaum, David. "Lévy Processes in Stochastic Differential Geometry." In Lévy Processes, 111–37. Boston, MA: Birkhäuser Boston, 2001. http://dx.doi.org/10.1007/978-1-4612-0197-7_6.
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Gliklikh, Yuri E. "Stochastic Differential Equations on Manifolds." In Ordinary and Stochastic Differential Geometry as a Tool for Mathematical Physics, 75–98. Dordrecht: Springer Netherlands, 1996. http://dx.doi.org/10.1007/978-94-015-8634-4_3.
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De Vecchi, Francesco C., and Massimiliano Gubinelli. "A Note on Supersymmetry and Stochastic Differential Equations." In Geometry and Invariance in Stochastic Dynamics, 71–87. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-87432-2_5.
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Gliklikh, Yuri E. "Elements of Coordinate — Free Differential Geometry." In Ordinary and Stochastic Differential Geometry as a Tool for Mathematical Physics, 1–44. Dordrecht: Springer Netherlands, 1996. http://dx.doi.org/10.1007/978-94-015-8634-4_1.
Full textConference papers on the topic "Stochastic Differential Geometry"
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Bhatnagar, Lakshya, Guillermo Paniagua, David G. Cuadrado, Papa Aye N. Aye-Addo, Antonio Castillo Sauca, Francisco Lozano, and Matthew Bloxham. "Uncertainty in High-Pressure Stator Performance Measurement in an Annular Cascade at Engine-Representative Reynolds and Mach." In ASME Turbo Expo 2021: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/gt2021-59702.
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Abstract The betterment of the turbine performance plays a prime role in all future transportation and energy production systems. Precise uncertainty quantification of experimental measurement of any performance differential is therefore essential for turbine development programs. In this paper, the uncertainty analysis of loss measurements in a high-pressure turbine vane are presented. Tests were performed on a stator geometry at engine representative conditions in a new annular turbine module called BRASTA (Big Rig for Annular Stationary Turbine Analysis) located within the Purdue Experimental Turbine Aerothermal Lab. The aerodynamic probes are described with emphasis on their calibration and uncertainty analysis, first considering single point measurement, followed by the spatial averaging implications. The change of operating conditions and flow blockage due to measurement probes are analyzed using CFD, and corrections are recommended on the measurement data. The test section and its characterization are presented, including calibration of the sonic valve. The sonic valve calibration is necessary to ensure a wide range of operation in Mach and Reynolds. Finally, the vane data are discussed, emphasizing their systematic and stochastic uncertainty.
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Zheng, Qian, Weichen Qiu, Noah Ergezinger, Yong Li, Nader Yoosef-Ghodsi, Matt Fowler, and Samer Adeeb. "Development of an Online Calculation Tool for Safety Evaluation of Pipes Subjected to Ground Movements." In 2022 14th International Pipeline Conference. American Society of Mechanical Engineers, 2022. http://dx.doi.org/10.1115/ipc2022-86485.
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Abstract Underground pipelines are inevitably installed in unstable geohazard areas associated with the possible development of significant ground deformations. Under ground movement, excessive strains can be generated in the pipe wall, which poses a threat to pipeline integrity. This study aims to develop an industry-oriented calculation tool for safety evaluation of pipes subjected to ground movements induced by a variety of nature and construction-related hazards. The tool, comprised of deterministic and reliability-based analyses, is designed within MecSimCalc which is an innovative online platform for creating and sharing web-based Apps for individuals and groups. Calculation flow behind the tool is developed according to a novel method proposed based upon the finite difference method (FDM). Given grid nodes along the pipe, a large set of simultaneous finite-difference equations are constructed based on nonlinear governing differential equations of the Euler-Bernoulli beam under large deflections. The nonlinearities arising from pipe material, pipe-soil interaction, and geometry of the pipe are considered within the model. As unknowns of the finite-difference equations, the axial and lateral displacement of the pipe at each grid node can be obtained using nonlinear equation solvers. This method is utilized to predict the strain demand in the limit state function for reliability-based assessment. Applying stochastic properties for each basic parameter, the probability of failure can be calculated using Monte Carlo Simulation. Meanwhile, the program is compiled using Numba in Python and then optimized by the parallelization technique to enhance computational efficiency.
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Yang, Xuechen, Shan Zhao, and Hongjun Li. "Investment Portfolio Strategy Based on Geometric Brownian Motion and Backward Stochastic Differential Equations." In the 2018 2nd International Conference. New York, New York, USA: ACM Press, 2018. http://dx.doi.org/10.1145/3180374.3181350.
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Devarajan, K., V. Shankaranarayanan, K. Nithishrajan, M. Gaouthaman, and B. Chandraditya. "Probabilistic Response of a Vibration Energy Harvester With Customized Nonlinear Force Driven by Random Excitation." In ASME 2021 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/imece2021-68647.
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Abstract Nonlinear restoring forces have been widely introduced into the harvesting of vibratory energy to enhance energy harvester performance. Generally, the various source of nonlinearity include magnetic forces, spring forces, geometric and material nonlinearity, etc. However, these kind of nonlinear forces cannot be manipulated in an arbitrary manner. The performance of the energy harvester can be further optimized if the nonlinear forces are manipulated according to the requirements. The aim of this work is to study the energy harvesting performance of vibration energy harvester that can customize the nonlinear forces subjected to Gaussian white noise. The approximate analytical solution is obtained by moment differential method for the vibration energy harvester and there is a good agreement between analytical and the numerical solution obtained by Euler Maruyama scheme. Mean squared output power is presented to illustrate the device output performance. The nonlinear forces are customized For monostable and bistable piezoelectric energy harvester and the influence of noise intensity, and effect of system parameters on the response of the energy harvester is analyzed both numerically and analytically. The effect of stochastic resonance on the performance of the system also analyzed numerically for the bistable energy harvester.
Reports on the topic "Stochastic Differential Geometry"
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Snyder, Victor A., Dani Or, Amos Hadas, and S. Assouline. Characterization of Post-Tillage Soil Fragmentation and Rejoining Affecting Soil Pore Space Evolution and Transport Properties. United States Department of Agriculture, April 2002. http://dx.doi.org/10.32747/2002.7580670.bard.
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Tillage modifies soil structure, altering conditions for plant growth and transport processes through the soil. However, the resulting loose structure is unstable and susceptible to collapse due to aggregate fragmentation during wetting and drying cycles, and coalescense of moist aggregates by internal capillary forces and external compactive stresses. Presently, limited understanding of these complex processes often leads to consideration of the soil plow layer as a static porous medium. With the purpose of filling some of this knowledge gap, the objectives of this Project were to: 1) Identify and quantify the major factors causing breakdown of primary soil fragments produced by tillage into smaller secondary fragments; 2) Identify and quantify the. physical processes involved in the coalescence of primary and secondary fragments and surfaces of weakness; 3) Measure temporal changes in pore-size distributions and hydraulic properties of reconstructed aggregate beds as a function of specified initial conditions and wetting/drying events; and 4) Construct a process-based model of post-tillage changes in soil structural and hydraulic properties of the plow layer and validate it against field experiments. A dynamic theory of capillary-driven plastic deformation of adjoining aggregates was developed, where instantaneous rate of change in geometry of aggregates and inter-aggregate pores was related to current geometry of the solid-gas-liquid system and measured soil rheological functions. The theory and supporting data showed that consolidation of aggregate beds is largely an event-driven process, restricted to a fairly narrow range of soil water contents where capillary suction is great enough to generate coalescence but where soil mechanical strength is still low enough to allow plastic deforn1ation of aggregates. The theory was also used to explain effects of transient external loading on compaction of aggregate beds. A stochastic forInalism was developed for modeling soil pore space evolution, based on the Fokker Planck equation (FPE). Analytical solutions for the FPE were developed, with parameters which can be measured empirically or related to the mechanistic aggregate deformation model. Pre-existing results from field experiments were used to illustrate how the FPE formalism can be applied to field data. Fragmentation of soil clods after tillage was observed to be an event-driven (as opposed to continuous) process that occurred only during wetting, and only as clods approached the saturation point. The major mechanism of fragmentation of large aggregates seemed to be differential soil swelling behind the wetting front. Aggregate "explosion" due to air entrapment seemed limited to small aggregates wetted simultaneously over their entire surface. Breakdown of large aggregates from 11 clay soils during successive wetting and drying cycles produced fragment size distributions which differed primarily by a scale factor l (essentially equivalent to the Van Bavel mean weight diameter), so that evolution of fragment size distributions could be modeled in terms of changes in l. For a given number of wetting and drying cycles, l decreased systematically with increasing plasticity index. When air-dry soil clods were slightly weakened by a single wetting event, and then allowed to "age" for six weeks at constant high water content, drop-shatter resistance in aged relative to non-aged clods was found to increase in proportion to plasticity index. This seemed consistent with the rheological model, which predicts faster plastic coalescence around small voids and sharp cracks (with resulting soil strengthening) in soils with low resistance to plastic yield and flow. A new theory of crack growth in "idealized" elastoplastic materials was formulated, with potential application to soil fracture phenomena. The theory was preliminarily (and successfully) tested using carbon steel, a ductile material which closely approximates ideal elastoplastic behavior, and for which the necessary fracture data existed in the literature.