Author: J. H. van ZantenPublisher:
ISBN:
Category :
Languages : en
Pages : 20
Book Description
New Limit Theorems for Regular Diffusion Processes with Finite Speed Measure
Author: J. H. van ZantenNew Limit Theorems for Regular Diffusion Processes with Finite Speed Measure
Author: Jacob Hendrik ZantenLimit Theorems for Markov-modulated and Reflected Diffusion Processes
Author: Publisher:
ISBN:
Category :
Languages : en
Pages : 120
Book Description
"In this thesis, asymptotic properties of two variants of one-dimensional diffusion processes, which are Markov-modulated and reflected Ornstein-Uhlenbeck processes, are studied. Besides the random term of the Brownian motion, the Markov-modulated diffusion process evolves in an extra random environment, namely the finite-state Markov chain. The reflected Ornstein-Uhlenbeck process behaves as an Ornstein-Uhlenbeck process which has instantaneous reflection at boundaries. They are widely used in modeling due to above mentioned their distinctive features and great analytical tractability. We obtain four limit theorems from the perspectives of weak convergence and large deviations. Firstly, we prove weak convergence of a sequence of Markov-modulated diffusion processes with rapid switching to an ordinary diffusion process by verifying its tightness property. Secondly, a sample-path large deviations principle for the coupling of the Markov-modulated diffusion process with small noise and the occupation measure of the rapid switching Markov chain is obtained. The large deviations principles for each individual term are derived by the contraction principle. Those results reveal interesting behavior of Markov-modulated diffusion process when the modulating Markov chain switches fast. Thirdly, transient asymptotics of large deviations type are acquired for reflected and doubly reflected Ornstein-Uhlenbeck processes. Fourthly, we prove central limit theorems and functional central limit theorems for the centered and scaled loss and idle processes of doubly reflected Ornstein-Uhlenbeck processes."--Samenvatting auteur.
Essentials of Brownian Motion and Diffusion
Author: Frank B. KnightPublisher: American Mathematical Soc.
ISBN: 0821815180
Category : Mathematics
Languages : en
Pages : 220
Book Description
Presents some gratuitous generalities on scientific method as it relates to diffusion theory. This book defines Brownian motion by the characterization of P Levy, and then constructed in three basic ways and these are proved to be equivalent in the appropriate sense.
Diffusions and Elliptic Operators
Author: Richard F. BassPublisher: Springer Science & Business Media
ISBN: 0387226044
Category : Mathematics
Languages : en
Pages : 240
Book Description
A discussion of the interplay of diffusion processes and partial differential equations with an emphasis on probabilistic methods. It begins with stochastic differential equations, the probabilistic machinery needed to study PDE, and moves on to probabilistic representations of solutions for PDE, regularity of solutions and one dimensional diffusions. The author discusses in depth two main types of second order linear differential operators: non-divergence operators and divergence operators, including topics such as the Harnack inequality of Krylov-Safonov for non-divergence operators and heat kernel estimates for divergence form operators, as well as Martingale problems and the Malliavin calculus. While serving as a textbook for a graduate course on diffusion theory with applications to PDE, this will also be a valuable reference to researchers in probability who are interested in PDE, as well as for analysts interested in probabilistic methods.
Measure Theory and its Applications
Author: J.M. BelleyPublisher: Springer
ISBN: 3540386904
Category : Mathematics
Languages : en
Pages : 335
Book Description
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Measure-Valued Branching Markov Processes
Author: Zenghu LiPublisher: Springer Nature
ISBN: 3662669102
Category : Mathematics
Languages : en
Pages : 481
Book Description
This book provides a compact introduction to the theory of measure-valued branching processes, immigration processes and Ornstein–Uhlenbeck type processes. Measure-valued branching processes arise as high density limits of branching particle systems. The first part of the book gives an analytic construction of a special class of such processes, the Dawson–Watanabe superprocesses, which includes the finite-dimensional continuous-state branching process as an example. Under natural assumptions, it is shown that the superprocesses have Borel right realizations. Transformations are then used to derive the existence and regularity of several different forms of the superprocesses. This technique simplifies the constructions and gives useful new perspectives. Martingale problems of superprocesses are discussed under Feller type assumptions. The second part investigates immigration structures associated with the measure-valued branching processes. The structures are formulated by skew convolution semigroups, which are characterized in terms of infinitely divisible probability entrance laws. A theory of stochastic equations for one-dimensional continuous-state branching processes with or without immigration is developed, which plays a key role in the construction of measure flows of those processes. The third part of the book studies a class of Ornstein-Uhlenbeck type processes in Hilbert spaces defined by generalized Mehler semigroups, which arise naturally in fluctuation limit theorems of the immigration superprocesses. This volume is aimed at researchers in measure-valued processes, branching processes, stochastic analysis, biological and genetic models, and graduate students in probability theory and stochastic processes.
Brownian Motion and Diffusion
Author: David FreedmanPublisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 268
Book Description
Diffusion Processes with Reflection
Author: Sebastian AndresPublisher: Sudwestdeutscher Verlag Fur Hochschulschriften AG
ISBN: 9783838109282
Category :
Languages : de
Pages : 120
Book Description
In recent years diffusion processes with reflection have been subject of active research in the field of probability theory and stochastic analysis, where such reflected processes arise in quite various manners. The present work deals with two rather different types of reflected diffusion processes. In the first part we prove pathwise differentiabilty results for Skorohod SDEs with respect to the initial condition, in particular we consider processes on convex polyhedrons with oblique reflection at the boundary as well as processes on bounded smooth domains with normal reflection. In the second part a particle approximation of the Wasserstein diffusion is established, where the approximating process can be intepreted as a system of interacting Bessel processes with small Bessel dimension. More precisely, we introduce a reversible particle system, whose associated empirical measure process converges weakly to the Wasserstein diffusion in the high-density limit. Moreover, we prove regularity properties of the approximating system, in particular Feller properties, using tools from harmonic analysis on weighted Sobolev spaces.
Stochastic Space—Time Models and Limit Theorems
Author: L. ArnoldPublisher: Springer Science & Business Media
ISBN: 9400953909
Category : Mathematics
Languages : en
Pages : 270
Book Description
Approach your problems from It isn't that they can't see the right end and begin with the solution. the answers. Then one day, It is that they can't see the perhaps you will find the problem. final question. G.K. Chesterton. The Scandal 'The Hermit Clad 1n Crane of Father Brown 'The Point of Feathers' in R. van Gulik's a Pin'. The Chinese Maze Murders. Growing specialisation and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches wich were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD" , "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.