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Author: Sebastian Andres Publisher: 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.
Author: Sebastian Andres Publisher: 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.
Author: Feng-yu Wang Publisher: World Scientific ISBN: 9814452661 Category : Mathematics Languages : en Pages : 392
Book Description
Stochastic analysis on Riemannian manifolds without boundary has been well established. However, the analysis for reflecting diffusion processes and sub-elliptic diffusion processes is far from complete. This book contains recent advances in this direction along with new ideas and efficient arguments, which are crucial for further developments. Many results contained here (for example, the formula of the curvature using derivatives of the semigroup) are new among existing monographs even in the case without boundary.
Author: Kazuaki Taira Publisher: Springer Nature ISBN: 9811910995 Category : Mathematics Languages : en Pages : 792
Book Description
This book is an easy-to-read reference providing a link between functional analysis and diffusion processes. More precisely, the book takes readers to a mathematical crossroads of functional analysis (macroscopic approach), partial differential equations (mesoscopic approach), and probability (microscopic approach) via the mathematics needed for the hard parts of diffusion processes. This work brings these three fields of analysis together and provides a profound stochastic insight (microscopic approach) into the study of elliptic boundary value problems. The author does a massive study of diffusion processes from a broad perspective and explains mathematical matters in a more easily readable way than one usually would find. The book is amply illustrated; 14 tables and 141 figures are provided with appropriate captions in such a fashion that readers can easily understand powerful techniques of functional analysis for the study of diffusion processes in probability. The scope of the author’s work has been and continues to be powerful methods of functional analysis for future research of elliptic boundary value problems and Markov processes via semigroups. A broad spectrum of readers can appreciate easily and effectively the stochastic intuition that this book conveys. Furthermore, the book will serve as a sound basis both for researchers and for graduate students in pure and applied mathematics who are interested in a modern version of the classical potential theory and Markov processes. For advanced undergraduates working in functional analysis, partial differential equations, and probability, it provides an effective opening to these three interrelated fields of analysis. Beginning graduate students and mathematicians in the field looking for a coherent overview will find the book to be a helpful beginning. This work will be a major influence in a very broad field of study for a long time.
Author: Mauricio Andrés Duarte Espinoza Publisher: ISBN: Category : Brownian motion processes Languages : en Pages : 114
Book Description
This dissertation studies two different types of interaction of diffusion processes with the boundary of a domain $DsubRR^n$, which is assumed to be bounded, and of class $C^2(RR^n)$. The first process that is studied is obliquely reflected Brownian motion, and it is constructed as the unique Hunt process $X$ properly associated with the following Dirichlet form: begin{align} label{eq:abs_df} tag{1} E(u, v) = frac12int_D nabla unabla(urho) dx + frac12int_D nabla u cdotvec{tau} v rho(x)sigma(dx), end{align} where $vectau:partial DtoRR^n$ is tangential to $partial D$, and $u, v$ belong to the Sobolev space $W^{1,2}(D)$. The reference measure $rho(x)dx$ is assumed to be given by a harmonic function $rho$ whose gradient $nablarho$ is uniformly bounded. It is shown that such process $X$ admits a Skorohod decomposition begin{align} label{eq:abs_skorohod} tag{2} dX_t = dB_t + [vec{n}+vectau](X_t)dL_t. end{align} Moreover, we show that the unique stationary distribution of $X$ is the measure given by $rho(x)dx$. In the second part of the dissertation, we present a new reflection process $X_t$ in a bounded domain $D$ of class $C^2(RR^n)$ that behaves very much like oblique reflected Brownian motion, except that the directions of reflection depend on an external parameter $S_t$ called spin. The spin is allowed to change only when the process $X_t$ is on the boundary of $D$. The pair $(X, S)$ is called spinning Brownian motion and is found as the unique strong solution to the following stochastic differential equation: % %Let $Dsubseteqmathbb{R}^n$ be an open $C^2$ domain, and let $B_t$ be a $n$-dimensional Brownian motion. A pair $(X_t, S_t)$ is called spinning Brownian motion (sBM) if it solves the following stochastic differential equation begin{align} label{eq:abs_sbm} tag{3} left{ begin{array}{rl} dX_t & = sigma(X_t)dB_t + vec{n}(X_t)dL_t + vec tau (X_t, S_t)dL_t \ dS_t & = spar{vec{g}(X_t) - S_t } dL_t end{array} right. end{align} where $L_t$ is the local time process of $X_t$, $vec{n}$ is the interior unit normal to $partial D$, and $vectau$ is a vector field perpendicular to $hat n$. The function $sigma(cdot)$ is a non-degenerate $(ntimes n)$-matrix valued function, and $vec{tau}(cdot)$ and $vec g(cdot)$ are Lipschitz bounded vector fields. % We prove that a unique strong solution to eqref{eq:abs_sbm} exists as the limit of a family of processes $(X^e, S^e)$ that satisfy an equation like eqref{eq:abs_sbm}, but in which the spin component $dS$ has a noise $e dW$. With this added noise, the process $(X^e, S^e)$ is an obliquely reflected Brownian motion in an unbounded domain. % It is also shown that spinning Brownian motion has a unique stationary distribution. The main tool of the proof is excursion theory, and an identification of the Local time of $X_t$ as a component of an exist system for $X_t$.
Author: N. Ikeda Publisher: Elsevier ISBN: 1483296156 Category : Mathematics Languages : en Pages : 572
Book Description
Being a systematic treatment of the modern theory of stochastic integrals and stochastic differential equations, the theory is developed within the martingale framework, which was developed by J.L. Doob and which plays an indispensable role in the modern theory of stochastic analysis.A considerable number of corrections and improvements have been made for the second edition of this classic work. In particular, major and substantial changes are in Chapter III and Chapter V where the sections treating excursions of Brownian Motion and the Malliavin Calculus have been expanded and refined. Sections discussing complex (conformal) martingales and Kahler diffusions have been added.
Author: Ari Arapostathis Publisher: Cambridge University Press ISBN: 1107079284 Category : Mathematics Languages : en Pages : 341
Book Description
This comprehensive volume on ergodic control for diffusions highlights intuition alongside technical arguments. A concise account of Markov process theory is followed by a complete development of the fundamental issues and formalisms in control of diffusions. This then leads to a comprehensive treatment of ergodic control, a problem that straddles stochastic control and the ergodic theory of Markov processes. The interplay between the probabilistic and ergodic-theoretic aspects of the problem, notably the asymptotics of empirical measures on one hand, and the analytic aspects leading to a characterization of optimality via the associated Hamilton–Jacobi–Bellman equation on the other, is clearly revealed. The more abstract controlled martingale problem is also presented, in addition to many other related issues and models. Assuming only graduate-level probability and analysis, the authors develop the theory in a manner that makes it accessible to users in applied mathematics, engineering, finance and operations research.
Author: Merkel H. Jacobs Publisher: Springer Science & Business Media ISBN: 3642864147 Category : Science Languages : en Pages : 165
Book Description
A basic tenet of present day biophysics is that flows in biological systems are causally related to forces. A large and growing fraction of membrane biophysics is devoted to an exploration of the quantitative relationship between forces and flows in order to understand both the nature of biological membranes and the processes that take place on and in these membranes. This is why the discussion of the nature of diffusion is so important in any formal development of membrane bio physics. This was equally true twenty years ago when tracers were just beginning to be used for the measurement of m.
Author: Sebastian Andres
Author: Sebastian Andres
Author: Zhen-Qing Chen
Author: Feng-yu Wang
Author: Andrey Pilipenko
Author: Kazuaki Taira
Author: Kiyosi Itō
Author: Mauricio Andrés Duarte Espinoza
Author: N. Ikeda
Author: Ari Arapostathis
Author: Merkel H. Jacobs