Uniqueness And Non Uniqueness Of Semigroups Generated By Singular Diffusion Operators

This book addresses both probabilists working on diffusion processes and analysts interested in linear parabolic partial differential equations with singular coefficients. The central question discussed is whether a given diffusion operator, i.e., a second order linear differential operator without zeroth order term, which is a priori defined on test functions over some (finite or infinite dimensional) state space only, uniquely determines a strongly continuous semigroup on a corresponding weighted Lp space. Particular emphasis is placed on phenomena causing non-uniqueness, as well as on the relation between different notions of uniqueness appearing in analytic and probabilistic contexts.

The book provides a systemic treatment of time-dependent strong Markov processes with values in a Polish space. It describes its generators and the link with stochastic differential equations in infinite dimensions. In a unifying way, where the square gradient operator is employed, new results for backward stochastic differential equations and long-time behavior are discussed in depth. The book also establishes a link between propagators or evolution families with the Feller property and time-inhomogeneous Markov processes. This mathematical material finds its applications in several branches of the scientific world, among which are mathematical physics, hedging models in financial mathematics, and population models.

This volume contains contributions on recent results in infinite dimensional harmonic analysis and its applications to probability theory. Some papers deal with purely analytic topics such as Frobenius reciprocity, diffeomorphism groups, equivariant fibrations and Harish-Chandra modules. Several other papers touch upon stochastic processes, in particular Lévy processes. The majority of the contributions emphasize on the algebraic-topological aspects of the theory by choosing configuration spaces, locally compact groups and hypergroups as their basic structures. The volume provides a useful survey of innovative work pertaining to a highly actual section of modern analysis in its pure and applied shapings.

This volume contains recent research papers presented at the international workshop on OC Probabilistic Methods in FluidsOCO held in Swansea. The central problems considered were turbulence and the NavierOCoStokes equations but, as is now well known, these classical problems are deeply intertwined with modern studies of stochastic partial differential equations, jump processes and random dynamical systems. The volume provides a snapshot of current studies in a field where the applications range from the design of aircraft through the mathematics of finance to the study of fluids in porous media."

This volume contains recent research papers presented at the international workshop on “Probabilistic Methods in Fluids” held in Swansea. The central problems considered were turbulence and the Navier–Stokes equations but, as is now well known, these classical problems are deeply intertwined with modern studies of stochastic partial differential equations, jump processes and random dynamical systems. The volume provides a snapshot of current studies in a field where the applications range from the design of aircraft through the mathematics of finance to the study of fluids in porous media. Contents:Probabilistic Approach to Hydrodynamic Equations (S Albeverio & Y Belopolskaya)A Mean Field Result for 3D Vortex Filaments (H Bessaih & F Flandoli)Semilinear Stochastic Wave Equations (P-L Chow)Some Remarks on a Statistical Theory of Turbulent Flows (F Flandoli)On the Dispersion of Sets Under the Action of an Isotropic Brownian Flow (H Lisei & M Scheutzow)A Version of the Law of Large Numbers and Applications (A Shirikyan)A Comparison Theorem for Solutions of Backward Stochastic Differential Equations with Two Reflecting Barriers and Its Applications (T-S Zhang)and other papers Readership: Research mathematicians with an interest in stochastic analysis, turbulence, fluid mechanics and stochastic partial differential equations. Keywords:Stochastic Analysis;Turbulence;Fluid Mechanics

Spectral theory of bounded linear operators teams up with von Neumann’s theory of unbounded operators in this monograph to provide a general framework for the study of stable methods for the evaluation of unbounded operators. An introductory chapter provides numerous illustrations of unbounded linear operators that arise in various inverse problems of mathematical physics. Before the general theory of stabilization methods is developed, an extensive exposition of the necessary background material from the theory of operators on Hilbert space is provided. Several specific stabilization methods are studied in detail, with particular attention to the Tikhonov-Morozov method and its iterated version.

The territory of preserver problems has grown continuously within linear analysis. This book presents a cross-section of the modern theory of preservers on infinite dimensional spaces (operator spaces and function spaces) through the author's corresponding results. Special emphasis is placed on preserver problems concerning some structures of Hilbert space operators which appear in quantum mechanics. In addition, local automorphisms and local isometries of operator algebras and function algebras are discussed in detail.

This volume is devoted to a beautiful object, called the valuative tree and designed as a powerful tool for the study of singularities in two complex dimensions. Its intricate yet manageable structure can be analyzed by both algebraic and geometric means. Many types of singularities, including those of curves, ideals, and plurisubharmonic functions, can be encoded in terms of positive measures on the valuative tree. The construction of these measures uses a natural tree Laplace operator of independent interest.