K. D. Elworthy

Bio: K. D. Elworthy is an academic researcher from University of Warwick. The author has contributed to research in topics: Stochastic differential equation & Malliavin calculus. The author has an hindex of 15, co-authored 25 publications receiving 1836 citations. Previous affiliations of K. D. Elworthy include Coventry Health Care.

Stochastic Differential Equations on Manifolds

Abstract: A. The title is designed to indicate those particular aspects of stochastic differential equations which will be considered here: these are almost equally valid when the manifold in question is ℝ n (although compactness is often a useful simplifying assumption). In fact one of the main themes here will be that stochastic differential equations, even on ℝ n , induce non-trivial differential-geometric structures and these structures are an important tool in analyzing the behaviour of the solutions of the s.d.e.

Formulae for the derivatives of heat semigroups

Abstract: We use a basic martingale method to show a differentiation formula for the derivatives $$d(P_tf)(x_0)(v_0)={1\over t} E f(x_t) \int_0^t \langle Y(x_s)(v_s),dB_t\rangle_{R^m}.$$ These are proved first on $R^n$, then on manifolds. Afterwards for solutions of heat equations on differential forms, and a second order formula.

On the Geometry of Diffusion Operators and Stochastic Flows

Abstract: Construction of connections.- The infinitesimal generators and associated operators.- Decomposition of noise and filtering.- Application: Analysis on spaces of paths.- Stability of stochastic dynamical systems.- Appendices.

Stochastic Equations in Infinite Dimensions

Abstract: Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Random Dynamical Systems

Abstract: I. Random Dynamical Systems and Their Generators.- 1. Basic Definitions. Invariant Measures.- 2. Generation.- II. Multiplicative Ergodic Theory.- 3. The Multiplicative Ergodic Theorem in Euclidean Space.- 4. The Multiplicative Ergodic Theorem on Bundles and Manifolds.- 5. The MET for Related Linear and Affine RDS.- 6. RDS on Homogeneous Spaces of the General Linear Group.- III. Smooth Random Dynamical Systems.- 7. Invariant Manifolds.- 8. Normal Forms.- 9. Bifurcation Theory.- IV. Appendices.- Appendix A. Measurable Dynamical Systems.- A.1 Ergodic Theory.- A.2 Stochastic Processes and Dynamical Systems.- A.3 Stationary Processes.- A.4 Markov Processes.- Appendix B. Smooth Dynamical Systems.- B.1 Two-Parameter Flows on a Manifold.- B.4 Autonomous Case: Dynamical Systems.- B.5 Vector Fields and Flows on Manifolds.- References.

Sobolev met Poincaré

Abstract: Introduction What are Poincare and Sobolev inequalities? Poincare inequalities, pointwise estimates, and Sobolev classes Examples and necessary conditions Sobolev type inequalities by means of Riesz potentials Trudinger inequality A version of the Sobolev embedding theorem on spheres Rellich-Kondrachov Sobolev classes in John domains Poincare inequality: examples Carnot-Caratheodory spaces Graphs Applications to PDE and nonlinear potential theory Appendix References.