Amarjit Budhiraja

Bio: Amarjit Budhiraja is an academic researcher from University of North Carolina at Chapel Hill. The author has contributed to research in topics: Rate function & Brownian motion. The author has an hindex of 28, co-authored 176 publications receiving 2961 citations. Previous affiliations of Amarjit Budhiraja include Brown University & Iowa State University.

Large deviations for infinite dimensional stochastic dynamical systems

Abstract: The large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation. The construction and justification of the approximations can be onerous, especially in the case where the process state is infinite dimensional. In this paper we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of Brownian noise. The approach is based on a variational representation for functionals of Brownian motion. Proofs of large deviations properties are reduced to demonstrating basic qualitative properties (existence, uniqueness and tightness) of certain perturbations of the original process.

Large deviations for infinite dimensional stochastic dynamical systems

Abstract: The large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation. The construction and justification of the approximations can be onerous, especially in the case where the process state is infinite dimensional. In this paper we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of Brownian noise. The approach is based on a variational representation for functionals of Brownian motion. Proofs of large deviations properties are reduced to demonstrating basic qualitative properties (existence, uniqueness and tightness) of certain perturbations of the original process.

Worst case propagated uncertainty of multidisciplinary systems in robust design optimization

Abstract: While simulation based design tools continue to be advanced at unprecedented rates, little attention has been paid to how these tools interact with other advanced design tools and how that interaction influences the multidisciplinary system analysis and design processes. In this research an investigation of how uncertainty propagates through a multidisciplinary system analysis subject to the bias errors associated with the disciplinary design tools and the precision errors in the inputs is undertaken. A rigorous derivation for estimating the worst case propagated uncertainty in multidisciplinary systems is developed and validated using Monte Carlo simulation in application to a small analytic problem and an Autonomous HoverCraft (AHC) problem. The method of worst case estimation of uncertainty is then integrated into a robust optimization framework. In robust optimization, both the objective function and the constraints consist of two parts, the original or conventional functions and an estimate of the variation of the functions. In robust optimization the engineer must trade off an increase in the objective function value for a decrease in variation. The robust optimization approach is tested in application to the AHC problem and the corresponding results are discussed.

Variational representations for continuous time processes

Abstract: Une formule variationnelle pour des fonctionnelles positives d’une mesure de Poisson aleatoire et d’un mouvement brownien est demontree. Cette formule provient de la representation des integrales exponentielles par l’entropie relative, et peut etre utilisee pour obtenir des estimees de grandes deviations. Un resultat de grandes deviations general est demontre.

Random graphs

Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Convergence of Probability Measures

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

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.

Convergence of probability measures

Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an