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Analysis and Probability

Students select Analysis and Probability I in the fall semester, and then pick a specialization in the spring; either Analysis II or Probability II.

Analysis and Probability I

I Measure Theory

  • Construction of the integral, limits and integration
  • Lp spaces of functions
  • Construction of measures, Lebesgue-Stieltjes product measures
  • Examples: ergodicity, Liouville measure, Hausdorff measure

II Elements of Probability

  • The coin-tossing or random walk model
  • Independent events and independent random variables
  • The Khintchin weak law and the Kolmogorov strong law of large numbers
  • Notions of convergence of random variables
  • The Central Limit Theorem

III Elements of Fourier Analysis

  • Fourier transforms of measures, Fourier-Lévy Inversion Formula
  • Convergence of distributions and characteristic functions
  • Proof of the Central Limit and Lindeberg Theorems
  • Fourier transforms on Euclidean spaces
  • Fourier series, the Poisson summation formula
  • Spectral decompositions of the Laplacian
  • The heat equation and heat kernel

IV Brownian Motion

  • Brownian motion as a Gaussian process
  • Brownian motion as scaling limit of random walks
  • Brownian motion as random Fourier series
  • Brownian motion and the heat equation
  • Elementary properties of Brownian paths

Analysis II: Partial Differential Equations and Functional Analysis

I First Order Partial Differential Equations

  • Cauchy’s Theorem for first order real partial differential equations
  • Completely integrable first order equations

II Implicit Function Theorems

  • Basic examples of linear and non-linear partial differential equations
  • The functional analytic framework, Banach and Hilbert spaces
  • Bounded linear operators, spectrum, invertibility
  • Implicit function theorems in Banach spaces
  • Sketch of subsequent applications to the basic examples

III Second Order Partial Differential Equations

  • Qualitative description: elliptic,parabolic, hyperbolic equation
  • The Cauchy problem
  • Maximum principles
  • Sobolev and Schauder spaces
  • A priori estimates and Green’s functions
  • Riesz-Schauder theory of compact operators
  • Detailed treatment of basic examples
  • The Laplace and heat equations on compact manifolds
  • Applications to de Rham and Hodge theory

IV Selected Topics, chosen from

  • Riemann-Roch and index theorems
  • Determinants of Laplacians, modular forms
  • Integral representations, Hilbert transforms, singular integral operators
  • Subelliptic equations
  • Nash-Moser implicit function theorems
  • Non-linear equations from geometry or physics

Probability II

Prerequisite: Analysis and Probability I.  Can be taken concurrently with Analysis II.

I Rare Events

  • Cramér’s Theorem
  • Introduction to the Theory of Large Deviations
  • The Shannon-Breiman-McMillan Theorem

II Conditional Distributions and Expectations

  • Absolute continuity and singularity of measures
  • Radon-Nikodým theorem. Conditional distributions
  • Conditional expectations as least-square projections
  • Notion of conditional independence
  • Introduction to Markov Chains. Harmonic functions

III Martingales

  • Definitions, basic properties, examples, transforms
  • Optional sampling and upcrossings theorems, convergence
  • Burkholder-Gundy and Azuma inequalities
  • Doob decomposition, square-integrable martingales
  • Strong laws of large numbers and central limit theorems

IV Applications

  • Optimal stopping
  • Branching processes and their limiting behavior. Urn schemes
  • Stochastic approximation. Probabilistic analysis of algorithms

V Stochastic Integrals and Stochastic Differential Equations

  • Detailed study of Brownian motion
  • Martingales in continuous time
  • Doob-Meyer decomposition, stopping times
  • Integration with respect to continuous martingales, Itô’s rule
  • Girsanov’s theorem and its applications
  • Introduction to stochastic differential equations. Diffusion processes

VI Elements of Potential Theory

  • The Dirichlet problem. Poisson integral formula
  • Solution in terms of Brownian motion
  • Detailed study of the heat equation; Cauchy and boundary-value problems
  • Feynman-Kac theorems, applications
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