Level (a), 0.5 unit , Term 2, 3 hours/week, starting in week from ...
Topics from: Probability measures. Random variables. Independence. Sums of independent random variables; weak and strong laws of large numbers. Weak convergence, characteristic function, central limit theorem. Brownian motion. Martingales.
Functional analysis with advanced study |
M3/ 4P7 | |
| Vector spaces. Existence of a Hamel basis. Normed vector spaces. Banach spaces. Finite dimensional spaces. Isomorphism. Separability. The Hilbert space. The Riesz-Fisher theorem. The Hahn-Banach theorem. Principle of uniform boundedness. Dual spaces. Operators, compact operators. Hermitian operators and the spectral theorem. Banach algebras. | ||
Stochastic Processes I |
B75 | |
Potential Theory |
B76 | |
| Basic theory of elliptic partial differential equations. The Laplace equation, the heat equation. The maximum principle, the mean value property and the Dirichlet problem. Relationship with Brownian motion and other diffusion processes. The reduite and its basic properties. Green function, heat kernel. Other topics could include Sobolev embedding theorems. Quasilinear and fully non-linear elliptic parabolic equations, viscosity solutions. | ||
Stochastic Processes II |
B77 | |
| Further properties of Brownian motion. Local time. Girsanov formulae and relationship with drifting brownian motion. Gaussian measures and Cameron Martin Spaces. Stochastic differential equations. Weak Solutions. Strong Solutions. Anticipating calculus. Other topics could include Malliavin Calculus: Integration by parts, and Hormanders Theorem. Wiener Chaos Expansions. The Kusuoka/Ramer Theorem. | ||
Advanced analysis |
B78 | |
| Fourier series and integrals; convergence and inversion theorems. $L^p$ spaces, interpolation. Applications in analysis (e.g. Tauberian theorems). Connections with complex variables; Hardy spaces, the Hilbert transform. Singular integral operators in n dimensions. Calderon-Zygmund theory, the maximal function. Wavelet decompositions. | ||
For other courses look also at
Undergraduate Courses p.277