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Teaching University of Bonn Institute for Applied Mathematics
 
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V5F1 - Advanced Topics in Stochastics

 

Analysis on Probability Spaces and SPDE

 

Lecture Notes

 

Andreas Eberle Summer Semester 2012

 

Time: Tuesdays 8-10, Thursdays 14-12, Room 0.006

 

 

Prerequisites: The course should be accessible to students with a background either in probability or in analysis. The material covered in Stochastic Analysis is not assumed, although this course may be taken as a follow-up to V4F1.

 

Contents: The course gives an introduction to analysis on function spaces endowed with a probability measure (e.g. the Wiener space), and to stochastic partial differential equations.

  • Gaussian measures on function spaces: Existence, regularity, Wiener processes, Gaussian free field, perturbations of Gaussian measures.

  • Semigroups and generators, Dirichlet forms.

  • Functional inequalities on Gaussian measure spaces: Logarithmic Sobolev Inequality, Isoperimetric Inequality, Concentration Inequalities.

  • Linear SPDE: Stochastic integral, stochastic convolutions, Ornstein-Uhlenbeck processes on Hilbert spaces, time and space regularity, long-time behaviour.

  • Semilinear SPDE: Reaction-diffusion equations, Navier-Stokes.

 

 

References:

 

  • Da Prato/Zabczyk: Stochastic equations in infinite dimensions, CUP

  • Prévôt/Röckner: A concise course on SPDE

  • Da Prato: An Introduction to Infinite-Dimensional Analysis, Springer

  • Royer: An Initiation to Logarithmic Sobolov inequalities, AMS