Introduction to Stochastic Partial Differential Equations
Introductory Workshop: Randomness and long time dynamics in nonlinear evolution differential equations August 24, 2015  August 28, 2015
Location: SLMath: Eisenbud Auditorium
KPZ
reactiondiffusion
Burger's equation
Brownian motion
Wiener process
definition of Ito integral
linear heat equation
35L35  Initialboundary value problems for higherorder hyperbolic equations
57Q50  Microbundles and block bundles [See also 55R60, 57N55]
57R56  Topological quantum field theories (aspects of differential topology)
14335
After a presentation of white noise and stochastic calculus in infinite dimension, I will explain how to solve classical SPDEs with white noise. I will focus on the stochastic Burgers and reactiondiffusion equations which will be first solved with spatially smooth noise and then with space time white noise. The case of the reactiondiffusion equation in dimension 2 is already not so obvious since the solutions are not expected to be function valued processes. The case of dimension 3 is much more difficult and has been solved only recently by Martin Hairer. I will not explain in details his theory on regularity structure, this would take too much time. However, I will explain why the problem is so difficult and give few hints on this difficult theory. No prerequisite on stochastic calculus is expected from the audience.
Debussche Notes

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