Pattern formation, energy landscapes, and scaling laws

Felix Otto

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+49 (0) 341 - 9959 - 950

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04103 Leipzig

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Katja Heid
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Lecture in the Sommer term 2015

Topics in stochastic partial differential equations

  • Lecturer: Felix Otto
  • Date: Tuesday 09.15 - 10.45
  • Room: MPI MiS, A 01
  • Language: English
  • Target audience: MSc students, PhD students, Postdocs
  • Content (Keywords): parabolic differential equations, stochastic differential equation, Concentration of measure, Schauder theory


We will introduce parabolic differential equations driven by white noise in time. We will be mostly interested in nonlinear parabolic equations with a nonlinearity π in the leading order term and a noise ξ that is white not only in time but also in space. The latter limits the space dimension to one, leading to $$\partial_tu-\partial_x^2\pi(u)=\xi$$.

We are interested in the path-wise regularity of solutions to such equations. In case of our model problem, a scaling argument suggests that the solutions are Hölder continuous (with almost exponent ½ in space and almost exponent ¼ in time). This is also the regularity in the linear case.
We shall show that this is indeed true. The argument relies on the following ingredients:

  • On the stochastic side:
    1. Arguments typical for stochastic differential equation (Martingale arguments) that give second-moment regularity estimates.
    2. Concentration of measure arguments on the level of the space-time white noise (Malliavin derivative) that upgrade the low-moment regularity results to Gaussian moments.
  • On the deterministic side:
    1. The Ḣ-1-contraction principle for nonlinear parabolic equations of the form $$\partial_tu-\partial_x^2\pi(u)=0$$
    2. Campanato-type arguments for a Schauder theory for non-constant coefficient parabolic equations of the form $$\partial_tu-\partial_x^2(au)=f$$.
Hence despite the specifics of the model problem, the arguments are fairly general and it is thus a good excuse for introducing the above-mentioned concepts.

20.03.2023, 09:09