Ergodicity in infinite dimensions: summary and some names and facese

Transcription

1 Ergodicity in infinite dimensions: summary and some names and facese

2 Notions from the main story Continuous dynamical system Ergodicity and mixing (weak, strong) Invariant set; angle variable Canonical dynamical system Properties of the transition semigroup: continuity; recurrence; irreducibility; Feller, strong Feller, etc. Mild solution vs. variational solutions Dissipative and m-dissipative operators Reaction-diffusion equation Chaos expansion

3 Results from the main story Koopman-von Neumann theorem Characterizations and properties of ergodic measures Krylov-Bogoljubov theorem: invariant measure from compactness; going forward in time Doob s theorem: regularity implies (a) ergodicity of every invariant measure, (b) uniqueness of an invariant measure, (c) strong mixing of the invariant measure, but it does not imply existence Strong Feller and irreducibility imply regularity (Suitable) dissipativity implies existence (going backward in time), uniqueness, and exponential mixing of an invariant measure Condition ImS(t) Im(Q 1/2 t ) is equivalent to the strong Feller property for the OU process

4 Questions from the main story Continuity of the semigroup vs continuity of the canonical process Strong mixing in Doob s theorem Properties of the pressure in the stochastic Navier-Stokes equation

5 Other notions Hilbert cube (as an example of an infinite-dimensional locally compact separable metric space) Cylindrical process; space-time white noise Hilbert-Schmidt and trace-class operators Hilbert space-valued martingales Support of a measure Distance in total variation and other ways to measure distance between measures ω-excessive functions: ρ(x) ω ρ(x) Yosida approximation of a dissipative operator Null controllability and approximate controllability Normal (Gel fand, evolution) triple of Hilbert spaces Gibbs measure Leray-Helmholtz projector and the Stokes operator

6 Other results Basic ergodic theorems: pointwise (Birkhoff), mean (von Neumann), maximal Stone s theorem (self-adjoint/skew-symmetric operators generate unitary groups) Lumer-Phillps theorem (m-dissipative operators generate contraction semigroups) Hille-Phillips theorem (generators of C 0 semigroups) Spectral decomposition of a self-adjoin operator Kolmogorov s continuity criterion Burkholder-Davis-Gundy inequality

7 More results Two versions of the Itô formula in infinite dimensions Spectrum of 1-d OU semigroup Cameron-Martin theorem Cameron-Martin formula Feldman-Hajek theorem Stability of delay equations Derivation of basic equations of fluid motion Elimination of pressure in the Navier-Stokes equation

8 Gronwall s inequality Thomas Hakon Grönwall ( ): Swedish-American (1919)

9 Sobolev (embedding theorems and spaces) Sergei Lvovich Sobolev ( ), Russian (Sobolev spaces: 1930s)

10 Neumann Problem: father or son? Franz Ernst Neumann (father) , Carl Gottfried Neumann (son) Both worked in math physics.

11 Gibbs Measure Josiah Willard Gibbs ( ), American

12 Euler s equations Leonhard Euler Born ( ), Swiss (fluids: 1750s)

13 Navier and Stokes Claude Louis Marie Henri Navier ( ), French (1822) George Gabriel Stokes ( ), British (1842)

14 Leray-Helmholtz projector Jean Leray ( ), French (1930s) Baron Hermann Ludwig Ferdinand von Helmholtz ( ), German

15 Burgers Equation Harry Bateman ( ), British-American (1915) Johannes Martinus Burgers ( ), Dutch ( )

16 Asymptotically Strong Feller Property Martin Hairer (b. 1975), Austrian. Ph.D (Physics, U. of Geneva). Jonathan Mattingly (b. 1969), American. Ph.D (Princeton).

17 The authors Giuseppe Da Prato (b (?)), Italian. Ph.D hits on MathSciNet. 22 Ph.D. students total, including S. Cerrai, F. Flandoli, M. Fuhrman, A. Lunardi, E. Priola, G. Tessitore, L. Tubaro, L. Zambotti. Jerzy Zabczyk (b. 1941), Polish. Ph.D. 1969, Habilitation hits on MathSciNet. His Ph.D. students: A. Chojnowska-Michalik, L. Stettner, T. Bielecki, S. Peszat, A. Milian, J. Sobczyk, W. Jachimiak.