## Antti Kupiainen

**Ph.D., Academy Professor**

**Group Leader**

**Email:** firstname.lastname(at)helsinki.fi

**Personal Homepage:** http://wiki.helsinki.fi/display/mathphys/antti

**Telephone:** +358-503088010

**Mailing Address:**

Department of Mathematics and Statistics

PL 68

FIN-00014 University of Helsinki

Finland

**Visiting Address:**

Gustaf Haellstroemin katu 2b, Helsinki

Room D334

### Scientific Activities: Publications

My background is in Constructive Quantum Field Theory and Statistical Mechanics. In the 80's I was developing the Renormalization Group (RG) method for the rigorous analysis of renormalizable field theories and phase tranistions in lattice spin systems. Some papers from this period are numbers 1-4 appearing on the list below.

I also did some work on Conformal Field Theory. See, e.g., 5-7 below.

Subsequently I applied the RG to various problems in probability, dynamical systems and PDE's. In the paper #8 of the list, random walk with random asymmetric transition probabilities was shown to be diffusive in dimensions greater than or equal to 3. In 9-16, asymptotics of solutions of nonlinear parabolic PDE's were studied using the RG in situations of decay to zero, blow-up in finite time and formation of spatial patterns and moving fronts.

In the papers 17-20, ideas from statistical mechanics (high temperature expansions) were used to study the Sinai-Ruelle-Bowen measures for chaotic spatially extended dynamical systems and the problem of space-time chaos and in 21-22, the Kolmogorov-Arnold-Moser and Melnikov theorems of invariant tori in Hamiltonian systems were revisited using the RG.

Later, I worked on various aspects dealing with the problem of turbulence. See Lessons for Turbulence, Geom. Funct. Analysis, GAFA2000, 316-333 (2000). In 23-25 a stochastic PDE describing the advection of a scalar quantity in a random Hölder continous velocity field was studied and anomalous scaling and breakdown of the Kolmogorov theory of turbulence was established. In #26 this phenomenon was shown to be connected to the non-uniqueness of particle trajectories in turbulent velocity fields.

The question of ergodicity and uniqueness of the invariant measure for 2 dimensional stochastically forced Navier-Stokes equation was discussed in the publications 27-29.

More recently I have been working on foundations of non-equilibrium statistical mechanics and the problem of deriving diffusion and transport from first principles i.e. deterministic Hamiltonian or Schrödinger dynamics and on issues in random geometry, random conformal welding.

#### Publications cited above

1. Massless Lattice {\phi}^{4}_{4} Theory: Rigorous Control of a Renormalizable Asymptotically Free Model, Commun. Math. Phys. **99**, 197-252 (1985) (with K. Gawedzki)

2. Gross-Neveu Model Through Convergent Perturbation Expansions, Commun. Math. Phys. **102**, 1-30 (1985) (with K. Gawedzki)

3. Renormalization of a Non-Renormalizable Quantum Field Theory, Nucl. Phys. B **262**, 33-48 (1985) (with K. Gawedzki)

4. Phase Transition in the 3d Random Field Ising model, Commun. Math. Phys. **116**, 539-572 (1987) (with J. Bricmont)

5. The Spectrum of WZW models with Arbitrary Simple Groups, Commun. Math. Phys. 117, 127-158 (1988) (with G. Felder, K. Gawedzki)

6. Coset Construction from Functional Integrals, Nucl. Phys. **B** 320 (1989), 625-661 (with K. Gawedzki)

7. SU(2) Chern-Simons Theory at Genus Zero, Commun. Math. Phys. **135**, 531-554 (1990) (with K. Gawedzki)

8. Random Walks in Asymmetric Random Environments, Commun. Math. Phys. **142**, 345-420 (1991) (with J. Bricmont)

9. Renormalization Group and Asymptotics of Solutions of Nonlinear Parabolic Equations, Comm. Pure Appl. Math. **47**, 893-922 (1994) (with J. Bricmont and G. Lin)

10. Renormalization Group and the Ginzburg-Landau Equation, Commun. Math. Phys. **150**, 193-208 (1992) (with J. Bricmont)

11. Renormalizing Partial Differential Equations, in Constructive Physics, ed. by V. Rivasseau, 83-117, Springer (1995)

12. Universality in Blow-up, Nonlinearity **7**, 1-37 (1994) (with J. Bricmont)

13. Stable Non-gaussian Diffusive Profiles, Nonlin. Analysis, Theory, Methods and Applications, **26**, 583-593 (1996) (with J. Bricmont)

14. Stability of Moving Fronts in the Ginzburg-Landau Equation, Commun. Math. Phys. **159**, 287-318 (1994) (with J. Bricmont)

15. Global large time self-similarity of a thermal-diffusive combustion system with critical nonlinearity, J. Diff Eqn, Vol. 130, No. 1, 1996, pp 9-35 (with J. Bricmont and J. Xin)

16. Stability of Cahn-Hilliard fronts, Comm. Pure Appl. Math. **52** (1999), 839-871 (with J. Bricmont, J. Taskinen)

17. Coupled Analytic Maps, Nonlinearity **8**, 379-393 (1995) (with J. Bricmont)

18. High Temperature Expansions and Dynamical Systems, Commun. Math. Phys. **178**, 703-732 (1996) (with J. Bricmont)

19. Infinite dimensional SRB measures, Physica **D 103** (1997) 18-33 (with J. Bricmont)

20. The spectrum of weakly coupled map lattices, J. Math. Pure. Appl. **77**, 539-584 (1998) (with V. Baladi, M. Degli Esposti, S. Isola, E. Järvenpää)

21. KAM theorem and quantum field theory, Commun. Math. Phys. **201** (1999) 3, 699-727 (with J. Bricmont, K. Gawedzki)

22. Renormalization Group and the Melnikov Problem for PDE's (with J. Bricmont and A. Schenkel)

23. Anomalous Scaling for Passive Scalar, Phys. Rev. Lett. **75** 3834 (1995) (with K. Gawedzki)

24. Anomalous scaling in the N-point functions of passive scalar, Phys. Rev. E **54**, 2564 (1996) (with D. Bernard and K. Gawedzki)

25. Some mathematical problems of passive advection, Contemporary Math. **217**, 83-99 (1998)

26. Slow modes in passive advection, J. Stat. Phys. (with D. Bernard and K. Gawedzki)

27. Probabilistic estimates for the two dimensional stochastic Navier-Stokes equations, J. Stat. Phys. **100** (3/4), 2000, 743-756(with J. Bricmont, R. Lefevere)

28. Ergodicity of the 2D Navier-Stokes Equations with Random Forcing, Commun. Math. Phys., to appear (with J. Bricmont, R. Lefevere)

29. Exponential Mixing of the 2D Stochastic Navier-Stokes Dynamics