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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

antti

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)

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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)

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

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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)

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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)

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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)

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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)

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

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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

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