Project 2: Intermittency phenomena in random media
Participants
Jean-Dominique Deuschel, Nicolas Dirr, Jürgen
Gärtner, Wolfgang
König, Barbara
Niethammer
Summary
We investigate in detail the effect of intermittency in spatially
random systems for two main models,
the parabolic Anderson model (PAM), and the Mullins-Sekerka evoluton
(MSE) for Ostwald ripening.
This effect says that the long-time behavior is determined by an
extremely inhomogeneous behavior of
the system, i.e., by extremely large contributions coming from
extremely small parts of the space.
In the PAM, we propose to study more refined questions like the number
of these small regions and the
precise time-evolution, and to apply advanced cluster techniques to
deal with contributions from large
time-dependent boxes, and to investigate time-dependent and drifted
random potentials.
For the MSE, we propose to construct a version of the model on the
entire space and study it with particular
consideration of the collisions of the randomly located particles, the
screening effect,
and asymptotic self-similarity of the system.
Some related earlier preprints
J. Gärtner and W. König:
The parabolic Anderson model,
in: J.-D. Deuschel and A. Greven (Eds.), Interacting Stochastic Systems,
pp. 153--179, Springer (2005).
R. van der Hofstad, W. König and
P. Mörters:
The universality classes in the parabolic Anderson model,
Commun. Math. Phys. 267:2, 307-353 (2006).
Preprint
ps,
revised version,
pdf.
J.
Gärtner, W. König and S. Molchanov:
Geometric characterization
of intermittency in the parabolic Anderson model,
Ann. Probab. 35 (2007), 439-499.
Publication
J.
Gärtner, M. Heydenreich:
Annealed asymptotics for the parabolic Anderson model with a moving catalyst,
Stoch. Proc. Appl. 116 (2006), 1511-1529.
Publication
J.
Gärtner and F. den Hollander:
Intermittency in a catalytic random medium,
Ann. Probab. 34, 2219-2287 (2006).
Preprint in Math ArXive.
Achievements of the Research Group
M. Birkner and R.~Sun :
Annealed vs. quenched critical points for a random walk pinning model,
preprint.
J.
Gärtner, F. den Hollander, and G. Maillard:
Intermittency on catalysts: symmetric exclusion,
Elec. Jour. Probab. 12 (2007), paper no. 18, 516-573.
Publication
J. Gärtner and R. Sun:
A quenched limit theorem for the local time of random walks on \Z^2,
preprint.
J. Gärtner, F. den Hollander and G. Maillard:
Intermittency on catalysts
In: J. Blath, P. Mörters and M. Scheutzow (eds), Trends in Stochastic
Analysis.
LMS 353, Cambridge Univ. Press. In print (2008).
Preprint.
W. König, H. Lacoin, P. Mörters and N. Sidorova:
A two cities theorem for the parabolic Anderson model,
Preprint,
ps, Preprint,
pdf, Ann. Probab., to appear.
G. Grüninger and W. König:
Potential confinement property of the parabolic
Anderson model,
Preprint,
ps, Preprint,
pdf.
B. Niethammer and J. Velasquez:
On screening induced fluctuations in Ostwald ripening,
Preprint, pdf, J. Stat. Phys. 130:3, 415-453 (2008).
B. Niethammer and J. Velasquez:
Screening in interacting particle systems,
preprint, pdf (2008).
B. Niethammer:
Effective theories for Ostwald ripening,
preprint, pdf (2008).
S. Conti, A. Hönig, B. Niethammer, F. Otto:
Non-universality in low-volume-fraction Ostwald ripening,
preprint, pdf (2006).
M. Herrmann, B. Niethammer and J. Velasquez:
Self-similar solutions for the LSW model with encounters,
preprint (2008).
G. Menon, B. Niethammer and R.L. Pego:
Dynamics and self-similarity in min-driven clustering,
preprint (2008)