 |
|
|
| | | | | |
|
|
|
|
|
A Microscopic Model for the Burgers Equation and Longest Increasing Subsequences
|
Timo Seppäläinen, Iowa State University
|
Abstract
We introduce an interacting random process related
to Ulam's problem, or finding the limit of the normalized longest
increasing subsequence of a random permutation. The process describes
the evolution of a configuration of sticks on the sites of the
one-dimensional integer lattice. Our main result is a hydrodynamic
scaling limit: The empirical stick profile converges to a weak
solution of the inviscid Burgers equation under a scaling of lattice
space and time. The stick process is also an alternative view of
Hammersley's particle system that Aldous and Diaconis used to give
a new solution to Ulam's problem. Along the way to the scaling limit
we produce another independent solution to this question. The heart
of the proof is that individual paths of the stochastic process evolve
under a semigroup action which under the scaling turns into the
corresponding action for the Burgers equation, known as the Lax
formula. In a separate appendix we use the Lax formula to give an
existence and uniqueness proof for scalar conservation laws with
initial data given by a Radon measure.
|
Full text: PDF
Pages: 1-51
Published on: March 6, 1996
|
Bibliography
-
D. Aldous and P. Diaconis,
Hammersley's interacting particle process and
longest increasing subsequences,
Probab. Theory Relat. Fields 103, (1995), 199--213.
Math Review article not available.
-
R. Durrett,
Probability: Theory and Examples,
Wadsworth, Pacific Grove (1991).
Math Review link
-
M. Ekhaus and T. Sepp"al"ainen,
Stochastic dynamics macroscopically
governed by the porous medium equation
for isothermal flow,
Ann. Acad. Sci. Fenn. Ser. A I Math (to appear).
Math Review article not available.
-
S. N. Ethier and T. G. Kurtz,
Markov Processes: Characterization and Convergence,
Wiley, New York (1985).
Math Review link
-
S. Feng, I. Iscoe, and T. Sepp"al"ainen,
A class of stochastic evolutions that scale
to the porous medium equation,
J. Statist. Phys. ( to appear).
Math Review article not available.
-
P. A. Ferrari,
Shocks in one-dimensional processes with drift,
Probability and Phase Transitions,
ed. G. Grimmett,
Kluwer Academic Publishers (1994).
Math Review link
-
J. M. Hammersley,
A few seedlings of research,
Proc. Sixth Berkeley Symp.
Math. Stat. Probab. Vol. I, (1972), 345--394.
Math Review link
-
C. Kipnis, Central limit theorems
for infinite series of queues and applications to simple
exclusion,
Ann. Probab. 14, (1986), 397--408.
Math Review link
-
P. Lax, Hyperbolic systems of
conservation laws II,
Comm. Pure Appl. Math. 10, (1957), 537--566.
Math Review link
-
P. Lax,
Hyperbolic Systems of
Conservation Laws and the Mathematical Theory of
Shock Waves,
SIAM, Philadelphia, (1973).
Math Review link
-
T. M. Liggett,
Interacting Particle Systems,
Springer-Verlag, New York, (1985).
Math Review link
-
P. L. Lions,
Generalized Solutions of Hamilton-Jacobi Equations,
Pitman, London (1982).
Math Review link
-
F. Rezakhanlou,
Hydrodynamic limit for attractive particle systems
on $mmZ^d$,
Comm. Math. Phys. 140, (1991), 417--448.
Math Review link
-
R. T. Rockafellar,
Convex Analysis,
Princeton University Press, Princeton, (1970).
Math Review link
-
H. Rost,
Non-equilibrium behaviour of a many particle
process: Density profile and local equilibrium,
Z. Wahrsch. Verw. Gebiete 58, (1981), 41--53.
Math Review link
-
J. Smoller,
Shock Waves and Reaction-Diffusion Equations,
Springer-Verlag, New York, (1983).
Math Review link
-
Y. Suzuki and K. Uchiyama,
Hydrodynamic limit for a spin system
on a multidimensional lattice,
Probab. Theory Relat. Fields 95, (1993), 47--74.
Math Review link
|
|
|
|
|
|
|
|
| | | | |
Electronic Journal of Probability. ISSN: 1083-6489 |
|
Context