105 research outputs found
Hitting properties of a random string
We consider Funaki's model of a random string taking values in R^d. It is
specified by the following stochastic PDE, du = u_{xx} + W, where W=W(x,t) is
two-parameter white noise, also taking values in R^d. We study hitting
properties, double points, and recurrence. The main difficulty is that the
process has the Markov property in time, but not in space. We find: (1) The
string hits points if d<6. (2) For fixed t, there are points x,y such that
u(t,x)=u(t,y) iff d < 4. (3) There exist points t,x,y such that u(t,x)=u(t,y)
iff d < 8. (4) There exist points s,t,x,y such that u(t,x)=u(s,y) iff d < 12.
(5) The string is recurrent iff d < 7
On the large time asymptotics of decaying Burgers turbulence
The decay of Burgers turbulence with compactly supported Gaussian "white
noise" initial conditions is studied in the limit of vanishing viscosity and
large time. Probability distribution functions and moments for both velocities
and velocity differences are computed exactly, together with the "time-like"
structure functions .
The analysis of the answers reveals both well known features of Burgers
turbulence, such as the presence of dissipative anomaly, the extreme anomalous
scaling of the velocity structure functions and self similarity of the
statistics of the velocity field, and new features such as the extreme
anomalous scaling of the "time-like" structure functions and the non-existence
of a global inertial scale due to multiscaling of the Burgers velocity field.
We also observe that all the results can be recovered using the one point
probability distribution function of the shock strength and discuss the
implications of this fact for Burgers turbulence in general.Comment: LATEX, 25 pages, The present paper is an extension of the talk
delivered at the workshop on intermittency in turbulent systems, Newton
Institute, Cambridge, UK, June 199
Pfaffian formulae for one dimensional coalescing and annihilating systems
The paper considers instantly coalescing, or instantly annihilating, systems of one-dimensional Brownian particles on the real line. Under maximal entrance laws, the distribution of the particles at a fixed time is shown to be Pfaffian point processes closely related to the Pfaffian point process describing one dimensional distribution of real eigenvalues in the real Ginibre ensemble of random matrices. As an application, an exact large time asymptotic for the n-point density function for coalescing particles is derived
Multi-point correlations for two dimensional coalescing random walks
This paper considers an infinite system of instantaneously coalescing rate
one simple random walks on , started from the initial condition
with all sites in occupied. We show that the correlation
functions of the model decay, for any , as as . This generalises the results for due to Bramson and
Griffeath and confirms a prediction in the physics literature for . An
analogous statement holds for instantaneously annihilating random walks.
The key tools are the known asymptotic due to
Bramson and Griffeath, and the non-collision probability , that no
pair of a finite collection of two dimensional simple random walks meets by
time , whose asymptotic was
found by Cox, Merle and Perkins. This paper re-derives the asymptotics both for
and by proving that these quantities satisfy {\it
effective rate equations}, that is approximate differential equations at large
times. This approach can be regarded as a generalisation of the Smoluchowski
theory of renormalised rate equations to multi-point statistics.Comment: 26 page
One dimensional annihilating particle systems as extended Pfaffian point processes
We prove that the multi-time particle distributions for annihilating Brownian
motions, under the maximal entrance law on the real line, are extended Pfaffian
point processesComment: An earlier version of this paper falsely stated that the multi-time
distributions of coalescing Brownian motions (CBM's) were also an extended
Pfaffian point process. This erroneous claim has been removed from the
current version. We do not have a simple description of the multi-time
distributions for CBM
Uniqueness for a class of one-dimensional stochastic PDEs using moment duality
We establish a duality relation for the moments of bounded solutions to a class of one-dimensional parabolic stochastic partial differential equations. The equations are driven by multiplicative space-time white noise, with a non-Lipschitz multiplicative functional. The dual process is a system of branching Brownian particles. The same method can be applied to show uniqueness in law for a class of non-Lipschitz finite dimensional stochastic differential equations
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