219 research outputs found
Rough paths and 1d sde with a time dependent distributional drift. Application to polymers
Motivated by the recent advances in the theory of stochastic partial
differential equations involving nonlinear functions of distributions, like the
Kardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability of
one-dimensional stochastic differential equations, the drift of which is a
distribution, by means of rough paths theory. Existence and uniqueness are
established in the weak sense when the drift reads as the derivative of a
H{\"o}lder continuous function. Regularity of the drift part is investigated
carefully and a related stochastic calculus is also proposed, which makes the
structure of the solutions more explicit than within the earlier framework of
Dirichlet processes
A forward--backward stochastic algorithm for quasi-linear PDEs
We propose a time-space discretization scheme for quasi-linear parabolic
PDEs. The algorithm relies on the theory of fully coupled forward--backward
SDEs, which provides an efficient probabilistic representation of this type of
equation. The derivated algorithm holds for strong solutions defined on any
interval of arbitrary length. As a bypass product, we obtain a discretization
procedure for the underlying FBSDE. In particular, our work provides an
alternative to the method described in [Douglas, Ma and Protter (1996) Ann.
Appl. Probab. 6 940--968] and weakens the regularity assumptions required in
this reference.Comment: Published at http://dx.doi.org/10.1214/105051605000000674 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Auxiliary SDEs for homogenization of quasilinear PDEs with periodic coefficients
We study the homogenization property of systems of quasi-linear PDEs of
parabolic type with periodic coefficients, highly oscillating drift and highly
oscillating nonlinear term. To this end, we propose a probabilistic approach
based on the theory of forward-backward stochastic differential equations and
introduce the new concept of ``auxiliary SDEs.''Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Probability
(http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000014
The Landau Equation for Maxwellian molecules and the Brownian Motion on SO_R(N)
In this paper we prove that the spatially homogeneous Landau equation for
Maxwellian molecules can be represented through the product of two elementary
processes. The first one is the Brownian motion on the group of rotations. The
second one is, conditionally on the first one, a Gaussian process. Using this
representation, we establish sharp multi-scale upper and lower bounds for the
transition density of the Landau equation, the multi-scale structure depending
on the shape of the support of the initial condition.Comment: 3
Information Transmission under Random Emission Constraints
We model the transmission of a message on the complete graph with n vertices
and limited resources. The vertices of the graph represent servers that may
broadcast the message at random. Each server has a random emission capital that
decreases at each emission. Quantities of interest are the number of servers
that receive the information before the capital of all the informed servers is
exhausted and the exhaustion time. We establish limit theorems (law of large
numbers, central limit theorem and large deviation principle), as n tends to
infinity, for the proportion of visited vertices before exhaustion and for the
total duration. The analysis relies on a construction of the transmission
procedure as a dynamical selection of successful nodes in a Galton-Watson tree
with respect to the success epochs of the coupon collector problem
Convergence order of upwind type schemes for transport equations with discontinuous coefficients
An analysis of the error of the upwind scheme for transport equation with
discontinuous coefficients is provided. We consider here a velocity field that
is bounded and one-sided Lipschitz continuous. In this framework, solutions are
defined in the sense of measures along the lines of Poupaud and Rascle's work.
We study the convergence order of the upwind scheme in the Wasserstein
distances. More precisely, we prove that in this setting the convergence order
is 1/2. We also show the optimality of this result. In the appendix, we show
that this result also applies to other "diffusive" "first order" schemes and to
a forward semi-Lagrangian scheme
Global solvability of a networked integrate-and-fire model of McKean-Vlasov type
We here investigate the well-posedness of a networked integrate-and-fire
model describing an infinite population of neurons which interact with one
another through their common statistical distribution. The interaction is of
the self-excitatory type as, at any time, the potential of a neuron increases
when some of the others fire: precisely, the kick it receives is proportional
to the instantaneous proportion of firing neurons at the same time. From a
mathematical point of view, the coefficient of proportionality, denoted by
, is of great importance as the resulting system is known to blow-up
for large values of . In the current paper, we focus on the
complementary regime and prove that existence and uniqueness hold for all time
when is small enough.Comment: Published at http://dx.doi.org/10.1214/14-AAP1044 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Weak solutions to the master equation of potential mean field games
The purpose of this work is to introduce a notion of weak solution to the
master equation of a potential mean field game and to prove that existence and
uniqueness hold under quite general assumptions. Remarkably, this is achieved
without any monotonicity constraint on the coefficients. The key point is to
interpret the master equation in a conservative sense and then to adapt to the
infinite dimensional setting earlier arguments for hyperbolic systems deriving
from a Hamilton-Jacobi-Bellman equation. Here, the master equation is indeed
regarded as an infinite dimensional system set on the space of probability
measures and is formally written as the derivative of the
Hamilton-Jacobi-Bellman equation associated with the mean field control problem
lying above the mean field game. To make the analysis easier, we assume that
the coefficients are periodic, which allows to represent probability measures
through their Fourier coefficients. Most of the analysis then consists in
rewriting the master equation and the corresponding Hamilton-Jacobi-Bellman
equation for the mean field control problem as partial differential equations
set on the Fourier coefficients themselves. In the end, we establish existence
and uniqueness of functions that are displacement semi-concave in the measure
argument and that solve the Hamilton-Jacobi-Bellman equation in a suitable
generalized sense and, subsequently, we get existence and uniqueness of
functions that solve the master equation in an appropriate weak sense and that
satisfy a weak one-sided Lipschitz inequality. As another new result, we also
prove that the optimal trajectories of the associated mean field control
problem are unique for almost every starting point, for a suitable probability
measure on the space of probability measures
The transition point in the zero noise limit for a 1D Peano example
The zero-noise result for Peano phenomena of Bafico and Baldi (1982) is revisited. The original proof was based on explicit solutions to the elliptic equations for probabilities of exit times. The new proof given here is purely dynamical, based on a direct analysis of the SDE and the relative importance of noise and drift terms. The transition point between noisy behavior and escaping behavior due to the drift is identified
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