1,002 research outputs found
Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation
We study a quasilinear parabolic Cauchy problem with a cumulative
distribution function on the real line as an initial condition. We call
'probabilistic solution' a weak solution which remains a cumulative
distribution function at all times. We prove the uniqueness of such a solution
and we deduce the existence from a propagation of chaos result on a system of
scalar diffusion processes, the interactions of which only depend on their
ranking. We then investigate the long time behaviour of the solution. Using a
probabilistic argument and under weak assumptions, we show that the flow of the
Wasserstein distance between two solutions is contractive. Under more stringent
conditions ensuring the regularity of the probabilistic solutions, we finally
derive an explicit formula for the time derivative of the flow and we deduce
the convergence of solutions to equilibrium.Comment: Stochastic partial differential equations: analysis and computations
(2013) http://dx.doi.org/10.1007/s40072-013-0014-
Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme
In the present paper, we prove that the Wasserstein distance on the space of
continuous sample-paths equipped with the supremum norm between the laws of a
uniformly elliptic one-dimensional diffusion process and its Euler
discretization with steps is smaller than where
is an arbitrary positive constant. This rate is intermediate
between the strong error estimation in obtained when coupling the
stochastic differential equation and the Euler scheme with the same Brownian
motion and the weak error estimation obtained when comparing the
expectations of the same function of the diffusion and of the Euler scheme at
the terminal time . We also check that the supremum over of the
Wasserstein distance on the space of probability measures on the real line
between the laws of the diffusion at time and the Euler scheme at time
behaves like .Comment: Published in at http://dx.doi.org/10.1214/13-AAP941 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Efficiency of the Wang-Landau algorithm: a simple test case
We analyze the efficiency of the Wang-Landau algorithm to sample a multimodal
distribution on a prototypical simple test case. We show that the exit time
from a metastable state is much smaller for the Wang Landau dynamics than for
the original standard Metropolis-Hastings algorithm, in some asymptotic regime.
Our results are confirmed by numerical experiments on a more realistic test
case
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