1,560 research outputs found
Comparison and maximum principles for a class of flux-limited diffusions with external force fields
In this paper, we are interested in a general equation that has finite speed
of propagation compatible with Einstein's theory of special relativity. This
equation without external force fields has been derived recently by means of
optimal transportation theory. We first provide an argument to incorporate the
external force fields. Then we are concerned with comparison and maximum
principles for this equation. We consider both stationary and evolutionary
problems. We show that the former satisfies a comparison principle and a strong
maximum principle. While the latter fulfils weaker ones. The key technique is a
transformation that matches well with the gradient flow structure of the
equation.Comment: 15 pages. Comments are welcom
The two-scale approach to hydrodynamic limits for non-reversible dynamics
In a recent paper by Grunewald et.al., a new method to study hydrodynamic
limits was developed for reversible dynamics. In this work, we generalize this
method to a family of non-reversible dynamics. As an application, we obtain
quantitative rates of convergence to the hydrodynamic limit for a weakly
asymmetric version of the Ginzburg-Landau model endowed with Kawasaki dynamics.
These results also imply local Gibbs behavior, following a method introduced in
a recent paper by the second author.Comment: 26 page
Coupled McKean-Vlasov diffusions: wellposedness, propagation of chaos and invariant measures
In this paper, we study a two-species model in the form of a coupled system
of nonlinear stochastic differential equations (SDEs) that arises from a
variety of applications such as aggregation of biological cells and pedestrian
movements. The evolution of each process is influenced by four different
forces, namely an external force, a self-interacting force, a cross-interacting
force and a stochastic noise where the two interactions depend on the laws of
the two processes. We also consider a many-particle system and a (nonlinear)
partial differential equation (PDE) system that associate to the model. We
prove the wellposedness of the SDEs, the propagation of chaos of the particle
system, and the existence and (non)-uniqueness of invariant measures of the PDE
system.Comment: 35 pages. Comments are welcom
An operator splitting scheme for the fractional kinetic Fokker-Planck equation
In this paper, we develop an operator splitting scheme for the fractional
kinetic Fokker-Planck equation (FKFPE). The scheme consists of two phases: a
fractional diffusion phase and a kinetic transport phase. The first phase is
solved exactly using the convolution operator while the second one is solved
approximately using a variational scheme that minimizes an energy functional
with respect to a certain Kantorovich optimal transport cost functional. We
prove the convergence of the scheme to a weak solution to FKFPE. As a
by-product of our analysis, we also establish a variational formulation for a
kinetic transport equation that is relevant in the second phase. Finally, we
discuss some extensions of our analysis to more complex systems
Weakly Non-Equilibrium Properties of Symmetric Inclusion Process with Open Boundaries
We study close to equilibrium properties of the one-dimensional Symmetric
Inclusion Process (SIP) by coupling it to two particle-reservoirs at the two
boundaries with slightly different chemical potentials. The boundaries
introduce irreversibility and induce a weak particle current in the system. We
calculate the McLennan ensemble for SIP, which corresponds to the entropy
production and the first order non-equilibrium correction for the stationary
state. We find that the first order correction is a product measure, and is
consistent with the local equilibrium measure corresponding to the steady state
density profile.Comment: 17 pages, revise
Analysis of the mean squared derivative cost function
In this paper, we investigate the mean squared derivative cost functions that
arise in various applications such as in motor control, biometrics and optimal
transport theory. We provide qualitative properties, explicit analytical
formulas and computational algorithms for the cost functions. We also perform
numerical simulations to illustrate the analytical results. In addition, as a
by-product of our analysis, we obtain an explicit formula for the inverse of a
Wronskian matrix that is of independent interest in linear algebra and
differential equations theory.Comment: 28 page
On the expected number of equilibria in a multi-player multi-strategy evolutionary game
In this paper, we analyze the mean number of internal equilibria in
a general -player -strategy evolutionary game where the agents' payoffs
are normally distributed. First, we give a computationally implementable
formula for the general case. Next we characterize the asymptotic behavior of
, estimating its lower and upper bounds as increases. Two important
consequences are obtained from this analysis. On the one hand, we show that in
both cases the probability of seeing the maximal possible number of equilibria
tends to zero when or respectively goes to infinity. On the other hand,
we demonstrate that the expected number of stable equilibria is bounded within
a certain interval. Finally, for larger and , numerical results are
provided and discussed.Comment: 26 pages, 1 figure, 1 table. revised versio
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