1,561 research outputs found
On the generalized almost periodic homogenization of stochastic conservation laws with multiplicative noise
We consider the generalized almost periodic homogenization problem for two
types of stochastic conservation laws with oscillatory coefficients and a
multiplicative noise, namely, the nonlinear transport equation and the equation
with a stiff oscillating external force. We use the notion of homogenization by
noise-approximation, introduced here, which amounts to the homogenization of
the equations with an approximate noise as well as showing that the solutions
of the approximate equations with an artificial viscosity term converge, as the
noise-approximation parameter goes to zero, to the solutions of the original
equation with artificial viscosity, whose solutions are shown to converge to
the solutions of the original equation as the viscosity parameter goes to zero.
Besides, the homogenization limits of the approximate equations are themselves
limits of a two-parameter sequence, when one of these parameters, representing
viscosity, goes to zero, and whose counterpart limits, obtained when the other
parameter, representing the noise-approximation, goes to zero, converge to a
well determined limit which we call the homogenization limit by
noise-approximation (b.n.a.). In both cases the multiplicative noise is
prescribed so that the corresponding stochastic equation has special solutions
that play the role of steady-state solutions in the deterministic case and are
crucial elements in the homogenization analysis. As a byproduct, our
prescription of the multiplicative noise provides a way to justify the noise
perturbation of the corresponding deterministic equation
Scalar conservation laws with rough (stochastic) fluxes; the spatially dependent case
We continue the development of the theory of pathwise stochastic entropy
solutions for scalar conservation laws in with quasilinear
multiplicative ''rough path'' dependence by considering inhomogeneous fluxes
and a single rough path like, for example, a Brownian motion. Following our
previous note where we considered spatially independent fluxes, we introduce
the notion of pathwise stochastic entropy solutions and prove that it is well
posed, that is we establish existence, uniqueness and continuous dependence in
the form of a (pathwise) -contraction. Our approach is motivated by the
theory of stochastic viscosity solutions, which was introduced and developed by
two of the authors, to study fully nonlinear first- and second-order stochastic
pde with multiplicative noise. This theory relies on special test functions
constructed by inverting locally the flow of the stochastic characteristics.
For conservation laws this is best implemented at the level of the kinetic
formulation which we follow here
Stochastic foundations of undulatory transport phenomena: Generalized Poisson-Kac processes - Part I Basic theory
This article introduces the notion of Generalized Poisson-Kac (GPK) processes
which generalize the class of "telegrapher's noise dynamics" introduced by Marc
Kac in 1974, usingPoissonian stochastic perturbations. In GPK processes the
stochastic perturbation acts as a switching amongst a set of stochastic
velocity vectors controlled by a Markov-chain dynamics. GPK processes possess
trajectory regularity (almost everywhere) and asymptotic Kac limit, namely the
convergence towards Brownian motion (and to stochastic dynamics driven by
Wiener perturbations), which characterizes also the long-term/long-distance
properties of these processes. In this article we introduce the structural
properties of GPK processes, leaving all the physical implications to part II
and part III
The Lifshitz-Slyozov-Wagner equation for reaction-controlled kinetics
We rigorously derive a weak form of the Lifshitz-Slyozov-Wagner equation as
the homogenization limit of a Stefan-type problem describing
reaction-controlled coarsening of a large number of small spherical particles.
Moreover, we deduce that the effective mean-field description holds true in the
particular limit of vanishing surface-area density of particles.Comment: 15 pages, LaTeX; minor revision, change of titl
Effective governing equations for poroelastic growing media
A new mathematical model is developed for the macroscopic behaviour of a porous, linear elastic solid, saturated with a slowly flowing incompressible, viscous fluid, with surface accretion of the solid phase. The derivation uses a formal two-scale asymptotic expansion to exploit the well-separated length scales of the material: the pores are small compared to the macroscale, with a spatially periodic microstructure. Surface accretion occurs at the interface between the solid and fluid phases, resulting in growth of the solid phase through mass exchange from the fluid at a prescribed rate (and vice versa). The averaging derives a new poroelastic model, which reduces to the classical result of Burridge and Keller in the limit of no growth. The new model is of relevance to a large range of applications including packed snow, tissue growth, biofilms and subsurface rocks or soils
Fast stochastic simulation of biochemical reaction systems by\ud alternative formulations of the Chemical Langevin Equation
The Chemical Langevin Equation (CLE), which is a stochastic differential equation (SDE) driven by a multidimensional Wiener process, acts as a bridge between the discrete Stochastic Simulation Algorithm and the deterministic reaction rate equation when simulating (bio)chemical kinetics. The CLE model is valid in the regime where molecular populations are abundant enough to assume their concentrations change continuously, but stochastic fluctuations still play a major role. The contribution of this work is that we observe and explore that the CLE is not a single equation, but a parametric family of equations, all of which give the same finite-dimensional distribution of the variables. On the theoretical side, we prove that as many Wiener processes are sufficient to formulate the CLE as there are independent variables in the equation. On the practical side, we show that in the case where there are m1 pairs of reversible reactions and m2 irreversible reactions only m1+m2 Wiener processes are required in the formulation of the CLE, whereas the standard approach uses 2m1 + m2. We illustrate our findings by considering alternative formulations of the CLE for a\ud
HERG ion channel model and the Goldbeter–Koshland switch. We show that there are considerable computational savings when using our insights
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