Homogenization of nonlinear stochastic partial differential equations in a general ergodic environment

In this paper, we show that the concept of sigma-convergence associated to stochastic processes can tackle the homogenization of stochastic partial differential equations. In this regard, the homogenization problem for a stochastic nonlinear partial differential equation is studied. Using some deep compactness results such as the Prokhorov and Skorokhod theorems, we prove that the sequence of solutions of this problem converges in probability towards the solution of an equation of the same type. To proceed with, we use a suitable version of sigma-convergence method, the sigma-convergence for stochastic processes, which takes into account both the deterministic and random behaviours of the solutions of the problem. We apply the homogenization result to some concrete physical situations such as the periodicity, the almost periodicity, the weak almost periodicity, and others.Comment: To appear in: Stochastic Analysis and Application

Interior error estimate for periodic homogenization

In a previous article about the homogenization of the classical problem of diff usion in a bounded domain with su ciently smooth boundary we proved that the error is of order $\epsilon^{1/2}$. Now, for an open set with su ciently smooth boundary $C^{1,1}$ and homogeneous Dirichlet or Neuman limits conditions we show that in any open set strongly included in the error is of order $\epsilon$. If the open set $\Omega\subset R^n$ is of polygonal (n=2) or polyhedral (n=3) boundary we also give the global and interrior error estimates

Stochastic Hybrid Control

The objective of this paper is to study the stochastic version of a previous paper of the authors, in which hybrid control for deterministic systems was considered. The modelling is quite similar to the deterministic case. We have a system whose state is composed of a continuous part and a discrete part. They are affected by a continuous type control and an impulse control. The dynamics is moreover perturbed by noise, also a continuous and a discrete noise process. The Markovian character of the state process is preserved. We develop the model and show how the dynamic programming approach leads to some involved quasi-variational inequality

Homogenization of the elliptic Dirichlet problem: operator error estimates in L2L_2

Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain of class $C^2$. In the Hilbert space $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $\mathcal{A}_{D,\varepsilon}$ with the Dirichlet boundary condition. Here $\varepsilon>0$ is the small parameter. The coefficients of the operator are periodic and depend on $\mathbf{x}/\varepsilon$. A sharp order operator error estimate $\|\mathcal{A}_{D,\varepsilon}^{-1} - (\mathcal{A}_D^0)^{-1} \|_{L_2 \to L_2} \leq C \varepsilon$ is obtained. Here $\mathcal{A}^0_D$ is the effective operator with constant coefficients and with the Dirichlet boundary condition.Comment: 13 page

Process of care in outpatient Integrative healthcare facilities: a systematic review of clinical trials

© 2015 Grant et al. Abstract Background: Patients currently integrate complementary medicine (CM) and allopathic, choosing a combination of therapies rather than a single therapy in isolation. Understanding integrative healthcare (IHC) extends beyond evaluation of specific therapies to encompass evaluations of multidisciplinary complex interventions. IHC is defined as a therapeutic strategy integrating conventional and complementary medical practices and practitioners in a shared care setting to administer an individualized treatment plan. We sought to review the outcomes of recent clinical trials, explore the design of the interventions and to discuss the methodological approaches and issues that arise when investigating a complex mix of interventions in order to guide future research. Method: Five databases were searched from inception to 30 March 2013. We included randomized and quasi-experimental clinical trials of IHC. Data elements covering process of care (initial assessment, treatment planning and review, means for integration) were extracted. Results: Six thousand two hundred fifty six papers were screened, 5772 were excluded and 484 full text articles retrieved. Five studies met the inclusion criteria. There are few experimental studies of IHC. Of the five studies conducted, four were in people with lower back pain. The positive findings of these studies indicate that it is feasible to conduct a rigorous clinical trial of an integrative intervention involving allopathic and CM treatment. Further, such interventions may improve patient outcomes. Conclusions: The trials in our review provide a small yet critical base from which to refine and develop larger studies. Future studies need to be adequately powered to address efficacy, safety and include data on cost effectiveness

Diffusion of a passive scalar by convective flows under parametric disorder

We study transport of a weakly diffusive pollutant (a passive scalar) by thermoconvective flow in a fluid-saturated horizontal porous layer heated from below under frozen parametric disorder. In the presence of disorder (random frozen inhomogeneities of the heating or of macroscopic properties of the porous matrix), spatially localized flow patterns appear below the convective instability threshold of the system without disorder. Thermoconvective flows crucially effect the transport of a pollutant along the layer, especially when its molecular diffusion is weak. The effective (or eddy) diffusivity also allows to observe the transition from a set of localized currents to an almost everywhere intense "global" flow. We present results of numerical calculation of the effective diffusivity and discuss them in the context of localization of fluid currents and the transition to a "global" flow. Our numerical findings are in a good agreement with the analytical theory we develop for the limit of a small molecular diffusivity and sparse domains of localized currents. Though the results are obtained for a specific physical system, they are relevant for a broad variety of fluid dynamical systems.Comment: 12 pages, 4 figures, the revised version of the paper for J. Stat. Mech. (Special issue for proceedings of 5th Intl. Conf. on Unsolved Problems on Noise and Fluctuations in Physics, Biology & High Technology, Lyon (France), June 2-6, 2008

Nonlinear dynamics of the viscoelastic Kolmogorov flow

The weakly nonlinear regime of a viscoelastic Navier--Stokes fluid is investigated. For the purely hydrodynamic case, it is known that large-scale perturbations tend to the minima of a Ginzburg-Landau free-energy functional with a double-well (fourth-order) potential. The dynamics of the relaxation process is ruled by a one-dimensional Cahn--Hilliard equation that dictates the hyperbolic tangent profiles of kink-antikink structures and their mutual interactions. For the viscoelastic case, we found that the dynamics still admits a formulation in terms of a Ginzburg--Landau free-energy functional. For sufficiently small elasticities, the phenomenology is very similar to the purely hydrodynamic case: the free-energy functional is still a fourth-order potential and slightly perturbed kink-antikink structures hold. For sufficiently large elasticities, a critical point sets in: the fourth-order term changes sign and the next-order nonlinearity must be taken into account. Despite the double-well structure of the potential, the one-dimensional nature of the problem makes the dynamics sensitive to the details of the potential. We analysed the interactions among these generalized kink-antikink structures, demonstrating their role in a new, elastic instability. Finally, consequences for the problem of polymer drag reduction are presented.Comment: 26 pages, 17 figures, submitted to The Journal of Fluid Mechanic

A class of non-zero-sum stochastic differential investment and reinsurance games

In this article, we provide a systematic study on the non-zero-sum stochastic differential investment and reinsurance game between two insurance companies. Each insurance company’s surplus process consists of a proportional reinsurance protection and an investment in risky and risk-free assets. Each insurance company is assumed to maximize his utility of the difference between his terminal surplus and that of his competitor. The surplus process of each insurance company is modeled by a mixed regime-switching Cramer–Lundberg diffusion approximation process, i.e. the coefficients of the diffusion risk processes are modulated by a continuous-time Markov chain and an independent market-index process. Correlation between the two surplus processes, independent of the risky asset process, is allowed. Despite the complex structure, we manage to solve the resulting non-zero sum game problem by applying the dynamic programming principle. The Nash equilibrium, the optimal reinsurance/investment, and the resulting value processes of the insurance companies are obtained in closed forms, together with sound economic interpretations, for the case of an exponential utility function.postprin

On the terminal velocity of sedimenting particles in a flowing fluid

The influence of an underlying carrier flow on the terminal velocity of sedimenting particles is investigated both analytically and numerically. Our theoretical framework works for a general class of (laminar or turbulent) velocity fields and, by means of an ordinary perturbation expansion at small Stokes number, leads to closed partial differential equations (PDE) whose solutions contain all relevant information on the sedimentation process. The set of PDE's are solved by means of direct numerical simulations for a class of 2D cellular flows (static and time dependent) and the resulting phenomenology is analysed and discussed.Comment: 13 pages, 2 figures, submitted to JP