Scalar conservation laws with rough (stochastic) fluxes

We develop a pathwise theory for scalar conservation laws with quasilinear multiplicative rough path dependence, a special case being stochastic conservation laws with quasilinear stochastic dependence. We introduce the notion of pathwise stochastic entropy solutions, which is closed with the local uniform limits of paths, and prove that it is well posed, i.e., we establish existence, uniqueness and continuous dependence, in the form of pathwise $L^1$-contraction, as well as some explicit estimates. 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 averaging lemmas for kinetic equations

We develop a class of averaging lemmas for stochastic kinetic equations. The velocity is multiplied by a white noise which produces a remarkable change in time scale. Compared to the deterministic case and as far as we work in $L^2$, the nature of regularity on averages is not changed in this stochastic kinetic equation and stays in the range of fractional Sobolev spaces at the price of an additional expectation. However all the exponents are changed; either time decay rates are slower (when the right hand side belongs to $L^2$), or regularity is better when the right hand side contains derivatives. These changes originate from a different space/time scaling in the deterministic and stochastic cases. Our motivation comes from scalar conservation laws with stochastic fluxes where the structure under consideration arises naturally through the kinetic formulation of scalar conservation laws

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 $\R^N$ 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) $L^1$-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

Speed of propagation for Hamilton-Jacobi equations with multiplicative rough time dependence and convex Hamiltonians

We show that the initial value problem for Hamilton-Jacobi equations with multiplicative rough time dependence, typically stochastic, and convex Hamiltonians satisfies finite speed of propagation. We prove that in general the range of dependence is bounded by a multiple of the length of the "skeleton" of the path, that is a piecewise linear path obtained by connecting the successive extrema of the original one. When the driving path is a Brownian motion, we prove that its skeleton has almost surely finite length. We also discuss the optimality of the estimate

Exploring Direct and Indirect Effects of English Proficiency on Access, Utilization, and Health Status among Californian Adults with Limited English Proficiency (LEP)

Background and Study Purpose: Findings from previous studies suggest that, in a health care delivery context, individuals with limited English proficiency (LEP) are adversely impacted by lack of patient-provider language concordance. Yet, the concept of LEP has been mostly studied in the context of cultural competence and language has been generally considered a demographic or cultural characteristic. There is a growing body of research concerning LEP and health status; however, it is limited. This study sought to evaluate the effects of LEP on access, utilization, and self-rated health status (SRHS) among LEP respondents to a large health interview survey by comparing LEPs to two groups: English only (EO) and English and another language (E+OL). Methods: The study design was retrospective, cross-sectional, and observational. Quantitative statistical analyses were required. Secondary data from the 2013-2014 California Health Interview Survey was used. N = 40,240 non-institutionalized Californian adults. The predictor was levels of English proficiency. EO was a reference group. The outcomes were access, utilization, and SRHS. Covariates were age, sex, race, income and education. Results. Logistic regressions showed that compared to the E+OLs, LEPs had: (1) Lower odds ratio on all observed variables measuring access with statistical significance for some variables and others no statistical significance. (2) Lower odds ratio on all observed variables measuring utilization with statistical significance. Further, correlations among the all measurement variables were positive and effect sizes ranged from low to medium. Finally, results from a path analysis for LEPs showed a recursive inverse effect on access (p \u3c .05, B = -0.27, 95% CI [-0.36, -0.18]), utilization (p \u3c .05, B = -.80, 95% CI [-0.97, 0-.62]), and SRHS (p \u3c .05, B = -.88, 95% CI [-1.04, -0.73]). In addition, there was a predictive effect of access on SRHS and access had a mediating effect related to LEP on SRHS (p = 0.003, 95% CI [0.01, 0.06]) and a predictive effect of utilization on SRHS and utilization had a mediating effect related to LEP on SRHS (p \u3c .05, 95% CI [0.03, 0.06]). Further analysis showed that, when levels of English proficiency was not allowed a direct path to SRHS and access and utilization had respective direct paths to SRHS, path loadings were equal across EOs, E+OLs, and LEPs and were statistically significant across groups (access: p \u3c .05; utilization: p \u3c .05). These results suggest that levels of English proficiency contribute to the disparities observed among LEPs. Conclusion: There are disparities in access, utilization, and SRHS among individuals with limited English proficiency. Those disparities can be reduced through decreasing barriers to access and utilization. Based on findings from this study, the LEP Health Outcomes Assessment and Decision model was developed and is being proposed for used in studying perceived health outcomes in LEPs

Exploring Direct and Indirect Effects of English Proficiency on Access, Utilization, and Health Status among Californian Adults with Limited English Proficiency (LEP)

Background and Study Purpose: Findings from previous studies suggest that, in a health care delivery context, individuals with limited English proficiency (LEP) are adversely impacted by lack of patient-provider language concordance. Yet, the concept of LEP has been mostly studied in the context of cultural competence and language has been generally considered a demographic or cultural characteristic. There is a growing body of research concerning LEP and health status; however, it is limited. This study sought to evaluate the effects of LEP on access, utilization, and self-rated health status (SRHS) among LEP respondents to a large health interview survey by comparing LEPs to two groups: English only (EO) and English and another language (E+OL). Methods: The study design was retrospective, cross-sectional, and observational. Quantitative statistical analyses were required. Secondary data from the 2013-2014 California Health Interview Survey was used. N = 40,240 non-institutionalized Californian adults. The predictor was levels of English proficiency. EO was a reference group. The outcomes were access, utilization, and SRHS. Covariates were age, sex, race, income and education. Results. Logistic regressions showed that compared to the E+OLs, LEPs had: (1) Lower odds ratio on all observed variables measuring access with statistical significance for some variables and others no statistical significance. (2) Lower odds ratio on all observed variables measuring utilization with statistical significance. Further, correlations among the all measurement variables were positive and effect sizes ranged from low to medium. Finally, results from a path analysis for LEPs showed a recursive inverse effect on access (p \u3c .05, B = -0.27, 95% CI [-0.36, -0.18]), utilization (p \u3c .05, B = -.80, 95% CI [-0.97, 0-.62]), and SRHS (p \u3c .05, B = -.88, 95% CI [-1.04, -0.73]). In addition, there was a predictive effect of access on SRHS and access had a mediating effect related to LEP on SRHS (p = 0.003, 95% CI [0.01, 0.06]) and a predictive effect of utilization on SRHS and utilization had a mediating effect related to LEP on SRHS (p \u3c .05, 95% CI [0.03, 0.06]). Further analysis showed that, when levels of English proficiency was not allowed a direct path to SRHS and access and utilization had respective direct paths to SRHS, path loadings were equal across EOs, E+OLs, and LEPs and were statistically significant across groups (access: p \u3c .05; utilization: p \u3c .05). These results suggest that levels of English proficiency contribute to the disparities observed among LEPs. Conclusion: There are disparities in access, utilization, and SRHS among individuals with limited English proficiency. Those disparities can be reduced through decreasing barriers to access and utilization. Based on findings from this study, the LEP Health Outcomes Assessment and Decision model was developed and is being proposed for used in studying perceived health outcomes in LEPs

Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions

We study periodic homogenization problems for second-order pde in half-space type domains with Neumann boundary conditions. In particular, we are interested in "singular problems" for which it is necessary to determine both the homogenized equation and boundary conditions. We provide new results for fully nonlinear equations and boundary conditions. Our results extend previous work of Tanaka in the linear, periodic setting in half-spaces parallel to the axes of the periodicity, and of Arisawa in a rather restrictive nonlinear periodic framework. The key step in our analysis is the study of associated ergodic problems in domains with similar structure

Eikonal equations and pathwise solutions to fully non-linear SPDEs

We study the existence and uniqueness of the stochastic viscosity solutions of fully nonlinear, possibly degenerate, second order stochastic pde with quadratic Hamiltonians associated to a Riemannian geometry. The results are new and extend the class of equations studied so far by the last two authors

Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications

We study the homogenization of fully nonlinear degenerate second-order pde, with “ellipticity” of the same order as the space oscillations, in periodic and almost periodic. As a special case we consider the class of quasi-linear, degenerate elliptic pde. The results apply to level sets equations describing the evolution of fronts with prescribed normal velocity. We also discuss an application about the averaged properties of interfacial motions in periodic and almost periodic environments.ou

The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation

We study the asymptotics of Allen-Cahn-type bistable reaction-diffusion equations which are additively perturbed by a stochastic forcing (time white noise). The conclusion is that the long time, large space behavior of the solutions is governed by an interface moving with curvature dependent normal velocity which is additively perturbed by time white noise. The result is global in time and does not require any regularity assumptions on the evolving front. The main tools are (i)~the notion of stochastic (pathwise) solution for nonlinear degenerate parabolic equations with multiplicative rough (stochastic) time dependence, which has been developed by the authors, and (ii)~the theory of generalized front propagation put forward by the second author and collaborators to establish the onset of moving fronts in the asymptotics of reaction-diffusion equations.Comment: arXiv admin note: substantial text overlap with arXiv:1809.0174