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  1. Abstract For a Brownian directed polymer in a Gaussian random environment, with q ( t , ⋅) denoting the quenched endpoint density and Q n ( t , x 1 , … , x n ) = E [ q ( t , x 1 ) … q ( t , x n ) ] , we derive a hierarchical PDE system satisfied by { Q n } n ⩾ 1 . We present two applications of the system: (i) we compute the generator of { μ t ( d x ) = q ( t , x ) d x } t ⩾ 0 for some special functionals, where { μ t ( d x ) } t ⩾ 0 is viewed as a Markov process taking values in the space of probability measures; (ii) in the high temperature regime with d ⩾ 3, we prove a quantitative central limit theorem for the annealed endpoint distribution of the diffusively rescaled polymer path. We also study a nonlocal diffusion-reaction equation motivated by the generator and establish a super-diffusive O ( t 2/3 ) scaling.
  2. Abstract. In response to the coronavirus disease of 2019 (COVID-19),California issued statewide stay-at-home orders, bringing about abrupt anddramatic reductions in air pollutant emissions. This crisis offers us anunprecedented opportunity to evaluate the effectiveness of emissionreductions in terms of air quality. Here we use the Weather Research and Forecastingmodel with Chemistry (WRF-Chem) in combination with surface observations tostudy the impact of the COVID-19 lockdown measures on air quality insouthern California. Based on activity level statistics and satelliteobservations, we estimate the sectoral emission changes during the lockdown.Due to the reduced emissions, the population-weighted concentrations of fineparticulate matter (PM2.5) decrease by 15 % in southernCalifornia. The emission reductions contribute 68 % of the PM2.5concentration decrease before and after the lockdown, while meteorologyvariations contribute the remaining 32 %. Among all chemical compositions,the PM2.5 concentration decrease due to emission reductions isdominated by nitrate and primary components. For O3 concentrations, theemission reductions cause a decrease in rural areas but an increase in urbanareas; the increase can be offset by a 70 % emission reduction inanthropogenic volatile organic compounds (VOCs). These findings suggest thata strengthened control on primary PM2.5 emissions and a well-balancedcontrol on nitrogen oxides and VOC emissions are needed to effectively andsustainably alleviate PM2.5 and O3 pollution in southernCalifornia.