We consider the stochastic thin-film equation with colored Gaussian Stratonovich noise and establish the existence of nonnegative weak (martingale) solutions. The construction is based on a Trotter-Kato-type decomposition into a deterministic and a stochastic evolution, which yields an easy to implement numerical algorithm. Compared to previous work, no interface potential has to be included and the Trotter-Kato scheme allows for a simpler proof of existence than in case of Ito noise.
We study pathwise entropy solutions for scalar conservation laws with inhomogeneous fluxes and quasilinear multiplicative rough path dependence. This extends the previous work of Lions, Perthame and Souganidis who considered spatially independent and inhomogeneous fluxes with multiple paths and a single driving singular path respectively. The approach is motivated by the theory of stochastic viscosity solutions which 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.
Data mining has a wide range of applications in the real world. However, it is possible to disclose the private information of users in the process of data mining. Therefore, it is of great significance to protect the users' privacy while mining the knowledge behind the data. In this paper, we propose a Naive Bayes classification method based on differential privacy. For nominal attributes, we add Laplace noise to the count. For numerical attributes, we add Laplace noise to the mean, standard deviation, and scale parameter, and then use the noisy parameters to calculate the prior probability and conditional probability. For numerical attributes, we assume that they follow Gaussian, Laplace, or lognormal distribution, and apply our algorithms to compare utilities.