This paper studies some analytical properties of weak solutions of 3D stochastic primitive equations with periodic boundary conditions. The martingale problem associated to this model is shown to have a family of solutions satisfying the Markov property, which is achieved by means of an abstract selection principle. The Markov property is crucial to extend the regularity of the transition semigroup from small times to arbitrary times. Thus, under a regular additive noise, every Markov solution is shown to have a property of continuous dependence on initial conditions, which follows from employing the weak-strong uniqueness principle and the Bismut-Elworthy-Li formula.
Over past decades,deceptive counterfeits which cannot be recognized by ordinary consumers when purchasing,such as counterfeit cosmetics,have posed serious threats on consumers’health and safety,and resulted in huge economic loss and inestimable brand damages to the genuine goods at the same time.Thus,how to effectively control and eliminate deceptive counterfeits in the market has become a critical problem to the local government.One of the principal challenges in combating the cheating action for the government is how to enhance the enforcement of relative quality inspection agencies like industrial administration office(IAO).In this paper,we formulate a two-stage counterfeit product model with a fixed checking rate from IAO and a penalty for holding counterfeits.Tominimize the total expected cost over two stages,the retailer adopts optimal ordering policies which are correlated with the checking rate and penalty.Under certain circumstances,we find that the optimal expected cost function for the retailer is first-order continuous and convex.The optimal ordering policy in stage two depends closely on the inventory level after the first sales period.When the checking rate in stage one falls into a certain range,the optimal ordering policy for the retailer at each stage is to order both kinds of products.Knowing the retailer’s optimal ordering policy at each stage,IAO can modify the checking rate accordingly to keep the ratio of deceptive counterfeits on the market under a certain level.
The limiting behavior of stochastic evolution processes with small noise intensityεis investigated in distribution-based approaches.Letμεbe a stationary measure for stochastic process Xεwith smallεand X0 be a semiflow on a Polish space.Assume that{με:0<ε≤ε0}is tight.Then all their limits in the weak sense are X0-invariant and their supports are contained in the Birkhoff center of X0.Applications are made to various stochastic evolution systems,including stochastic ordinary differential equations,stochastic partial differential equations,and stochastic functional differential equations driven by Brownian motion or Levy processes.
