The two-dimensional primitive equations with Lévy noise are studied in this paper.We prove the existence and uniqueness of the solutions in a fixed probability space which based on a priori estimates,weak convergence method and monotonicity arguments.
This paper provides a mathematically rigorous foundation for self-consistent mean field theory of the polymeric physics. We study a new model for dynamics of mono-polymer systems. Every polymer is regarded as a string of points which are moving randomly as Brownian motions and under elastic forces. Every two points on the same string or on two different strings also interact under a pairwise potential V. The dynamics of the system is described by a system of N coupled stochastic partial differential equations （SPDEs）. We show that the mean field limit as N -＋ c~ of the system is a self-consistent McKean-Vlasov type equation, under suitable assumptions on the initial and boundary conditions and regularity of V. We also prove that both the SPDE system of the polymers and the mean field limit equation are well-posed.
Living cells are open systems that exist far away from a state of thermodynamical equilibrium. They utilize the high-grade chemical energy provided by food to produce ATP and re- lease ADP and Pi together with heat dissipation. Living cells exist in a non-equilibrium steady state （NESS）, they replicate themselves and respond to various environmental changes via signal transduction pathways. Because the majority of cells exist at room temperature, the stochasticity of chemical reac- tions in the cells is unavoidable. Recent research into fluores- cent proteins and microscopy techniques have enabled us to observe the dynamic process of mRNA and proteins in single living bacterial cells [1], and these have resulted in new in- sights into regulation mechanisms in molecular biology, i.e., in cellular signal transduction pathways.
