Analysis On Stationary Properties Of Stochastic Systems Driven By Cross-correlated Multiplicative Noises

The method of calculation of average properties of a dynamical system which has been hitherto used is to construct the Fokker-Planck equation in phase space from the underlying Langevin equation for the random variables representing say the position and velocity of a Brownian particle.The Langevin equation refers to paths of random variables while the Fokker-Planck equation is an equation for the evolution of the distribution function of fluctuating macroscopic variables. The main use of the Fokker-Planck equation is as an approximate description for any Markov process.In general, Langevin's method is far easier to comprehend than that of the Fokker-Planck equation as it is based directly on the concept of the time evolution of the underlying probability distribution.However, we would rather focus on predicting the statistical properties of these paths than on predicting these paths themselves .i.e. studying the evolution regulation of the distribution function for random variables by means of the Fokker-Planck equation.Chapter 2 begins with introducing the historical background of the concept for Brownian motion. And then, as to the cause of the phenomenon, we introduce the concept of the equation of motion of a random variable (in the case the position of a Brownian particle) by Langevin, namely the Langevin equation. We also construct the Fokker-Planck equation from the Langevin equation and get the underlying stationary probability distribution eventually.The study of effects on a cell growth model by cross-correlated white noises makes up the chapter 3 of this paper. Two white noises influence the growth rate and the decay rate of the cell mass, respectively. We found, in the end, that the cross-correlated white noises introduced before could make remarkable effects on the macroscopic variable of cell population. Either the two white noises themselves or the cross-correlation between them can eventually decide the destiny of cell population.Chapter 4 refers to the effects on a bistable system by the same cross-correlated white noises as chapter 3. First, some bistability phenomena exist extensively in the world such as optical bistability and their potential are mentioned. And the introducing of a bistable kinetic model, also a LE we interested in follows. Through the procedure as mentioned above, we find cross-correlated white noises play an important role in deciding the properties of the bistable system, too. Cross-correlated noises imposed on the bistable kinetic model could transform the former bistable state into a monostable state by undergoing a tri-stable state during this process, or the converse.