Symplectic Method And Effective Diffusivity Of Stochastic Differential Equation Driven By Fractional Browian Motion

Fractional Brownian motion has two important properties:self-similarity and nonMarkovianity,and is an intrinsic property of many natural and social phenomena.Around the 21st century,mathematical models were characterized by stochastic differential equations driven by fractional Brownian motion in many disciplines(especially Finance,Physics,Control theory and Biology).The study of the uniqueness and existence for the solution of the stochastic differential equations driven by fractional Brownian motion and the qualities of the solution have been very mature.Meanwhile,the effective diffusivity and asymptotic properties of the stochastic differential equations driven by Brownian motion have also been concluded theoretically and numerically.This paper is based on the study of stochastic differential equations driven by standard Brownian motion,and speculates whether stochastic differential equations driven by fractional Brownian motion have similar properties and theories.The symplectic method enables the discretized difference equation to maintain the symplectic structure of the original system,with good properties of long-term stability and tracking ability.So from the second-order implicit midpoint method,this paper first proposes a more general symplectic method,which can solve stochastic different ial equations driven by standard Brownian motion and those driven by fractional Brownian motion numerically.Secondly,taking the two-dimensional and three-dimensional Taylor-Green velocity fields as examples,the existence and asymptotic results of the effective diffusivity for the stochastic differential equations driven by fractional Brownian motion are numerically proved.At the same time,the symplectic method and Euler’s method are used to calculate the effective diffusivity.and it is found that the symplectic method can simulate its asymptotic properties when the molecular diffusion coefficient tends to 0.while Euler’s method cannot.Finally,in the two-dimensional and three-dimensional cases,the symplectic method proposed in this paper is used to solve the stochastic different ial equations driven by fractional Brownian motion.and the influence of different Hurst parameters when calculating its effective diffusivity is compared.