Singular Perturbation Methods In The Diffusion Model Of Chemical Reactions

The exit problem of random dynamical systems is an important issue in the field ofdynamical systems research. Its core content is the probability distribution of exit points andthe exit time of the trajectory, which was located in the attract domain of the deterministicsteady state dynamical system, and afected by the noise, leaving the domain of attraction.This problem due to its wide application background and thus subject to mathematicians,physicists, chemists and engineers have conducted attention and in-depth study.In1905, Einstein proposed the mathematical theory of Brownian Motion, and applied itto explain the molecular difusion problem. Since then, many mathematicians have devoteda lot of work in the exploration of random system. And now the applying scope of randomsystem contains many subjects of natural science and social science, like atomic and molecu-lar physics, chemical kinetics, filter theory, wave propagation in random medium, populationgenetics and so forth. The Ito formula, has become an important tool in the research of thetheory of stochastic diferential equations in bounded domains. The close relationship be-tween first passage time and boundary value problem for partial diferential equations madeasymptotic analysis of partial diferential equation become a powerful method in researchingfirst passage time. By calculating the singular perturbation solution of the boundary valueproblem, we can obtain the first exit time of the solutions of these equations starting from aparticular domain.The exit problem of a random system is that the trajectories of a deterministic dynamicalsystem exit from the certain domination under the efect of the small random perturbationsof the type as Gaussian white noise. Assuming the deterministic system isdx=b(x(t), t)dt,then the corresponding random dynamic system after the noise interference isdx=b(x(t), t)dt+σ(x(t), t)dω(t), where ω(t) is k dimensional independent Brownian Motion (k≤n), σ is a n x k matrix, and named diffusion coefficient matrix, b is a drift vector, x, b∈Rn.For the domain Q c Rn, the first exit time tx which the trajectory of the system first exit from Q, i.e. Τx=inf{t|x(t)∈dQ, x(0)=x}, is the solution of the following boundary value problemIn this paper, we mainly researched the n dimension system and the n dimension random dynamic system which affected by white noisewhere x=(x1,X2,…,xn)T denotes the position coordinates of the particle, Ω={x e Rn|x12+x22+…+xn2<R2}, w=(w1,w2,…,wn)T denotes a vector of independent Brownian Motions, s>0is small parameter.If the system satisfies the following assumptions:(Al) System (1) has a unique asymptotically stable equilibrium point X0in Ω, Ω is the domain of attraction of X0, and minΩ?=?(x0)=0;(A2) b(x)=-▽?(x) is the smooth vector field in Ω, and the minimum of <p is achieved at isolated points on dΩ.In the case of (x1,x2,…,xn)∈(?)Ω, b·γ<0, we obtainTheorem1Let system(2) satisfies the assumptions (A1)-(A2), and {x1,X2,…,xn) e dQ, b·γ<0. If x{t) is the solution of system (2) which is starting off at point x0in Ω, then the asymptotic expansion of the exit time of x(t) first hits the boundary (?)Ω is the determinant of the Hessian matrix of? at θi, b10and b11is the coefficient of the first two terms in Taylor expansion of b·γ on the boundary, b20is the coefficient of the first term in Taylor expansion of b·σ on the boundary,? is the minimum of? which is achieved at isolated points on (?)Ω, n is the dimension of the system.If the system satisfies another assumption:(A3) Extε is slow variable on tangential direction of (?)Ω.In the case of (x1, x2,…, xn) e (?)Ω, b·γ=0, we obtainTheorem2Let system(2) satisfies the assumptions (A1)-(A3), and (x1,x2,???,xn)∈(?)Ω, b·γ=0. If x(t) is the solution of system (2) which is starting off at point x0in Ω, then the asymptotic expansion of the exit time of x(t) first hits the boundary (?)Ω is the determinant of the Hessian matrix of <p at θi, b11, b12is the coefficient of the first two terms in Taylor expansion of b·γ on the boundary,? is the minimum of? which is achieved at isolated points on (?)Ω, n is the dimension of the system.