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We apply the Hamilton-Jacobi equation (HJE) formalism to solve the dynamicsof the Chemical Master Equation (CME). We found exact analytical expressions(in large system-size limit) for the probability distribution, includingexplicit expression for the dynamics of variance of distribution. We also givethe solution for some simple cases of the model with time-dependent rates. Wederived the results of Van Kampen method from HJE approach using a specialansatz. Using the Van Kampen method, we give a system of ODE to define thevariance in 2-d case. We performed numerics for the CME with stationary noise.We give analytical criteria for the disappearance of bi-stability in case ofstationary noise in 1-d CME.

Exact results in the large system size limit for the dynamics of the Chemical Master Equation, a one dimensional chain of equations