Numerical-statistical and analytical study of asymptotics for the average multiplication particle flow in a random medium

It is well known that, under rather general conditions, the particle flux density in a multiplying medium is asymptotically exponential in time $t$ with a parameter $\lambda$, i.e., with an exponent $\lambda t$. If the medium is random, then $\lambda$ is a random variable, and the time asymptotics of the average number of particles (over medium realizations) can be estimated in some approximation by averaging the exponent with respect to the distribution of $\lambda$. Assuming that this distribution is Gaussian, an asymptotic “superexponential” estimate for the average flux expressed by an exponential with the exponent $t\mathrm{E}\lambda+t^2\mathrm{D}\lambda/2$ can be obtained in this way. To verify this estimate in a numerical experiment, a procedure is developed for computing the probabilistic moments of $\lambda$ based on randomizations of Fourier approximations of special nonlinear functionals. The derived new formula is used to study the COVID-19 pandemic.