Vol 225, No 5 (2017)

We propose two approaches which allow us to construct probabilistic representations of classical and viscosity solutions of the Cauchy problem for a system of quasilinear parabolic equations. Based on these representations, we develop two numerical algorithms to construct the required solution. The system under consideration arises as a mathematical model of parabolic conservation laws. Bibliography: 14 titles.

An old conjecture states that among all simplices inscribed in the unit sphere, the regular one has the maximal mean width. We restate this conjecture probabilistically and prove its asymptotic version. We also show that the mean width of the regular simplex with 2n vertices is remarkably close to the mean width of the regular crosspolytope with the same number of vertices. We establish several formulas conjectured by S. Finch on the projection length W of the regular cube, simplex, and crosspolytope onto a line with random direction. Finally, we prove distributional limit theorems for W as the dimension of the regular polytope goes to ∞. Bibliography: 25 titles.

We construct analogs of symmetric α-stable distributions for noninteger indices α > 2 and study their relations to solutions of the Cauchy problem for some evolution equations.

We consider a pseudo-Poisson process of the following simple type. This process is a Poissonian subordinator for a sequence of i.i.d. random variables with finite variance. Further we consider sums of i.i.d. copies of a pseudo-Poisson process. For a family of distributions of these random sums, we prove the tightness (relative compactness) in the Skorokhod space. Under the conditions of the Central Limit Theorem for vectors, we establish the weak convergence in the functional Skorokhod space of the examined sums to the Ornstein–Uhlenbeck process.

We consider a system of two independent alternating renewal processes with states 0 and 1 and an initial shift t₀ of one process relative to the other one. An integral equation with respect to the expectation of time T (the first time when both processes have state 0) is derived. To derive this equation, we use the method of so-called minimal chains of overlapping 1-intervals. Such a chain generates some breaking semi-Markov process of intervals composing the interval (0, T ). A solution of the integral equation is obtained for the case where the lengths of 1-intervals have exponential distributions and lengths of 0-intervals have arbitrary distributions. For more general distributions of 1-intervals, the Monte Carlo method is applied when both processes are simulated numerically by a computer. A histogram for estimates of the expectation of T as a function of t₀ is demonstrated. Bibliography: 4 titles.