Investigation of solutions to higher-order dispersive equations with φ-sub-Gaussian initial conditions

Abstract

In this paper, there are studied sample paths properties of stochastic processes representing solutions of higher-order dispersive equations with random initial conditions given by φ-sub-Gaussian harmonizable processes. The main results are the bounds for the rate of growth of such stochastic processes considered over unbounded domains. The class of φ-sub-Gaussian processes with φ(x) = |x|^α/α, 1 < α <= 2, is a natural generalization of Gaussian processes. For such initial conditions the bounds for the distribution of supremum of solutions can be calculated in rather simple form. The bounds for the rate of growth of solution to higher-order partial differential equations with random initial conditions in the case of general φ were obtained in [9], the derivation was based on the sults stated in [1]. Here we use another approach, which allows us, for the particular case φ(x) = |x|^α/α, α є (1, 2], to present the expressions for the bounds in the closed form.

Bounds for the Tail Distributions of Suprema of Sub-Gaussian Type Random Fields

“…, where σ k are introduced in the next theorem, θ = inf k γ k ε k . Theorem 3.1 below is an extension of the result stated in Sakhno and Vasylyk (2021)(Theorem 1), see also Hopkalo and Sakhno (2021)(Theorem 4). The proof presented in Sakhno and Vasylyk (2021)(Theorem 1) for the case of metrics d = d 1 works for a general metrics d as well.…”

“…Proof. The proof follows the same lines as that of Corollary 1 in Sakhno and Vasylyk (2021). First, we need to derive another bound for the integral (3.4) which will be used to evaluate the expression (3.1).…”

“…We consider the specification of Theorem 3.1 for the cases of metrics d 1 and d 3 . Case d 1 was studied in Sakhno and Vasylyk (2021) and the following result was stated.…”

“…which is more convenient for estimation of the expression (3.1) (needed to apply Theorem 3.1) than the estimate from Theorem 2.1. We refer for more details to Hopkalo and Sakhno (2021), Sakhno and Vasylyk (2021).…”