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.
Highlights
Introduction harmonizable processLet y(t), t ∈ R, be a real strictly φ-sub-Gaussian process2, determining constant Cy and its covariance Ey(t)y(s) =Γy(s, t) has finite variation
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 results stated in [12]
Conditions of existence of the classical solution to the problem (1.4)–(1.5) with random φ-subGaussian initial condition for the case of general φ are stated in [1, 9], properties of solutions are investigated in [9, 10], in particular, different estimates for the distribution of supremum of solution, and in [9] bounds are presented for the rate of growth of solution
Summary
Оцiнки для швидкостi росту розв’язкiв диференцiальних рiвнянь з частковими похiдними старших порядкiв у випадку загального вигляду N-функцiї Орлiча φ отримано в [9], де виведення грунтувалося на результатах роботи [12]. 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 results stated in [12]. We use another approach, which allows us, for the particular case φ(x).
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