Estimates for distribution of suprema of solutions to higher-order partial differential equations with random initial conditions

Abstract

In the paper we consider higher-order partial differential equations from the class of linear dispersive equations. We investigate solutions to these equations subject to random initial conditions given by harmonizable ϕ-sub-Gaussian processes. The main results are the bounds for the distributions of the suprema for solutions. We present the examples of processes for which the assumptions of the general result are verified and bounds are written in the explicit form. The main result is also specified for the case of Gaussian initial condition.Keywords Higher-order dispersive equations, random initial conditions, harmonizable processes, sub-Gaussian processes, distribution of sumpremum of solution, entropy methods 2010 MSC 35G10, 35R60, 60G20, 60G60

Investigation of sample paths properties for some classes of φ-sub-Gaussian stochastic processes

“…The results obtained in Section 4 concerning the distribution of supremum for the processes related to the heat equations with ϕ-sub-Gaussian initial conditions provide the generalization and extension of results from papers [13,14] where the cases of Gaussian and sub-Gaussian initial conditions were considered. In [16,17] similar results were obtained for the case of higher order heattype equations, but under the conditions stated in terms of a different entropy integral (see also [16] for more references on the theory of ϕ-sub-Gaussian processes and additional references on partial differential equations with random initial conditions).…”

“…Denote γ 0 = σ (sup t,s∈T ρ(t, s)). Theorem 1 below is a variant of the result stated in [4, Theorem 4.4, p. 107] (see also [17,Theorem 2.3]). The analogous result for a Gaussian process is presented in [6].…”

“…The process (17) can be interpreted as the mean-square or L 2 ( ) solution to the Cauchy problem ( 11)-( 12), as justified in [14]. Note that the random solution in the form (17) to the heat equation with stationary initial condition appears already in the paper by Rosenblatt [26] (see Formula (1.11) therein).…”

“…Theorem 5. Let u(t, x), t > 0, x ∈ R, be the stochastic process given by (17) and Assumption H.1 hold. Then the following statements hold:…”

“…In [13,14] the heat equations with sub-Gaussian stationary initial conditions were studied, exponential bounds for the distribution of supremum of the solution were presented. Higherorder heat-type equations with ϕ-sub-Gaussian harmonizable initial conditins were investigated in [2,16,17].…”