On a Measurable Solution of a Class of higher-order Stochastic Heat-type Equation

Abstract

We give a generalized measurable–predictable solution to a higher–order stochastic parabolic initial–value problem in terms of the further generalized Hermite polynomials. Condition and estimates on the existence and uniqueness of the solution are given. We prove the upper second moment growth bound estimate for the solution and consequently show that the second moment of the solution grows exponentially in time with respect to the parameter λ at the precise rate of 2 + c₃λ²Lip²_σ, c₃ > 0 and ; c₃Lip²_σ as the noise level increases.

Key words and phrases. Generalized Hermite polynomials, growth moment, generalized solution, measurable solution, mild solution, higher-order initial-value problems.

2010 Mathematics Subject Classification. 35R60, 60H15, 82B44