The Initial-Final Value Problem for the Linear Stochastic Hoff Model

Abstract

The stochastic linear Hoff model of buckling of I-beam constructions amounts to a set of linear one-dimensional Hoff equations defined on the edges of a geometric graph with continuity and balance-of-flows conditions at its vertices. The deterministic model has been studied in various aspects by many mathematicians. We study the stochastic model for the first time. Our tool is the classical Ito - Stratonovich - Skorokhod approach extended to Hilbert spaces and Sobolev-type equations. The main result is an existence and uniqueness theorem for solutions to the initial-final value problem with additive white noise, understood as the generalized derivative of the $K$-Wiener process. The formulas expressing the solution are suitable for computer simulations.