Narrow Results

Colombeau generalized stochastic processes are introduced in order to solve nonlinear wave and Klein–Gordon equations with stochastic processes. Particular cases include one- and three-dimensional wave and Klein–Gordon equations with Lipschitz and cubic nonlinearities. In all cases, considered as Colombeau generalized stochastic processes, solutions are obtained and proved to be unique. The normalizations of the initial data and stochastic processes appear to be the crucial points for the existence of solutions to the above equations.

In this paper we study the solution of the stochastic heat equation where the potential V and the initial condition f are generalized stochastic processes. We construct explicitly the solution and we prove that it belongs to the generalized function space .

This work introduces a new notion of solution for the KPZ equation, in particular, our approach encompasses the Cole–Hopf solution. We set in the context of the distribution theory the proposed results by Bertini and Giacomin from the mid '90s.

This new approach provides a pathwise notion of solution as well as a structured approximation theory. The developments are based on regularization arguments from the theory of distributions.

Complex systems are usually under the influences of noises. Appropriately modeling these noises requires knowledge about generalized time derivatives and generalized stochastic processes. To this end, a brief introduction to generalized functions theory is provided. Then this theory is applied to fractional Brownian motion and its derivative, both regarded as generalized stochastic processes, and it is demonstrated that the “time derivative of fractional Brownian motion” is correlated and thus is a mathematical model for colored noise. In particular, the “time derivative of the usual Brownian motion” is uncorrelated and hence is an appropriate model for white noise.