Stochastic Bernoulli Equation on the Algebra of Generalized Functions

Abstract

Based on the topological dual space \({\mathcal{F}}_{\theta }^{*}\left({\mathcal{S}{\prime}}_{\mathbb{C}}\right)\) of the space of entire functions with θ-exponential growth of finite type, we introduce a generalized stochastic Bernoulli–Wick differential equation (or a stochastic Bernoulli equation on the algebra of generalized functions) by using the Wick product of elements in \({\mathcal{F}}_{\theta }^{*}\left({\mathcal{S}{\prime}}_{\mathbb{C}}\right)\). This equation is an infinite-dimensional analog of the classical Bernoulli differential equation for stochastic distributions. This stochastic differential equation is solved and exemplified by several examples.