A differential equation in a Banach space containing the variational derivative
The article deals with a Cauchy problem of the type \[ {\partial y \over \partial t} = Ay + a {\delta \over \delta v(t)} y + b(v,t), \quad y(v,t_ 0) = y_ 0(v) \tag{1} \] with respect to the unknown \(y(v,t)\): \(L([0,T]) \times [0,T] \to Y\) in a complex Banach space \(Y\) with linear closed operator \(A\) and variational derivative \(\delta/ \delta v(t)\) defined by \[ dy(v,h) = \int_ Ty(v,t) h(t)dt. \] The main result is a formula for the solution to this problem in terms of the semigroup \(U(t)\) generated by \(A\). As an application the Cauchy problem \[ {dx \over dt} = Ax + \varepsilon (t)x + f(t), \quad x(0) = x_ 0 \] with the stochastic process \(\varepsilon (t)\) is solved. It turns out that the moment function of this equation satisfies equations of type (1); the corresponding equations are presented and investigated and explicit formulas for moment functions of first and second order are obtained.
scientific article; zbMATH DE number 130775