Generalized solutions of a class of nuclear-space-valued stochastic evolution equations (Q751045)

A Langevin-type equation \(dY_ t=A^*dt+dZ_ t\) in \({\mathcal S}'(R^ d)\) is considered, where Z is an \({\mathcal S}'(R^ d)\)-valued semimartingale, A is the generator of a semigroup \((T_ t)\) on \(L^ 2(R^ d)\) such that \({\mathcal S}(R^ s)\subset Dom(A)\), but it is not assumed that either A or \(T_ i\) map \({\mathcal S}(R^ d)\) into itself. A typical example is \(A=-(- \Delta)^{\alpha /2}\), \(0<\alpha <2\). The authors give a rigorous meaning to this equation by introducing appropriate intermediate subspaces in the triple \({\mathcal S}\subset L^ 2\subset {\mathcal S}'\) and derive the corresponding variation of constant formula. An application is given to the investigation of the fluctuation limit of a system of branching particles, migrating according to a symmetric stable process.