Solutions of affine stochastic functional differential equations in the state space

Abstract.

We consider solutions of affine stochastic functional differential equations on ${\mathbb{R}}^d$. The drift of these equations is specified by a functional defined on a general function space ${\mathcal{B}}$ which is only described axiomatically. The solutions are reformulated as stochastic processes in the space ${\mathcal{B}}$. By representing such a process in the bidual space of ${\mathcal{B}}$ we establish that the transition functions of this process form a generalized Gaussian Mehler semigroup on ${\mathcal{B}}$. This way the process is characterized completely on ${\mathcal{B}}$ since it is Markovian.

Moreover we derive a sufficient and necessary condition on the underlying space ${\mathcal{B}}$ such that the transition functions are even an Ornstein-Uhlenbeck semigroup. We exploit this result to associate a Cauchy problem in the function space ${\mathcal{B}}$ to the stochastic functional differential equation.