Path-dependent PDEs (PPDEs) are natural objects to study when one deals with non Markovian models. Recently, after the introduction of the so-called pathwise (or functional or Dupire) calculus (see [15]), in the case of finite-dimensional underlying space various papers have been devoted to studying the well-posedness of such kind of equations, both from the point of view of regular solutions (see e.g. [15,9]) and viscosity solutions (see e.g. [16]). In this paper, motivated by the study of models driven by path-dependent stochastic PDEs, we give a first well-posedness result for viscosity solutions of PPDEs when the underlying space is a separable Hilbert space. We also observe that, in contrast with the finite-dimensional case, our well-posedness result, even in the Markovian case, applies to equations which cannot be treated, up to now, with the known theory of viscosity solutions.

We study the optimal control of path-dependent McKean-Vlasov equations valued in Hilbert spaces motivated by non Markovian mean-field models driven by stochastic PDEs. We first establish the well-posedness of the state equation, and then we prove the dynamic programming principle (DPP) in such a general framework. The crucial law invariance property of the value function V is rigorously obtained, which means that V can be viewed as a function on the Wasserstein space of probability measures on the set of continuous functions valued in Hilbert space. We then define a notion of pathwise measure derivative, which extends the Wasserstein derivative due to Lions [41], and prove a related functional Itô formula in the spirit of Dupire [24] and Wu and Zhang [51]. The Master Bellman equation is derived from the DPP by means of a suitable notion of viscosity solution. We provide different formulations and simplifications of such a Bellman equation notably in the special case when there is no dependence on the law of the control.

We present and apply a theory of one parameter C 0 -semigroups of linear operators in locally convex spaces. Replacing the notion of equicontinuity considered by the literature with the weaker notion of sequential equicontinuity, we prove the basic results of the classical theory of C 0 -equicontinuous semigroups: we show that the semigroup is uniquely identified by its generator and we provide a generation theorem in the spirit of the celebrated Hille-Yosida theorem. Then, we particularize the theory in some functional spaces and identify two locally convex topologies that allow to gather under a unified framework various notions C 0semigroup introduced by some authors to deal with Markov transition semigroups. Finally, we apply the results to transition semigroups associated to stochastic differential equations.

We consider an optimal stochastic impulse control problem over an infinite time horizon motivated by a model of irreversible investment choices with fixed adjustment costs. By employing techniques of viscosity solutions and relying on semiconvexity arguments, we prove that the value function is a classical solution to the associated quasi-variational inequality. This enables us to characterize the structure of the continuation and action regions and construct an optimal control. Finally, we focus on the linear case, discussing, by a numerical analysis, the sensitivity of the solution with respect to the relevant parameters of the problem.A.M.S. Subject Classification: 93E20 (Optimal stochastic control); 35Q93 (PDEs in connecton woth control and optimization); 35D40 (Viscosity solution); 35B65 (Smoothness and regularity of solutions).Related literature. First of all, it is worth noticing that the stochastic impulse control setting has been widely employed in several applied fields: e.g., exchange and interest rates [21,51,56], portfolio optimization with transaction costs [34,49,57], inventory and cash management [12,20,27,30,31,44,45,58,62,67,68,71], real options [47,54], reliability theory [7]. More recently, games of stochastic impulse control have been investigated with application to pollution [39].From a modeling point of view, the closest works to ours can be considered [3,6,26,35,49]. On the theoretical side, starting from the classical book [17], several works investigated QVIs associated to stochastic impulse optimal control in R n . Among them, we mention the recent [43] in a diffusion setting and [14,29] in a jump-diffusion setting. In particular, [17, Ch. 4] deals with Sobolev type solutions, whereas [43] deals with viscosity solutions. These two works prove a W 2,pregularity, with p < ∞, for the solution of QVI, which, by classical Sobolev embeddings, yields a C 1 -regularity. However, it is typically not easy to obtain by such regularity information on the structure of the so called continuation and action regions, hence on the candidate optimal control. If this structure is established, then one can try to prove a verificiation theorem to prove that the candidate optimal control is actually optimal. In a stylized one dimensional example, [43, Sec. 5] successfully employs this method by exploiting the regularity result proved in [43, Sec. 4] to depict the structure of the continuation and action region for the problem at hand. Concerning verification, we need to mention the recent paper [15], which provides a non-smooth verification theorem in a quite general setting based on the stochastic Perron method to construct a viscosity solution to QVI; also this paper, in the last section, provides and application of the results to a one dimensional problem with an implementable solution. In dimension one other approaches, based on excessive mappings and iterated optimal stopping schemes, have been successfully employed in the context of stochastic impulse control (see [3,6,35,46]). More recently, these ...