This volume of the Edizioni Della Normale is based on the authors’s thesis about the relationship of the solutions of a class stochastic partial differential equations (SPDE) with additive noise, the associated Kolmogorov operator \(K_0\) and the associated Markov generator \(K\) on spaces of continuous functions. The material is organized in seven chapters. After two chapters that set the abstract framework the remaining five chapters are applications of the results.
In Chapter 2, the author proves existence and uniqueness results under suitable conditions for the measure valued Kolmogorov equation of a general stochastically continuous Markov semigroup \(P_t\) with the an abstract generator \(K\). Chapter 3 is devoted to the linear case where \(P_t\) is the Ornstein-Uhlenbeck semigroup, which is extended to bounded and Lipschitz continuous perturbations \(F\). In these cases \(K\) it can be identified as the closure of \(K_0\) which is given explicitely by the coefficients of the SPDE in the space \(C_b(H;\mathbb{R})\) of uniformly continuous and bounded functions on a separable Hilbert space \(H\). In order to compensate the lack of regularity of the semigroups, he extends the so-called \(\pi\)-semigroup approach introduced by E. Priola [Stud. Math. 136, No. , 271–295 (1999; Zbl 0955.47024)]. Chapter 5 treats the case of general Lipschitz nonlinearities, here \(K\) is the closure of \(K_0\) in \(C_{1,b}(H;\mathbb{R})\) the space of bounded, continuously Frechet differentiable functions over \(H\). This improves results by G. Da Prato and L. Tubaro [J. Differ. Equations 173, No. 1, 17–39 (2001; Zbl 1003.60070)]. Moreover he solves the associated Kolmogorov equation for suitable initial conditions. In Chapter 6 the author discusses the concrete case of reaction diffusion equations over a compact domain with additive noise, where he can identify \(K\) as the closure of \(K_0\) in some weighted \(L^p\) space and solve the respective Kolmogorov equations. In the last chapter, Manca applies the results to the stochastic Burger’s equation with space-time white noise. He show that \(K\) is \(K_0\) in an appropriate nontrivial \(L^p\) space and proves existence an uniqueness for the associated Kolmogorov equations.
This monography gives a nice introduction to the field of measure valued Kolmogorov equations for a quite general class SPDE. The results link the abstract knowledge about the Markov behaviour of the solutions SPDE to the concrete shape of \(K_0\) on a natural space of test functions proving the strength of Priola’s approach.