Second Quantization and Evolution Operators in infinite dimension
Davide Addona, Paolo De Fazio
TL;DR
The work addresses the infinite-dimensional Ornstein-Uhlenbeck evolution in a separable Hilbert space by representing $P_{s,t}$ through a generalized second quantization operator acting on evolving Gaussian measures $gamma_t$. The authors develop a series and integral framework for the second quantization, extend hypercontractivity and compactness results to degenerate Gaussian settings, and derive asymptotic estimates in terms of the Cameron–Martin structure of the evolution. This yields precise criteria for hypercontractivity, compactness, and Hilbert–Schmidt properties of $P_{s,t}$ and connects the abstract theory to non-autonomous parabolic SPDEs and diagonal operator models. The results enhance understanding of smoothness, long-time behavior, and spectral properties of infinite-dimensional OU-type evolutions with time-dependent coefficients.
Abstract
In an infinite dimensional separable Hilbert space $X$, we study compactness properties and the hypercontractivity of the Ornstein-Uhlenbeck evolution operators $P_{s,t}$ in the spaces $L^p(X,γ_t)$, $\{γ_t\}_{t\in\R}$ being a suitable evolution system of measures for $P_{s,t}$. Moreover, we study the asymptotic behavior of $P_{s,t}$. Our results are produced thanks to a representation formula for $P_{s,t}$ through the second quantization operator. Among the examples, we consider the transition evolution operator associated to a non-autonomous stochastic parabolic PDE.