Second Quantization and Evolution Operators in infinite dimension

TL;DR

This paper investigates the properties and asymptotic behavior of evolution operators in infinite-dimensional spaces, using second quantization, with applications to stochastic PDEs.

Contribution

It introduces a representation formula for evolution operators via second quantization and analyzes their compactness, hypercontractivity, and asymptotics in infinite dimensions.

Findings

Established hypercontractivity of $P_{s,t}$ in $L^p$ spaces.

Derived a representation formula for $P_{s,t}$ using second quantization.

Analyzed the asymptotic behavior of the evolution operators.

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 $L^p(X,\gamma_t)$, $\{\gamma_t\}_{t\in\R}$ 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.