Quantization and process

Abstract

This article is concerned with generalizations of pseudo-differential operators in $L^2(\mathbb{R}^n)$, $n\geq 1$. The definition of the new calculi depends only on bounded measures on the phase space $\mathbb{R}^{2n}$ and each measure gives rise to a specific calculus. Quantizations of anti-Wick, Weyl, classical and Born-Jordan are particular cases of the general calculi. Classes of symbols in this framework are then studied. The Gevrey class of parameter 1/2 is a class of symbol that is common to all the general calculi, that is, a class of symbols independent on the bounded measures parametrizing the quantizations. Precise additional hypotheses on the measures are necessary in the aim to consider the larger class of symbols $L^{\infty}(\mathbb{R}^{2n})$. This result can be applied for anti-Wick but not for Weyl quantization. Concerning Weyl pseudo-differential calculus, we recover the standard class of Sj\"ostrand and Gr\"ochening. Then, we prove that probability measures of L\'evy processes on the phase space $\mathbb{R}^{2n}$ with diffusion larger than 1/4 are natural examples of measures satisfying the latter additional hypotheses in order to consider $L^{\infty}(\mathbb{R}^{2n})$ symbols. This relation is derived using the L\'evy-Khintchine formula. Composition laws in that general context are next investigated. In that purpose, we give a formula for the composition of two symbols in some precise class of symbols valid for all general quantizations. General calculi are relying on Wick quantization which is therefore primarily examined for some precise classes of symbols. Additional results in that context are provided, such as Mizrahi series expansions and Banach algebra isomorphisms between operators and symbol classes.