Hierarchy of Lévy–laplacians and Quantum Stochastic Processes

Abstract

Communicated by O. SmolyanovWe consider a family of infinite dimensional Laplace operators which contains the classical Lévy-Laplacian. We prove a representation of these operators as a quadratic functions of quantum stochastic processes. Particularly, for the classical Lévy-Laplacian, the following formula is proved: ∆ L = lim ε→0 R s−t <ε bsbtdsdt, where bt is the annihilation process.

Lévy Laplacians in Hida Calculus and Malliavin Calculus

“…For the first time formula (4) was used for study of the exotic Lévy Laplacians in [9]. The Lévy Laplacians ∆ {en},s L are the particular case of nonclassical Lévy Laplacians (see [8,30]).…”

“…is the Schwartz space of rapidly decreasing functions and E * C = S * (R, C d ) is the space of generalized functions of slow growth. The case, where T = [0, 1] and {e n } is a basis consisting of trigonometric functions, is also often considered (see [3,4,5,30] where ξ ∈ E C , is called a coherent state. The unitary Wiener-Ito-Segal isomorphism j 2 between Γ(H C ) and L 2 (E * R , µ I ; C) is uniquely determined by the values on coherent states…”

“…The chain of the exotic Lévy Laplacians generated by a basis of trigonometric functions in the Hida calculus has been considered in papers [3,4,5]. Its extension for negative orders has been considered in the paper [30]. In fact, the Lévy Laplacian of order (−1) was first considered in [8].…”

“…This operator will be called the classical Lévy Laplacian. An extensive literature is devoted to the study of this operator (see the review [21], and also [30,5] and the papers cited there). The second Laplacian is defined using the Malliavin calculus.…”

Stochastic Lévy differential operators and Yang–Mills equations

“…[16,18,17]). This operator is equal to zero on Γ(L 2 ([0, 1], R d )) (see [18] [8,26] 4 Stochastic Lévy Divergence…”