Lévy Laplacians in Hida Calculus and Malliavin Calculus

Abstract

Some connections between different definitions of Lévy Laplacians in the stochastic analysis are considered. Two approaches are used to define these operators. The standard one is based on the application of the theory of Sobolev-Schwartz distributions over the Wiener measure (the Hida calculus). One can consider the chain of Lévy Laplacians parametrized by a real parameter with the help of this approach. One of the elements of this chain is the classical Lévy Laplacian. Another approach to define the Lévy Laplacian is based on the application of the theory of Sobolev spaces over the Wiener measure (the Malliavin calculus). It is proved that the Lévy Laplacian defined with the help of the second approach coincides with one of the elements of the chain of Lévy Laplacians, which is not the classical Lévy Laplacian, under the imbedding of the Sobolev space over the Wiener measure into the space of generalized functionals over this measure. It is shown which Lévy Laplacian in the stochastic analysis is connected to the gauge fields.

Lévy differential operators and Gauge invariant equations for Dirac and Higgs fields

“…[19] and in Ref. [31]. In the next paper we develop some results of the current paper for the case of the Riemannian and the pseudo-Riemannian manifold.…”

“…The non-classical Lévy Laplacian ∆ [6]. As we show below, such operators can be useful in the study of gauge fields (see also [31]). Moreover, the exotic Lévy Laplacians (see Refs.…”

“…In Refs. [30,31] the Levy Laplacian defined as the Chesaro mean of the directional derivatives in the stochastic case was studied. 2 It was shown that, unlike the deterministic case, the equivalence of the Yang-Mills equations and the Levy-Laplace equation is not valid for such Laplacian.…”

Levy Laplacian on manifold and Yang-Mills heat flow

“…The relationship of this Levy Laplacian and the Yang-Mills equations was studied in [40]. The relationship between the Yang-Mills equations and different Levy Laplacians was also studied in [41,42,43,44,45].…”

Lévy Laplacians, holonomy group and instantons on 4-manifolds

“…Different approaches to the Yang-Mills fields based on the parallel transport but not based on the Lévy Laplacian were used in [24,25,26,27,28,29]. For a recent development in the study of the Lévy Laplacian in the white noise theory, see [30,31].…”