Infinite Dimensional Calculus Research Articles

Stochastic systems with memory and jumps

Stochastic systems with memory naturally appear in life science, economy, and finance. We take the modelling point of view of stochastic functional delay equations and we study these structures when the driving noises admit jumps. Our results concern existence and uniqueness of strong solutions, estimates for the moments and the fundamental tools of calculus, such as the Itô formula. We study the robustness of the solution to the change of noises. Specifically, we consider the noises with infinite activity jumps versus an adequately corrected Gaussian noise. The study is presented in two different frameworks: we work with random variables in infinite dimensions, where the values are considered either in an appropriate Lp-type space or in the space of càdlàg paths. The choice of the value space is crucial from the modelling point of view, as the different settings allow for the treatment of different models of memory or delay. Our techniques involve tools of infinite dimensional calculus and the stochastic calculus via regularisation.

A consistent measure for lattice Yang–Mills

The construction of a consistent measure for Yang–Mills is a precondition for an accurate formulation of nonperturbative approaches to QCD, both analytical and numerical. Using projective limits as subsets of Cartesian products of homomorphisms from a lattice to the structure group, a consistent interaction measure and an infinite-dimensional calculus have been constructed for a theory of non-Abelian generalized connections on a hypercubic lattice. Here, after reviewing and clarifying past work, new results are obtained for the mass gap when the structure group is compact.

Infinite-Dimensional Calculus Under Weak Spatial Regularity of the Processes

Two generalizations of Itô formula to infinite-dimensional spaces are given. The first one, in Hilbert spaces, extends the classical one by taking advantage of cancellations when they occur in examples and it is applied to the case of a group generator. The second one, based on the previous one and a limit procedure, is an Itô formula in a special class of Banach spaces having a product structure with the noise in a Hilbert component; again the key point is the extension due to a cancellation. This extension to Banach spaces and in particular the specific cancellation are motivated by path-dependent Itô calculus.

Infinite dimensional weak Dirichlet processes and convolution type processes

The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of weak Dirichlet process in this context. Such a process X, taking values in a Banach space H, is the sum of a local martingale and a suitable orthogonal process.The concept of weak Dirichlet process fits the notion of convolution type processes, a class including mild solutions for stochastic evolution equations on infinite dimensional Hilbert spaces and in particular of several classes of stochastic partial differential equations (SPDEs).In particular the mentioned decomposition appears to be a substitute of an Itô’s type formula applied to f(t,X(t)) where f:[0,T]×H→R is a C0,1 function and X a convolution type process.

An infinite-dimensional calculus for generalized connections on hypercubic lattices

A space for gauge theories is defined, using projective limits as subsets of Cartesian products of homomorphisms from a lattice on the structure group. In this space, non-interacting and interacting measures are defined as well as functions and operators. From projective limits of test functions and distributions on products of compact groups, a projective gauge triplet is obtained, which provides a framework for the infinite-dimensional calculus in gauge theories. The gauge measure behavior on non-generic strata is also obtained.

REGULARIZED CALCULUS: AN APPLICATION OF ZETA REGULARIZATION TO INFINITE DIMENSIONAL GEOMETRY AND ANALYSIS

A method of regularization in infinite dimensional calculus, based on spectral zeta function and zeta regularization is proposed. As applications, a mathematical justification of appearance of Ray–Singer determinant in Gaussian Path integral, regularized volume form of the sphere of a Hilbert space with the determinant bundle, eigenvalue problems of regularized Laplacian, are investigated. Geometric counterparts of regularization procedure are also discussed applying arguments from noncommutative geometry.

A Class of generalized Lévy Laplacians in infinite dimensional calculus

A class of generalized Levy Laplacians which contain as a special case the ordinary Levy Laplacian are considered. Topics such as limit average of the second order functional derivative with respect to a certain equally dense (uniformly bounded) orthonormal base, the relations with Kuo’s Fourier transform and other infinite dimensional Laplacians are studied.

On the Kakutani-Itô-Segal-Gross and Segal-Bargmann-Hall Isomorphisms

Recently, Gross has shown that the Kakutani-Itô-Segal isomorphism theorem has an extension from the setting of Gaussian measure on a vector space to "heat kernel" measure (pt) on a simply connected Lie group (G) of compact type. The isomorphism relates L2(pt) to a certain completion of the universal enveloping algebra of g=̇ Lie(G). Gross proves this result using the Kakutani-Itô-Segal theorem and an infinite dimensional calculus associated to G-valued Brownian motion. Hijab has greatly simplified and clarified Gross′ proof. Hiiab′s proof avoids most, but not all, of the "infinite dimensional" analysis in the original proof. In this paper, we will build on Hijab′s proof to give a completely "finite dimensional" non-probabilistic proof of Gross′ isomorphism theorem. The proof given here relies heavily on Hall′s beautiful "extension" of the Segal-Bargmann transformation to the setting of compact Lie groups. This theorem relating L2(pt) to a certain L2-space of holomorphic functions (L2(GC) ∩ H) on the complexified Lie group GC will also be generalized to Lie groups of compact type. In the process, it is shown how to characterize, in terms of summability conditions on all the derivatives at the identity in GC, those holomorphic functions which are in L2(GC).

Sobolev derivatives and It� decomposition formula for Gaussian processes using properties of their RKHSs

Malliavin calculus has been extensively developed for abstract Wiener spaces. It is of interest to develop the basic concepts of an infinite-dimensional calculus for an arbitrary Gaussian processX=(Xt), wheret\( \in\)T (T being a multiparameter set or, more generally, a complete separable metric space), bringing into evidence the properties of the covariance kernel (or, equivalently, the reproducing kernel Hilbert space) ofX. In this paper a definition of thekth Sobolev derivative is given and thekth chaos expansion of a functional is shown to be thekth-order divergence operator. An extension of Ito's decomposition formula is derived.

Some Remarks on Hu and Meyer’s Paper and Infinite-Dimensional Calculus on Finitely Additive Canonical Hilbert Space

The theory of accessible random variables developed in [4] is used to derive a formula suggested in [2]. A beginning of a calculus on a finitely additive canonical Hilbert space is outlined, including a definition of the Skorokhod integral in the finitely additive setup.

Integration by Parts, Homogeneous Chaos Expansions and Smooth Densities

By iterating a martingale representation result a homogeneous chaos expansion is obtained. Using the martingale representation, the integration-by-parts formula of the Malliavin calculus is derived using properties of stochastic flows. The infinite-dimensional calculus of variations is not required.