[Paper Review] Equivalence of the Brownian and energy representations

Sergio Albeverio, Bruce K. Driver+2 more|arXiv (Cornell University)|Nov 23, 2015

advanced mathematical theories3 citations

TL;DR

This paper establishes the unitary equivalence between the Brownian motion representation and the energy representation of the group of smooth paths in a compact Lie group. Using stochastic analysis and Fourier-Wiener transforms, it proves that both representations—defined on Wiener space with Wiener measure and on path space with Gaussian measure, respectively—are unitarily equivalent via an intertwining operator, resolving a long-standing question about cyclicity of the constant function in these representations.

ABSTRACT

We consider two unitary representations of the infinite-dimensional groups of smooth paths with values in a compact Lie group. The first representation is induced by quasi-invariance of the Wiener measure, and the second representation is the energy representation. We define these representations and their basic properties, and then we prove that these representations are unitarily equivalent.

Motivation & Objective

• To establish the unitary equivalence between the Brownian representation induced by quasi-invariance of Wiener measure and the energy representation on path space.
• To resolve the question of whether the constant function 1 is a cyclic vector for the Brownian and energy representations.
• To provide a rigorous construction of the energy representation using Stratonovich and Itô integrals on Lie group-valued paths.
• To demonstrate that the energy representation is unitarily equivalent to both left and right Brownian representations via the inverse Itô map and Fourier-Wiener transform.
• To unify two distinct approaches to unitary representations of infinite-dimensional path groups: stochastic processes and energy-based constructions.

Proposed method

• The Brownian representation is constructed on $ L^2(W(G), \mu) $, where $ \mu $ is the Wiener measure on continuous paths in a compact Lie group $ G $, using quasi-invariance under left and right multiplication by elements of $ H(G) $, the group of smooth paths in $ G $.
• The energy representation is defined on $ L^2(W(\mathfrak{g}), \nu) $, where $ \mathfrak{g} $ is the Lie algebra of $ G $, and $ \nu $ is the standard Gaussian measure on $ W(\mathfrak{g}) $, using the adjoint action and stochastic integrals.
• The key technical tool is the inverse Itô map $ B^L $, which transforms the Brownian representation into a form on $ W(\mathfrak{g}) $, allowing comparison with the energy representation.
• The Fourier-Wiener transform $ \mathcal{F} $ is used as an intertwining operator to relate the transformed Brownian representation to the energy representation.
• The proof relies on the equality of Itô and Stratonovich integrals for deterministic integrands, ensuring consistency in the stochastic calculus framework.
• The unitary equivalence is established by showing that the representation $ u^{R}_{\varphi} $, induced by the right multiplication on $ W(G) $, is unitarily equivalent to the energy representation $ E_{\varphi} $ via the composition $ \mathcal{F} \circ (B^L)^* $.

Experimental results

Research questions

• RQ1 Is the constant function 1 cyclic for the Brownian representation on $ L^2(W(G), \mu) $?
• RQ2 Are the Brownian representation (induced by quasi-invariance of Wiener measure) and the energy representation (on $ L^2(W(\mathfrak{g}), \nu) $) unitarily equivalent?
• RQ3 Can the energy representation be realized as a unitary transformation of the Brownian representation via stochastic calculus tools?
• RQ4 What is the precise intertwining operator between the Brownian and energy representations?
• RQ5 Does the inverse Itô map $ B^L $ provide a canonical isomorphism between the Brownian and energy representation spaces?

Key findings

• The constant function 1 is a cyclic vector for both the left and right Brownian representations on $ L^2(W(G), \mu) $, as established in Section 3.
• The Brownian representation $ U^R $ and the energy representation $ E $ are unitarily equivalent via the composition $ \mathcal{F} \circ (B^L)^* $, as proven in Theorem 5.8.
• The energy representation $ E_{\varphi} $ is given by $ (E_{\varphi}f)(w) = e^{i\int_0^T \langle \varphi^{-1}\varphi'(s), dw_s \rangle} f(O_{\varphi^{-1}}w) $, where $ O_{\varphi^{-1}}w = \int_0^\cdot \operatorname{Ad}_{\varphi} dw_s $, and this defines a unitary representation on $ L^2(W(\mathfrak{g}), \nu) $.
• The representation $ u^{R}_{\varphi} $ on $ L^2(W(\mathfrak{g}), \nu) $, obtained via the inverse Itô map, is unitarily equivalent to the energy representation $ E_{\varphi} $, with the intertwining operator being the Fourier-Wiener transform $ \mathcal{F} $.
• The proof relies on the fact that for deterministic integrands, the Itô and Stratonovich integrals coincide, ensuring consistency in the stochastic calculus framework used.
• Corollary 5.9 confirms that the constant function 1 is a cyclic vector for the energy representation, following from the cyclicity in the Brownian representation and the unitary equivalence established.