Infinite Dimensional Heat Kernel Measure

• Infinite Dimensional Heat Kernel Measure is a mathematical construct that extends heat kernel techniques to infinite-dimensional Lie groups and path spaces.
• It is constructed via stochastic processes like Brownian motion on Banach spaces, yielding smooth, quasi-invariant densities through finite-dimensional projections.
• This measure underpins the analysis of holomorphic function spaces and Malliavin calculus, linking infinite-dimensional analysis with quantum field theory and free probability.

Infinite dimensional heat kernel measure is a mathematical construct central to analysis on infinite-dimensional spaces, particularly infinite-dimensional Lie groups and path spaces where no Lebesgue measure exists. These measures arise as the endpoint laws of stochastic processes (such as Brownian motion) on infinite-dimensional manifolds or groups, exhibiting subelliptic or hypoelliptic regularity and reflecting a rich interplay between probability theory, functional analysis, differential geometry, and representation theory.

1. Structural Foundations: Infinite-Dimensional Heisenberg Groups and Abstract Wiener Space

The prototypical setting for the infinite dimensional heat kernel measure is the infinite-dimensional Heisenberg-like group $G$, constructed as a central extension $G = W \times C$ where:

• $W$ is a separable Banach space with a Gaussian (Wiener) measure, typically realized as the state space for abstract Wiener processes.
• $C$ is a finite-dimensional complex Hilbert space, forming the center of the group.

The group law for $G$ is defined via a continuous, skew-symmetric bilinear form $w: W \times W \to C$,

$$(w_1, c_1) \cdot (w_2, c_2) = (w_1 + w_2, c_1 + c_2 + \tfrac{1}{2} w(w_1, w_2)),$$

retaining the step-2 nilpotent structure of the classical Heisenberg group but allowing for infinite-dimensionality in $W$ (Gordina et al., 2011).

2. Construction and Properties of the Heat Kernel Measure

The heat kernel measure $v_t$ is induced by subelliptic or hypoelliptic Brownian motion on $G$:

• The process $(B_t, M_t)$ is defined where $B_t$ is Brownian motion on $W$ and $M_t = \int_0^t w(B_s, dB_s)$ is an $L^2$-martingale capturing the noncommutative extension.
• The endpoint law,

$$v_t = \text{Law}(g_t) = \text{Law}\left(B_t, \tfrac{1}{2} M_t\right),$$

serves as the subelliptic heat kernel measure on $G$.

This measure is "formally subelliptic": for every finite-dimensional projection, the pushforward of $v_t$ yields a smooth (hypoelliptic) density, displaying the familiar smoothing properties of classical heat kernels despite the infinite-dimensional setting (Gordina et al., 2011).

A version of the Fernique theorem ensures exponential integrability, guaranteeing that the associated $L^2$ space of (holomorphic) functions is nontrivial.

3. Analytic Framework: Holomorphic Functions and Square Integrability

Analysis is performed in the Hilbert space $H^2_t(G)$ consisting of holomorphic functions $f$ on $G$ that are square-integrable with respect to $v_t$: $$\|f\|_{L^2(v_t)} = \left( \int_G |f(g)|^2 v_t(dg) \right)^{1/2}.$$ Functions are constructed as limits (in $L^2(v_t)$) of holomorphic cylinder polynomials—functions depending on finite-dimensional projections of $G$—which allow for approximation and renormalization within an infinite-dimensional context. The Cameron-Martin subgroup $G_{CM} \subset G$, corresponding to the directions along which measure changes are meaningful, plays a central role and has zero measure with respect to $v_t$ (Gordina et al., 2011).

4. Algebraic Structure: Unitary Isomorphism via Taylor Expansion

A fundamental result is the establishment of a unitary isomorphism between $H^2_t(G_{CM})$ and a completed noncommutative Fock space $J_t$:

• The Taylor map

$T_t : H^2_t(G_{CM}) \to J_t$

associates to each $f$ the sequence of derivatives at the group identity, regarded as elements in the universal enveloping algebra of the Cameron-Martin Lie algebra.

• This map is isometric, $\|f\|_{H^2_t(G_{CM})} = \|T_t f\|_{J_t}$, and invertible, showing that the analytic data of a function is entirely captured by its Taylor coefficients (the group-differential data at the identity).
• A restriction map $R: H^2_t(G) \to H^2_t(G_{CM})$ further connects functions on the full group to the subgroup, establishing a unitary equivalence between $H^2_t(G)$ and $J_t$ (Gordina et al., 2011).

These constructions generalize the Segal-Bargmann and classical Taylor isomorphism to the infinite-dimensional, subelliptic setting.

5. Regularity, Quasi-Invariance, and Malliavin Calculus

The regularity properties of the infinite-dimensional heat kernel measure are deeply related to quasi-invariance and smoothness:

• Quasi-invariance results show that $v_t$ is absolutely continuous under translation by elements of $G_{CM}$ (Cameron-Martin type theorem), with explicit $L^p$ bounds on Radon-Nikodym derivatives,

$$\left\| \frac{d(v_t \circ R_y^{-1})}{d v_t} \right\|_{L^q(v_t)} \le \exp\left( \frac{(1 + 2 \|w\|^2) q d^2(e, y)}{4t} \right),$$

where $d(e, y)$ is the sub-Riemannian distance, and $\|w\|$ is the Hilbert-Schmidt norm of $w$ (Baudoin et al., 2011, Driver et al., 2013).

• The density and the Radon-Nikodym derivative are Malliavin smooth, i.e., derivatives along Cameron-Martin directions exist in all $L^p$ spaces, enabling robust versions of integration by parts and construction of Sobolev-type spaces in the absence of Lebesgue measure (Driver et al., 2013, Dobbs et al., 2012).

6. Implications: Infinite-Dimensional Analysis, Stochastic Processes, Quantum Field Theory

The infinite dimensional heat kernel measure furnishes a rigorous analytic and probabilistic foundation for:

• The paper of square-integrable holomorphic and smooth functions on infinite-dimensional Lie groups, facilitating harmonic analysis in absence of Haar measure.
• Applications in stochastic analysis (including Malliavin calculus), path integrals, subelliptic SPDEs, and quantization schemes (Segal-Bargmann transforms).
• Construction and analysis of infinite-dimensional Sobolev spaces, regularity properties of solutions to the heat equation, and the paper of quasi-invariance and integration by parts formulae for measures on path spaces and configuration spaces.
• Extension of these methods to infinite graphs and discretizations, where parametrix constructions provide explicit Taylor series expansions of the heat kernel and facilitate comparison between discrete and continuous models (Jorgenson et al., 17 Apr 2024).

7. Connections to Broader Geometric and Probabilistic Frameworks

The approach is closely related to:

• Curvature-dimension inequalities in metric measure spaces (Bakry-Émery Ricci curvature), ensuring Gaussian upper bounds, stability properties, and functional inequalities such as dimension-free Harnack inequalities in both finite and infinite dimensions (Wu et al., 2014, Jiang et al., 2014).
• Noncommutative distributional limits in random matrix theory, where empirical laws of eigenvalues converge to flows on infinite-dimensional polynomial algebras, elucidating the free probabilistic structure of infinite-dimensional heat kernel measures (Kemp, 2013, Klevtsov et al., 2015).
• Infinite-dimensional determinants, Fredholm and zeta regularization, appearing in the asymptotics of heat kernels and path integrals, with direct links to quantum field theoretic computations and higher-order operator expansions (Ludewig, 2016, Wachowski et al., 2018, Barvinsky et al., 2021).

8. Summary Table: Key Mathematical Features

Property	Mathematical Context	Reference
Group Structure	$G = W \times C$, step-2 nilpotent	(Gordina et al., 2011, Driver et al., 2013)
Heat Kernel Measure	Law of $(B_t, \int_0^t w(B_s, dB_s))$	(Gordina et al., 2011, Driver et al., 2013)
Formal Subellipticity	Finite-dimensional projections are smooth	(Gordina et al., 2011)
Quasi-Invariance under $G_{CM}$	$L^p$ bounds on Radon-Nikodym derivatives	(Baudoin et al., 2011, Driver et al., 2013)
Square-Integrable Holomorphic Functions	$L^2(v_t)$ closure, Taylor isomorphism	(Gordina et al., 2011)
Noncommutative Fock Space	Universal enveloping algebra completion	(Gordina et al., 2011)
Malliavin Smoothness	Density and derivatives in all $L^p$	(Dobbs et al., 2012, Driver et al., 2013)
Gaussian Heat Kernel Bounds	Curvature-dimension condition, Harnack inequalities	(Wu et al., 2014, Jiang et al., 2014)
Infinite-Dimensional Path Integral	Fredholm/zeta determinants in asymptotics	(Ludewig, 2016)

9. Conclusion

The infinite dimensional heat kernel measure encapsulates both analytic and probabilistic structure on infinite-dimensional spaces where traditional measure theoretic concepts (Lebesgue/Haar measure) fail. Through subelliptic and hypoelliptic constructions, quasi-invariance, integration by parts, Malliavin calculus, and connections to infinite-dimensional representation theory and free probability, these measures serve as essential tools for modern analysis, geometry, and mathematical physics in contexts where infinite-dimensionality and noncommutativity are intrinsic to the structure.