Free Multiplicative Brownian Motion

• Free multiplicative Brownian motion is a noncommutative analogue of classical Brownian motion defined via free stochastic differential equations.
• It employs advanced transform techniques, such as the η-transform and Σ-transform, to analyze spectral laws and convolution semigroups in free probability.
• Its study reveals novel spectral phenomena and universality properties with significant applications in random matrix theory and operator algebras.

Free multiplicative Brownian motion is a fundamental concept in free probability theory, serving as the multiplicative analogue of classical Brownian motion for noncommutative random variables. It arises naturally as the large-dimensional limit of Brownian motion on matrix groups, and its paper interconnects advanced techniques from operator algebras, random matrix theory, stochastic analysis, and harmonic analysis on Lie groups. This process provides a rich source of new phenomena in noncommutative probability, including explicit connections to free convolution semigroups, spectral theory for non-normal operators, and universality in random matrix ensembles.

1. Definition and Stochastic Differential Equation

The free multiplicative Brownian motion (FMBM) is most succinctly defined via a free stochastic differential equation (SDE) in a noncommutative probability space. For the canonical process on the general linear group, denote the FMBM by $(b_t)_{t\geq 0}$; it satisfies

$db_t = b_t\, dc_t,\qquad b_0 = 1,$

where $c_t$ is a free circular Brownian motion, which is the large-$N$ limit of Brownian motion in the Ginibre ensemble on $M_N(\mathbb{C})$ (Hall et al., 2018, Driver et al., 2019, Ho et al., 2019). In the unitary setting, the process $u_t$ on the unit circle $\mathbb{T}$ is defined by

$$du_t = i\, dX_t\, u_t - \frac{1}{2} u_t\, dt, \qquad u_0 = 1,$$

where $X_t$ is a free additive Brownian motion (Ulrich, 2014).

A positive version, the free positive multiplicative Brownian motion $(h_t)_{t\geq0}$, is constructed by $h_t := g_{t/2} g_{t/2}^*$, where $(g_t)$ is a FMBM as above (Auer, 9 May 2025).

2. Transform Techniques and Semigroup Structure

The analysis of FMBM employs the $\eta$-transform and the $\Sigma$-transform. For a probability measure $\mu$, the $\eta$-transform is defined on the complex unit disk, and the $\Sigma$-transform satisfies a key multiplicative property: $$\Sigma_{\mu\boxtimes\nu}(z) = \Sigma_\mu(z)\Sigma_\nu(z).$$ A central equation for the unitary case linking two measures $\mu$, $\nu$ via the $\Sigma$- and $\eta$-transforms is (1210.6090): $$\Sigma_{\lambda}(\eta_{\nu}(z)) = \frac{z}{\eta_{\mu}(z)},$$ where $\lambda$ is the free multiplicative analogue of the normal distribution. Under FMBM evolution, this transforms into

$$\Sigma_{\lambda}(\eta_{\nu\boxtimes\lambda_t}(z)) = \frac{z}{\eta_{\mathbb{M}_t(\mu)}(z)},$$

where $\mathbb{M}_t$ is a Bercovici–Pata type bijection. This framework extends to measures on $\mathbb{R}_+$, utilizing a modified $S$-transform for measures lacking nonzero mean.

The spectral distributions of FMBM form a semigroup under free multiplicative convolution: $\nu_s \boxtimes \nu_t = \nu_{s+t}.$ In the positive case, the law can be expressed as the exponential image measure of an additive free convolution: $$\nu_t = \exp\left(\mu_{sc, 2\sqrt{t}} \boxplus \mathrm{Unif}_{[-t/2, t/2]}\right),$$ where $\mu_{sc, R}$ is the semicircle law of radius $R$ and $\mathrm{Unif}_I$ is the uniform measure on interval $I$ (Auer, 9 May 2025, Auer et al., 1 Aug 2024).

3. Brown Measures, Spectral Domains, and Explicit Density Formulas

The spectral analysis of (non-normal) FMBM operators depends on the Brown measure, a noncommutative analogue of the eigenvalue distribution. For $b_t$, the Brown measure $\mu_{b_t}$ is supported on a planar domain $\Sigma_t$, explicitly constructed using conformal mapping: $$f_t(z) = z\,\exp\left[\frac{t}{2} \frac{1+z}{1-z}\right],$$ with $\Sigma_t$ the component containing $1$ in the complement of $\{z\in\mathbb C \setminus \mathbb T : |f_t(z)|=1\}$ (Hall et al., 2018). In polar coordinates $(r, \theta)$, the Brown measure density is given by (Driver et al., 2019, Ho et al., 2019): $W_t(r, \theta) = \frac{1}{r^2 w_t(\theta)},$ where $w_t(\theta)$ is an analytic function of the angle determined by the geometry of $\Sigma_t$.

For general initial conditions and parameter families $(b_{s,\tau})$, the Brown measures of FMBM exhibit a "model deformation phenomenon"—densities in logarithmic coordinates that are constant along the parameter $\tau$, and explicit formulas can be established for their transformation under natural push-forwards (Hall et al., 2021).

4. Asymptotics, Convergence, and Universality

FMBM is the large-dimension limit of Brownian motion on $GL(N,\mathbb C)$. The almost sure strong convergence (joint moments and operator norm) of matrix-valued multiplicative Brownian motions $G_{\lambda,\tau}$ to their free counterparts is now established for a general family of $(\lambda, \tau)$—this includes convergence of the spectra of any noncommutative polynomial evaluated at these random matrices (Banna et al., 18 Jul 2025). The technical core is a multiplicative interpolation strategy and sharp variance bounds: $$\mathrm{Var}\left[\mathrm{tr}_N(f(PP^*(\mathcal G_t, A^N)))\right] = O(1/N^2).$$ And for any deterministic family $A^N$ converging strongly,

$$\lim_{N\to\infty} \|P(G_t, A^N)\| = \|P(g_t, a)\| \quad\text{a.s.}$$

Local limit theorems for FMBM show "superconvergence": not only do empirical spectral measures converge weakly, but their densities converge locally uniformly (even analytically) to the density of the FMBM's law, both for $\mathbb{R}_+$ and the unit circle. In unitary cases, convergence to the Haar measure is uniform over the circle, enabling free entropic central limit theorems and universality phenomena (Anshelevich et al., 2013).

5. Explicit Formulas via Harmonic Analysis

Recent results connect large-$N$ asymptotics for eigenvalues of Brownian motions on matrix groups to classical harmonic analysis. The limiting empirical measures of the logarithmic singular values (i.e., eigenvalues of $g_t g_t^*$) can be explicitly described as push-forwards of free additive convolutions of a semicircle law and a uniform measure on an interval (Auer et al., 1 Aug 2024).

For the $GL(N, \mathbb{C})$ case, the density of the limiting spectral measure can be obtained from the densities of drifted Brownian motions on Hermitian spaces $H(N, \mathbb C)$, with explicit dependence on determinants and products of hyperbolic sines inherited from spherical function expansions. Analogous results extend to root systems $B_N, C_N, D_N$, yielding a unified view of spectral limits for FMBM across classical Lie groups.

6. Regularity, Log-Unimodality, and Long-Time Behavior

For the free multiplicative analogue of the normal distribution on the circle, the density is analytic on its support and strictly unimodal. For $0 < t < 4$, the support is a closed arc; for $t \geq 4$, it is the entire circle, and as $t \to \infty$ the law tends to the Haar measure, mirroring behavior in the additive case (1210.6090).

On $\mathbb{R}_+$, marginal laws of free positive multiplicative Brownian motion $\sigma_t \boxtimes \nu$ are log-unimodal for all $t > 0$ if $\nu$ is log-unimodal and multiplicatively symmetric; if $\nu$ merely has bounded support, log-unimodality is attained for large enough $t$ (Hasebe et al., 2020). This demonstrates the smoothing effect of the convolution semigroup.

7. Comparative and Theoretical Context

The FMBM can be seen as the multiplicative analogue of the classical (additive) Brownian motion and the Wigner–Dyson and Ginibre universality for eigenvalue statistics. Relations among the Brown measure of $b_t$ and the spectral measure of free unitary Brownian motion $u_t$ directly mirror the relationship of Wigner's semicircle law and Ginibre’s circular law (Driver et al., 2019).

Analytically, the functional calculus, SDE techniques, and Hamilton–Jacobi PDEs developed for FMBM yield new tools for analyzing non-normal operators in operator algebras. Connections to Yang–Mills theory, planar limits, and the master field arise from the ability to describe asymptotic eigenvalue distributions and their domains of support with great precision (Hall et al., 2018).

---

Summary Table: Core Structures and Results

Object / Formula	Context / Significance	Reference
$db_t = b_t dc_t,\ b_0=1$	FMBM SDE on GL$(N,\mathbb C)$	(Hall et al., 2018, Driver et al., 2019)
Spectral law via $$\nu_t = \exp(\mu_{sc,2\sqrt t} \boxplus \operatorname{Unif}_{[-t/2,t/2]})$$	Spectral distribution of positive FMBM	(Auer, 9 May 2025, Auer et al., 1 Aug 2024)
Brown measure on $\Sigma_t$ with $W_t(r,\theta) = 1/(r^2 w_t(\theta))$	Planar support and density for Brown measure	(Driver et al., 2019, Ho et al., 2019)
$\Sigma_\mu(z)$, $\eta_\mu(z)$	Transform techniques for convolutions and subordination	(1210.6090, Anshelevich et al., 2013)
$\nu_s \boxtimes \nu_t = \nu_{s+t}$	Multiplicative (free) convolution semigroup	(Auer, 9 May 2025)
Strong convergence in operator norm	Matrix Brownian motions $G_{\lambda,\tau}$ to FMGBM	(Banna et al., 18 Jul 2025)

---

Collectively, these developments establish free multiplicative Brownian motion as a central paradigm in noncommutative probability, providing both structural analogues to classical diffusion and novel spectral phenomena unique to the free setting. The explicit formulas, spectral domain analysis, and universality properties render it a key object for ongoing research in probability, random matrix theory, operator algebras, and mathematical physics.