Liouville Brownian Motion: Constructions & Properties

• Liouville Brownian Motion (LBM) is a time-changed Brownian process constructed via Gaussian multiplicative chaos and the Liouville measure.
• LBM bridges probabilistic geometry and two-dimensional quantum gravity, enabling rigorous analysis of fractal random geometries and spectral properties.
• LBM exhibits multifractality, strong symmetry, and unique heat kernel estimates, providing insights into scaling limits and random planar maps.

Liouville Brownian Motion (LBM) is the canonical diffusion process in the random geometry induced by a Gaussian free field via the exponential "Liouville measure," central to rigorous approaches to Liouville quantum gravity (LQG) and fractal random geometry. In its archetypal setting on the plane or a Riemann surface, LBM is constructed as a time-changed Brownian motion, with the clock determined by a positive continuous additive functional whose Revuz measure is the (subcritical or critical) Gaussian multiplicative chaos associated to the field. LBM thus provides the stochastic analytic bridge between probabilistic geometry and the physics of two-dimensional quantum gravity, with direct mathematical connections to Dirichlet form theory, spectral analysis, random matrix theory, and random planar maps.

1. Construction of Liouville Brownian Motion

Let $X$ be a mass-$m>0$ Gaussian free field (GFF) on $\mathbb{R}^2$, a log-correlated generalized Gaussian distribution with covariance kernel $G^{(m)}$ solving $(m^2 - \Delta)G^{(m)}(x,y) = 2\pi \delta_x(y)$. The Liouville measure (subcritical regime $\gamma\in(0,2)$) is defined via Gaussian multiplicative chaos,

$$M_\gamma(dz) = \lim_{n\to\infty} \exp\bigl(\gamma X_n(z) - \tfrac{\gamma^2}{2} \mathbb{E}[X_n(z)^2]\bigr) dz,$$

where $X_n$ is a smooth regularization of $X$ (e.g., via spectral truncation). $M_\gamma$ is a non-degenerate, atomless Radon measure, almost surely singular with respect to Lebesgue measure and supported off capacity-zero sets (Shin, 2019).

Given a planar Brownian motion $m>0$0 independent of $m>0$1, define its (strict) positive continuous additive functional (PCAF) $m>0$2 with Revuz measure $m>0$3 via

$m>0$4

The Liouville Brownian motion is the time-changed process

$m>0$5

This construction is almost sure with respect to the randomness of the GFF, and $m>0$6 is a strong Markov process with continuous sample paths, infinite lifetime, and symmetry with respect to $m>0$7 (Garban et al., 2013, Shin, 2019, Garban et al., 2013, Andres et al., 2014).

At criticality ($m>0$8), the measure is constructed using the derivative martingale (Rhodes et al., 2013). For general weight functions, the Dirichlet form is modified accordingly, yielding "Liouville distorted Brownian motion" with respect to the weighted chaos measure (Shin, 2019).

2. Dirichlet Forms, Symmetry, and Generator Structure

LBM is precisely the Hunt process associated to the closure in $m>0$9 of the symmetric, strongly local pre-Dirichlet form

$\mathbb{R}^2$0

(Garban et al., 2013, Garban et al., 2013). Its infinitesimal generator on $\mathbb{R}^2$1 is the elliptic operator

$\mathbb{R}^2$2

subject to closure in $\mathbb{R}^2$3. The form is regular, symmetric in $\mathbb{R}^2$4, and ensures the process is $\mathbb{R}^2$5-symmetric and reversible (Garban et al., 2013). For distorted LBM, the base measure incorporates a weight $\mathbb{R}^2$6, modifying the Dirichlet form and resulting process accordingly (Shin, 2019).

The strong Feller property holds for both the resolvent and the semigroup; the resolvent kernel is absolutely continuous and admits a strictly positive, jointly continuous density on compacts (Garban et al., 2013). The intrinsic metric associated to the Dirichlet form is identically zero, a manifestation of the extreme singularity of the Liouville metric geometry (Garban et al., 2013).

3. Heat Kernel and Spectral Properties

The LBM admits a jointly continuous heat kernel $\mathbb{R}^2$7 with respect to the Liouville measure, satisfying symmetry $\mathbb{R}^2$8 and Chapman-Kolmogorov equations. The semigroup is strong Feller, and strict positivity on compacts holds for all $\mathbb{R}^2$9 (Andres et al., 2014).

For $G^{(m)}$0, there are sub-Gaussian upper bounds: $G^{(m)}$1 and on-diagonal lower bounds

$G^{(m)}$2

$G^{(m)}$3-almost everywhere (Andres et al., 2014). For $G^{(m)}$4 (Brownian map case), the heat kernel obeys sharp two-sided estimates with off-diagonal exponents reflecting LQG metric volume growth: for example,

$G^{(m)}$5

in the $G^{(m)}$6-LQG metric (Andres et al., 17 Jul 2025).

The spectral dimension is almost surely 2, both pointwise and globally, reflecting the preservation of "quantum" diffusive scaling relative to the Liouville geometry (Andres et al., 2014, Andres et al., 17 Jul 2025). Short-time asymptotics of the heat kernel at $G^{(m)}$7 are governed by the multifractal thickness of $G^{(m)}$8 at $G^{(m)}$9 (Bertacco et al., 27 Jan 2025).

4. Multifractality and Interactions with Thick Points

LBM exhibits deep multifractal behavior stemming from the underlying GFF structure. The time spent by LBM in $(m^2 - \Delta)G^{(m)}(x,y) = 2\pi \delta_x(y)$0-thick points (where the GFF is atypically large) forms random sets $(m^2 - \Delta)G^{(m)}(x,y) = 2\pi \delta_x(y)$1 whose Hausdorff dimension is given by

$(m^2 - \Delta)G^{(m)}(x,y) = 2\pi \delta_x(y)$2

valid for $(m^2 - \Delta)G^{(m)}(x,y) = 2\pi \delta_x(y)$3 (Jackson, 2014). For $(m^2 - \Delta)G^{(m)}(x,y) = 2\pi \delta_x(y)$4, the LBM path is Lebesgue-almost everywhere differentiable, with subdiffusive or superdiffusive scaling in the Euclidean metric depending on the field thickness at the starting point.

The multifractal structure is encoded in the local exponents of the time-change functional and reflects in local times, level sets, and occupation measures of LBM (Jackson, 2014). In $(m^2 - \Delta)G^{(m)}(x,y) = 2\pi \delta_x(y)$5, the spectral dimension at $(m^2 - \Delta)G^{(m)}(x,y) = 2\pi \delta_x(y)$6 depends both on $(m^2 - \Delta)G^{(m)}(x,y) = 2\pi \delta_x(y)$7 and local field thickness $(m^2 - \Delta)G^{(m)}(x,y) = 2\pi \delta_x(y)$8 (Bertacco et al., 27 Jan 2025).

5. Extensions: Distorted, Discrete, and High-Dimensional Variants

Weighted and Distorted Liouville BM

For weight functions $(m^2 - \Delta)G^{(m)}(x,y) = 2\pi \delta_x(y)$9, the Dirichlet form and resulting process generalize to "Liouville distorted Brownian motion," with the chaos measure taking the form $\gamma\in(0,2)$0. The associated Hunt diffusion is constructed via Dirichlet-form theory, positive capacity of sets, and explicit PCAFs (Shin, 2019).

Regularization and SDE Representation

On regularized geometries (truncated GFF), n-regularized LBM admits solutions as weak diffusions driven by random, oscillatory diffusion coefficients. The stochastic differential equation for $\gamma\in(0,2)$1 is

$\gamma\in(0,2)$2

where $\gamma\in(0,2)$3 is a standard planar Brownian motion (Shin, 2016).

Discrete Approximations and Scaling Limits

LBM arises as the quenched scaling limit of random walk on random planar maps (mated-CRT maps, Poisson-Voronoi tessellations), when graphs are embedded using Peano or Tutte/Harmonic embeddings. The Markov process with invariant measure (vertex degree in the discrete, $\gamma\in(0,2)$4 in the continuum) converges in local-uniform topology on curves to LBM (Berestycki et al., 2020, Gwynne et al., 2018).

One-Dimensional LBM and Excursions

The theory extends to dimension one: given a Liouville measure $\gamma\in(0,2)$5 (from boundary GMC), the LBM is the scale-invariant linear diffusion in $\gamma\in(0,2)$6 with speed measure $\gamma\in(0,2)$7, constructed via time-change of Brownian motion by the local-time additive functional. Its spectral representation, excursion theory, and multifractal properties are accessible via Kreĭn's string theory (Jin, 2017).

Higher-Dimensional Analogues ($\gamma\in(0,2)$8)

LBM in $\gamma\in(0,2)$9 for $$M_\gamma(dz) = \lim_{n\to\infty} \exp\bigl(\gamma X_n(z) - \tfrac{\gamma^2}{2} \mathbb{E}[X_n(z)^2]\bigr) dz,$$0 is constructed using the log-correlated Gaussian field and time-change by whole-space GMC. The spectral dimension at $$M_\gamma(dz) = \lim_{n\to\infty} \exp\bigl(\gamma X_n(z) - \tfrac{\gamma^2}{2} \mathbb{E}[X_n(z)^2]\bigr) dz,$$1 is

$$M_\gamma(dz) = \lim_{n\to\infty} \exp\bigl(\gamma X_n(z) - \tfrac{\gamma^2}{2} \mathbb{E}[X_n(z)^2]\bigr) dz,$$2

for an $$M_\gamma(dz) = \lim_{n\to\infty} \exp\bigl(\gamma X_n(z) - \tfrac{\gamma^2}{2} \mathbb{E}[X_n(z)^2]\bigr) dz,$$3-thick point, interpolating between quantum (2) and Euclidean ($$M_\gamma(dz) = \lim_{n\to\infty} \exp\bigl(\gamma X_n(z) - \tfrac{\gamma^2}{2} \mathbb{E}[X_n(z)^2]\bigr) dz,$$4) behavior. Quantum cones and shift-invariance under LBM are established for characteristic weights (Bertacco et al., 27 Jan 2025).

6. Critical Case and Further Developments

At the critical value $$M_\gamma(dz) = \lim_{n\to\infty} \exp\bigl(\gamma X_n(z) - \tfrac{\gamma^2}{2} \mathbb{E}[X_n(z)^2]\bigr) dz,$$5, both the measure and LBM require additional renormalization, leading to construction via the derivative martingale and modified additive functional. The critical LBM is supported on a random set of Hausdorff dimension zero (Rhodes et al., 2013). The associated Dirichlet form remains strongly local and regular, but heat kernel estimates become subtler, with only weak Feller properties available. Full off-diagonal bounds for the heat kernel in the critical regime remain an open area (Rhodes et al., 2013).

Extension of LBM to other settings, such as distorted Dirichlet forms, higher topologies (torus/sphere), and boundaries (quantum disks), is active. The random geometry perspective continues to produce profound links between probability, random matrix theory, SLE/CLE, and mathematical physics (Gwynne et al., 2018, Andres et al., 17 Jul 2025).

7. Significance and Connections

LBM provides the analytic and probabilistic machinery to probe the metric and spectral properties of Liouville quantum gravity surfaces and random planar geometries. It is the unique strong Markov Hunt diffusion symmetric with respect to the Liouville measure, characterized by time-changed Brownian motion, reversibility, and strong Feller properties. Its probabilistic construction, Dirichlet form, and spectral kernel structure reflect the pathologies and richness of the random geometry—vanishing intrinsic metric, nontrivial scaling, multifractality, and universality under scaling limits of discrete random walks (Garban et al., 2013, Berestycki et al., 2020, Andres et al., 17 Jul 2025, Rhodes et al., 2013).