Exponentially Tilted Planar Brownian Motion

Updated 1 February 2026

Exponentially tilted planar Brownian motion is a diffusion process reweighted by singular potentials, Gaussian multiplicative chaos, or exponential functionals to modify standard Brownian behavior.
The construction employs techniques such as Doob transforms and point interaction kernels to yield unique drift terms and local time properties, which are rigorously characterized via heat kernel estimates.
Key analytical insights include sub-Gaussian transition density bounds, scaling limits converging to Liouville Brownian motion, and connections to integrable structures in random geometry.

Exponentially tilted planar Brownian motion describes a class of planar diffusion processes obtained via an exponential reweighting (tilt) of the law of Brownian motion in the plane, most notably through the insertion of singular potentials, multiplicative functionals, or, in the geometric context, through coupling to random geometry or random measures. The principal manifestations of this concept fall into three interconnected paradigms: (1) diffusions with point interactions (zero-range potential at the origin), (2) Liouville (Gaussian multiplicative chaos) tilting, and (3) exponential functionals related to integrable structures, such as Whittaker processes. Each framework sharply modifies the underlying Brownian motion, producing new objects of central importance in probability, analysis, and mathematical physics.

1. Exponential Tilting via Point Interactions

A central example of exponentially tilted planar Brownian motion is the family of diffusions with a point interaction at the origin. This is constructed via a Doob transform of the heat kernel of the Schrödinger operator with a delta potential: $H^\alpha = -\frac{1}{2}\Delta + \alpha\,\delta_0,$ where $\alpha > 0$ is the coupling constant. The resulting semigroup possesses integral kernel

$f_t(x,y) = g_t(x-y) + h_t(x,y),$

with the Gaussian kernel $g_t(z) = (2\pi t)^{-1} e^{-|z|^2/(2t)}$ and a Volterra-type correction $h_t(x,y)$ reflecting the point interaction. Exponentially tilting planar Brownian motion corresponds to the unique law on $C([0,T],\mathbb{R}^2)$ with transition densities

$p^\alpha_{s,t}(x,y) = f_{t-s}(x,y) \frac{1+H_{T-t}(y)}{1+H_{T-s}(x)}, \quad 0 \le s < t \le T,$

where $H_t(x) = \int_{\mathbb{R}^2} h_t(x,z)\,dz$ . This process possesses a drift

$b_t^\alpha(x) = \nabla_x\log(1+H_{T-t}(x)),$

which exhibits a singularity of order $1/(|x|\log^2|x|)$ as $|x|\to 0$ . The process admits local time $L_t^0$ at the origin, and the probability of hitting the origin before time $T$ is strictly between $0$ and $1$ for $x\neq 0$ , in contrast to standard planar Brownian motion. The construction and detailed analysis of this process are given in (Clark et al., 2023) and further placed in an axiomatic framework in (Mian, 24 Jan 2026).

2. The Lebesgue-Driven (Doob-Transformed) Planar Diffusion

A general axiomatic development of planar diffusions with point interactions is provided via the so-called Lebesgue-driven family: $K_t^\theta(x,y) = g_t(x-y) + 2\pi\theta\int_{0<r<s<t} g_r(x)\nu'(\theta(s-r))g_{t-s}(y)\,dr\,ds,$ where $\nu(a)=\int_0^\infty a^s/\Gamma(s+1)\,ds$ (the Volterra function). The key harmonic function is

$\varphi_t(x) = \int_{\mathbb{R}^2} K_t^\theta(x,y)dy = 1 + \int_0^t r^{-1} e^{-|x|^2/(2r)}\nu(\theta(t-r))dr.$

The exponentially tilted process is then the unique Markov diffusion with transition density

$p_{s,t}(x,y) = \frac{\varphi_{T-t}(y)}{\varphi_{T-s}(x)}K_{t-s}^{\theta}(x,y).$

The associated SDE on $\mathbb{R}^2\setminus \{0\}$ is

$dX_t = dW_t + b_{T-t}(X_t)\,dt, \qquad b_t(x) = \nabla_x\log\varphi_t(x),$

which admits unique weak solutions. The origin is non-sticky: the process spends zero Lebesgue measure time at $0$, but can visit the origin with strictly positive probability for finite $T$ (Mian, 24 Jan 2026).

3. Liouville Quantum Gravity and Gaussian Multiplicative Chaos Tilt

In random geometry, exponentially tilted planar Brownian motion arises as Liouville Brownian motion (LBM) on a random geometry governed by Gaussian multiplicative chaos (GMC). The Liouville quantum gravity (LQG) area measure is constructed from a Gaussian free field $h$ as

$\mu_h(dz) = \lim_{\varepsilon \to 0} \varepsilon^{\gamma^2/2} \exp(\gamma h_\varepsilon(z))\,dz,$

where $h_\varepsilon(z)$ is the circle average. The quantum clock associated to planar Brownian motion $B^z$ is

$\phi(t) = \lim_{\varepsilon\to 0} \int_0^{t\wedge\tau} \varepsilon^{\gamma^2/2}\exp(\gamma h_\varepsilon(B_s^z))ds,$

and LBM is the time-change $X^z_t = B^z_{\phi^{-1}(t)}$ . An equivalent viewpoint is via the Dirichlet form

$\mathcal{E}(f,f) = \frac{1}{2}\int_D |\nabla f(z)|^2\,dz,$

with invariant measure $\mu_h$ . The scaling limit of random walks on certain random planar maps converges, under appropriate embeddings, to LBM, establishing its universality as the canonical planar diffusion under exponential GMC tilt (Berestycki et al., 2020).

4. Exponential Functionals and Integrable Processes

Another class of exponentially tilted planar Brownian motions involves conditioning on exponential functionals along specific linear forms, leading to connections with integrable systems and Whittaker functions. For Brownian motion with drift in the plane and for $\alpha_i$ (simple roots) spanning a root system (e.g., $A_2$ ),

$F_i = \int_0^\infty \exp(-2\alpha_i(B_s+\mu s))\,ds,$

and the Laplace transform of their joint law is shown to solve a Schrödinger PDE providing explicit connections to class-one Whittaker functions: $\mathcal{L}_\mu(\theta_1,\theta_2) = \mathbb{E}\left[\exp(-\theta_1 F_1 - \theta_2 F_2)\right].$ The corresponding conditioned diffusion is Markovian, with generator given by a Doob $h$ -transform involving the Whittaker eigenfunction, yielding an integrable diffusion process (0809.2506).

5. Local Time, Excursions, and Fine Structure

Exponentially tilted planar Brownian motions admit subtle behaviors at isolated points such as the origin. For point-interaction processes, a local time $L_t^0$ at $0$ can be constructed, and the joint law of $(X_t, L_t^0)$ is explicitly computable. The inverse local time process is a Volterra subordinator, with excursion decompositions reflecting the stochastic nature of the origin's interaction: each encounter corresponds to a stochastic "kick," neither sticky nor absorbing. Limiting cases interpolate between standard planar Brownian motion ( $\alpha\to 0$ ) and instantaneous absorption at the origin ( $\alpha\to\infty$ ) (Clark et al., 2023).

6. Scaling Limits and Universality

Random walks on random planar maps, specifically the mated-CRT maps, converge under scaling to LBM, highlighting the role of exponentially tilted Brownian motion as a universal scaling limit in two-dimensional random geometry. Both the SLE/LQG embedding and the Tutte barycentric embedding yield convergence in the local uniform topology, matching the quantum time parametrization arising from GMC (Berestycki et al., 2020).

7. Analytical Properties: Heat Kernel and Green's Function

Sharp analytic estimates are available for transition densities and Green's functions for exponentially tilted planar Brownian motions. The transition density $p_h(t;x,y)$ of LBM, for example, satisfies two-sided sub-Gaussian bounds, while the Green's function behaves as $-\log|x-y| + O(1)$ as $|x-y|\to 0$ . These properties are essential for understanding the fine analytic and probabilistic structure of these processes (Berestycki et al., 2020). For point interactions, explicit formulae involve Bessel and Hankel functions, with singularities precisely capturing the interaction strength and localization.