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Expected signature of stopped Brownian motion on $d$ -dimensional $C^{2, α}$ -domains has finite radius of convergence everywhere: $2\leq d \leq 8$

This paper proves that for any bounded $C^{2,\alpha}$ -domain in dimensions $2 \leq d \leq 8$ , the expected signature of Brownian motion stopped at the first exit time has a finite radius of convergence everywhere, a result established by introducing a "domain-averaging hyperbolic development" to symmetrize the underlying PDE system.

Original authors: Siran Li, Hao Ni

Published 2026-02-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Siran Li, Hao Ni

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Fingerprint" of Randomness

Imagine you are watching a drunk person (a Brownian motion) wandering randomly through a city. They start at a specific point and keep walking until they hit a wall and leave a specific neighborhood (the domain).

In mathematics, we want to describe this entire random journey with a single, unique "fingerprint." This fingerprint is called the Signature. It's like a super-detailed summary of every twist, turn, and loop the person took.

The big question mathematicians have been asking is: If we know the "average fingerprint" of this journey, can we perfectly reconstruct the rules of the game? In other words, does the average tell us everything about the probability of the path?

To answer this, mathematicians look at the Radius of Convergence. Think of this as the "stability limit" of the fingerprint.

Infinite Radius: The fingerprint is perfectly stable and predictable forever. (This happens if the person walks for a fixed amount of time, like 1 hour).
Finite Radius: The fingerprint eventually becomes unstable or "blows up" if you try to calculate it too precisely. This instability actually tells us something profound: the average fingerprint does uniquely determine the path's law.

The Discovery:
For a long time, mathematicians knew that if the drunk person walks for a fixed time, the fingerprint is stable (infinite radius). But if they walk until they hit a wall (stopped Brownian motion), it was a mystery.

We knew it was stable somewhere.
We knew it was unstable (finite radius) for a perfect circle (the 2D unit disc).
But what about any other shape? (A square? A weird blob? A 3D room?)

The Answer:
This paper proves that for almost any shape (in dimensions 2 through 8), the fingerprint always has a finite radius of convergence. It is unstable everywhere. This confirms that the average signature is a perfect, unique identifier for the path, no matter the shape of the room they are trapped in.

The Challenge: The "Shape" Problem

Why was this so hard to prove?

Imagine you are trying to describe the average path of the drunk person in a perfectly round room. Because the room is round, it looks the same no matter how you spin it. This symmetry makes the math easy; you can use a special tool called Hyperbolic Development to turn the messy path into a simple, solvable equation.

But what if the room is a lumpy, irregular blob?

It doesn't look the same if you spin it.
The "Hyperbolic Development" tool gets confused because the shape breaks the symmetry.
Previous methods relied on the room being a perfect circle, so they couldn't be used for general shapes.

The Solution: The "Symmetry Blender"

The authors, Siran Li and Hao Ni, invented a clever trick to solve this. They didn't try to solve the problem for the weird shape directly. Instead, they used a "Domain-Averaging" technique.

The Analogy: The Smoothie Maker
Imagine you have a weirdly shaped room (the domain).

Spin it: Take that room and rotate it around its center in every possible direction (like spinning a globe).
Blend it: Instead of looking at the room in one specific orientation, you take the "average" of the room as it spins through all those angles.
The Result: Even though the original room was lumpy, the average of all its rotations creates a perfectly symmetrical, round "ghost room."

By doing this, the authors created a new mathematical object (the Domain-Averaging Development) that behaves as if it were in a perfect circle, even though the original domain was irregular.

The Proof: Finding the "Blow-Up" Point

Once they created this symmetrical "ghost room," they could use the same powerful tools they used for the perfect circle.

They set up a system of equations (ODEs) that describe the fingerprint.
They looked for a specific point where the math breaks down (where the answer goes to infinity).
They proved that for dimensions 2 through 8, this "breakdown point" always exists.

Why is "breaking" good?
In this context, "breaking" (finite radius of convergence) is the smoking gun. It proves that the average signature contains enough information to uniquely identify the path. If the radius were infinite, the signature might be too smooth to tell different paths apart. The fact that it "blows up" at a specific point confirms the signature is a perfect fingerprint.

Why Stop at Dimension 8?

The paper works for dimensions 2 through 8. Why not 9 or 10?

Think of the equations they solved as a complex machine with gears.

For dimensions 2–8, the gears mesh together in a way that creates a specific type of "wobble" (mathematically, complex numbers with specific properties) that guarantees the breakdown point exists.
At dimension 9, the gears change. The "wobble" disappears, and the math behaves differently. The authors suspect the result might still be true for higher dimensions, but their current "Symmetry Blender" tool doesn't work the same way. They leave this as a puzzle for future mathematicians.

Summary

The Problem: Can we uniquely identify a random path that stops when it hits a wall, just by looking at its average "fingerprint"?
The Obstacle: Previous methods only worked for perfect circles. Real-world shapes are irregular.
The Innovation: The authors invented a "Symmetry Blender" that averages the shape over all rotations, turning any irregular room into a perfect circle for the sake of calculation.
The Result: They proved that for 2D to 8D shapes, the fingerprint is always "unstable" (finite radius), which confirms it is a unique and perfect identifier for the path.

This work bridges the gap between the simple, perfect world of circles and the messy, irregular world of real shapes, showing that the fundamental rules of randomness hold true everywhere.

1. Problem Statement and Context

The Moment Problem for Rough Paths:
A central question in rough path theory is whether the expected signature of a stochastic process uniquely determines the law (probability distribution) of its signature. A sufficient condition for this uniqueness (the "moment problem") is that the expected signature has an infinite radius of convergence.

The Specific Challenge:
While it is known that Brownian motion on a fixed time interval $[0, T]$ has an expected signature with an infinite radius of convergence, the case of stopped Brownian motion (Brownian motion stopped at the first exit time $\tau_\Omega$ from a bounded domain $\Omega$ ) remained an open problem.

Previous work (Boedihardjo et al., 2020) proved that for the specific case of the 2-dimensional unit disc, the radius of convergence is finite.
However, that proof relied heavily on the rotational symmetry of the disc, making it difficult to extend to general domains or higher dimensions.
It was unknown whether the radius of convergence is finite for general bounded domains or for dimensions $d > 2$ .

The Main Question:
Does the expected signature of a Brownian motion stopped upon exiting a bounded domain $\Omega \subset \mathbb{R}^d$ have a finite radius of convergence for all points in $\Omega$ , for general domains and dimensions?

2. Methodology

The authors develop a novel technique called "Domain-Averaging Hyperbolic Development" to overcome the lack of rotational symmetry in general domains. The methodology proceeds in several key stages:

A. PDE Formulation

The expected signature $\Phi_\Omega(z)$ satisfies a system of nested Partial Differential Equations (PDEs) derived from the generator of Brownian motion. The authors establish that for $C^{2,\alpha}$ domains, the solution exists and has a geometric decay bound, ensuring a strictly positive (but potentially finite) radius of convergence.

B. Hyperbolic Development

They utilize the hyperbolic development, a morphism $H: T((E)) \to \mathfrak{gl}(d+1; \mathbb{R})$ that maps the signature to a matrix group. This transformation converts the tensor-valued PDE system into a matrix-valued PDE system. Crucially, the hyperbolic development of a path lies on a specific hyperboloid, providing geometric constraints (e.g., the last component is $\ge 1$ ).

C. Domain-Averaging (Symmetrization)

To handle general domains $\Omega$ that lack rotational invariance, the authors introduce a symmetrization argument:

Rotation: They consider the domain $\Omega$ rotated by all possible elements $R \in SO(d)$ .
Averaging: They define a domain-averaged development $H_{\lambda, \epsilon}(z)$ by integrating the hyperbolic development over all rotated domains centered at a point $z$ .
Properties: This averaged quantity inherits the PDE structure of the original problem but gains rotational invariance. Consequently, the PDE for $H_{\lambda, \epsilon}$ reduces to an Ordinary Differential Equation (ODE) system in the radial variable $r = |z|$ .

D. Reduction to ODEs and Bessel Functions

The symmetrized PDE reduces to a coupled system of ODEs for the first and last components of the hyperbolic development (denoted $h^{(1)}_\lambda$ and $h^{(d+1)}_\lambda$ ).

The solutions are expressed in terms of Bessel functions of the first kind ( $J_\nu$ ).
The radius of convergence is determined by the location of the first singularity (pole) of these solutions, which corresponds to the first zero of the Wronskian determinant appearing in the denominator of the explicit solution.

E. Numerical Verification

For dimensions $d \in \{2, \dots, 8\}$ , the authors explicitly analyze the Wronskian determinant. They prove that:

The denominator (Wronskian) has a real root $\lambda^*$ within a specific interval (e.g., $(2.5, 3)$ ).
The numerator at this root is non-zero.
Therefore, the solution blows up at $\lambda^*$ , implying a finite radius of convergence.

3. Key Contributions

Generalization to Arbitrary Domains: The paper resolves the finiteness problem for any bounded $C^{2,\alpha}$ domain, not just the unit disc. This is achieved via the "domain-averaging" technique, which effectively "symmetrizes" the problem without requiring the domain itself to be symmetric.
Extension to Higher Dimensions: The result is proven for all dimensions $2 \le d \le 8$ .
Explicit ODE Analysis: The authors derive explicit ODE systems for the hyperbolic development in general dimensions and solve them using Bessel functions with complex parameters $\beta_\pm$ and $\eta_\pm$ .
Computer-Assisted Proofs: The proof relies on rigorous numerical estimates (using Maple) to verify that the Wronskian has a root and the numerator does not vanish in the critical interval. These computations are verified to high precision.

4. Main Results

Theorem 1.1:
Let $\Omega \subset \mathbb{R}^d$ be a bounded $C^{2,\alpha}$ -domain with $2 \le d \le 8$ . The expected signature $\Phi$ of a Brownian motion stopped upon the first exit time from $\Omega$ has a finite radius of convergence at every point $z \in \Omega$ .

Implications:

Since the radius of convergence is finite, the expected signature does not satisfy the sufficient condition for the moment problem (infinite radius).
This suggests that the expected signature of stopped Brownian motion on such domains does not uniquely determine the law of the signature (though it does not strictly prove non-uniqueness, it removes the primary sufficient condition for uniqueness).

5. Significance and Limitations

Significance:

Theoretical Impact: It fundamentally alters the understanding of the moment problem for rough paths. It shows that even for "nice" processes like stopped Brownian motion on regular domains, the expected signature may fail to uniquely characterize the law due to finite convergence.
Methodological Innovation: The "domain-averaging hyperbolic development" is a powerful new tool that allows the application of symmetry-based techniques (like those used for the unit disc) to general, non-symmetric geometries.
Applications: This has implications for machine learning applications using signatures (e.g., signature-based hypothesis testing), suggesting that the metric based on normalized expected signatures might not be injective for stopped processes.

Limitations and Future Work:

Dimensional Restriction ( $d \le 8$ ): The proof relies on the parameters $\beta_\pm = \sqrt{2d - 6 \pm 2\sqrt{d^2 - 10d + 9}}$ $β_{\pm} = 2 d - 6 \pm 2 d^{2} - 10 d + 9$ being complex.
- For $d \in [2, 8]$ , the term under the square root ( $d^2 - 10d + 9$ ) is negative, making $\beta_\pm$ complex conjugates. This leads to oscillatory behavior in Bessel functions, allowing for the existence of zeros (roots) in the Wronskian.
- For $d \ge 9$ , the parameters become real, leading to qualitatively different behavior (exponential growth/decay) where the existence of a root is not guaranteed by the same mechanism. The authors leave the case $d \ge 9$ for future investigation.
Regularity: The result assumes the domain boundary is $C^{2,\alpha}$ .

In summary, this paper provides a definitive negative answer to the question of infinite convergence for stopped Brownian motion on a wide class of domains and dimensions, utilizing a sophisticated blend of PDE theory, geometric analysis, and numerical verification.