Asymptotics of Brownian occupation measures with unusually large intersections

Jiyun Park

arxiv: 2604.08127 · v2 · submitted 2026-04-09 · 🧮 math.PR

Asymptotics of Brownian occupation measures with unusually large intersections

Jiyun Park This is my paper

Pith reviewed 2026-05-10 18:01 UTC · model grok-4.3

classification 🧮 math.PR

keywords Brownian motionoccupation measureslarge deviationsGagliardo-Nirenberg inequalityintersection measuresweak convergenceconditioned processes

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The pith

Brownian occupation measures conditioned on large intersections converge weakly, after shifts, to the square of a Gagliardo-Nirenberg optimizer.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that Brownian paths forced to intersect far more than usual have their occupation measures converging to a deterministic shape. That shape is the square of the function achieving equality in the Gagliardo-Nirenberg inequality. The result is proved by first obtaining a compact large-deviation principle for ordinary Brownian occupation measures, then extending it to measures tilted by their intersection size, and finally using an approximation that lets the intersection functional be tested against bounded functions. A reader should care because the limit supplies a concrete probabilistic representation of the optimizer that appears in Sobolev-type inequalities. The argument also yields large-deviation principles for the intersection measure itself.

Core claim

The occupation measures of Brownian motions conditioned to have large intersections converge weakly, up to spatial shifts, to the measure whose density is the square of an optimizer of the Gagliardo-Nirenberg inequality. This follows from a large-deviation principle for Brownian occupation measures conditioned on large self-intersections or large mutual intersections, which in turn rests on a compact LDP for the unconditioned occupation measures and an exponentially good approximation of the intersection measure against bounded measurable functions.

What carries the argument

The exponentially good approximation of the intersection measure tested against all bounded measurable functions, which transfers large-deviation control from occupation measures to their intersections and enables the conditioned limit.

If this is right

The large-deviation principle extends to occupation measures tilted by their mutual or self-intersection size.
The intersection measure of two independent Brownian motions itself satisfies a large-deviation principle in the same topology.
The compact LDP for unconditioned occupation measures generalizes the earlier result of Mukherjee and Varadhan to the topologies required here.
The limiting measure is invariant under spatial translations only after explicit centering.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The optimizer of the Gagliardo-Nirenberg inequality acquires a direct interpretation as the typical density of paths realizing rare high-intersection events.
The approximation tool may transfer to other additive functionals of paths, such as local times or occupation times in moving windows.
Numerical checks of the conditioned density could provide an independent way to locate the Gagliardo-Nirenberg optimizer without solving the Euler-Lagrange equation directly.

Load-bearing premise

The unconditioned Brownian occupation measures obey a compact large-deviation principle in the topology used for conditioning, and the intersection functional can be approximated exponentially well by test functions against bounded measurable sets.

What would settle it

Monte Carlo sampling of Brownian paths conditioned on intersection size exceeding a large threshold, followed by centering and checking whether the empirical density converges to the square of the known Gagliardo-Nirenberg optimizer rather than to some other profile.

read the original abstract

We prove that the occupation measures of Brownian motions conditioned to have large intersections converge weakly, up to spatial shifts, to the measure whose density is the square of an optimizer of the Gagliardo-Nirenberg inequality. We do so by proving a large deviation principle (LDP) for Brownian occupation measures conditioned either on large self-intersections or large mutual intersections. To this end, we derive a compact LDP for unconditioned Brownian occupation measures, generalizing the work of Mukherjee and Varadhan. We also prove the LDP for Brownian occupation measures tilted by their intersections in the same topology. A key tool of independent interest is an exponentially good approximation of the intersection measure tested against all bounded measurable functions, from which we further get the LDP for the intersection measure of independent Brownian motions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Park gives new LDPs for Brownian occupation measures under large intersection constraints and derives the expected convergence to the squared Gagliardo-Nirenberg optimizer after shifts.

read the letter

The main result is that occupation measures of Brownian motions conditioned on large self- or mutual intersections converge weakly, up to spatial shifts, to the measure whose density is the square of a Gagliardo-Nirenberg optimizer. Park reaches this by first generalizing the Mukherjee-Varadhan compact LDP for unconditioned occupation measures, then proving LDPs for the intersection-tilted and conditioned versions, and using an auxiliary approximation lemma for the intersection functional against bounded measurable functions. The approximation is presented as a tool of independent interest that also yields an LDP for the intersection measure itself. These steps are new relative to the cited prior work and give explicit limiting profiles that link the probabilistic conditioning to a concrete variational problem. The argument structure separates the unconditioned LDP, the tilting, and the conditioning in a way that looks logically ordered. The connection to the Gagliardo-Nirenberg inequality follows once the rate function is identified, which is a natural outcome rather than an ad-hoc fit. The central claims rest on the compact LDP holding in a topology compatible with shifts and on the approximation being exponentially good; the manuscript states both are established, and the stress-test note finds no internal inconsistency or missing step in the outline. That said, the details of the topology choice and the uniformity of the approximation will need close checking in the proofs, as those are the load-bearing technical ingredients. Minor gaps in exposition could appear once the full derivations are read, but nothing in the abstract or argument skeleton suggests a circular reduction or hidden fitting. This paper is for readers already working with large deviations for occupation measures or with intersection properties of Brownian paths. Someone familiar with Mukherjee-Varadhan will see the extensions immediately and can judge whether the new approximation lemma travels to other settings. It deserves a serious referee because the results are new, the strategy is explicit, and the claims are checkable against the stated assumptions. I would send it to peer review.

Referee Report

1 major / 2 minor

Summary. The paper proves that the occupation measures of Brownian motions conditioned to have large self- or mutual intersections converge weakly, up to spatial shifts, to the measure whose density is the square of an optimizer of the Gagliardo-Nirenberg inequality. This is achieved by establishing a compact LDP for unconditioned Brownian occupation measures (generalizing Mukherjee-Varadhan), an LDP for measures tilted by their intersections, and an exponentially good approximation of the intersection measure against bounded measurable test functions, from which the LDP for the intersection measure itself follows.

Significance. If the results hold, the work furnishes a probabilistic asymptotic description of Gagliardo-Nirenberg optimizers via conditioned Brownian paths and strengthens the link between large-deviation theory for occupation measures and variational problems in analysis. The compact LDP generalization and the new approximation lemma are tools of independent interest that could apply to other intersection or occupation functionals.

major comments (1)

The central convergence result rests on the compact LDP for unconditioned occupation measures holding in a topology compatible with spatial shifts and on the exponentially good approximation of the intersection functional against bounded measurable functions. These are stated to be established in the manuscript, but the precise topology (e.g., the precise form of the weak topology or the role of shifts in the rate function) is load-bearing for transferring the LDP to the conditioned setting; a short explicit verification that the approximation passes to the shifted measures would strengthen the argument.

minor comments (2)

The abstract and introduction would benefit from a brief statement of the ambient dimension and the precise form of the Gagliardo-Nirenberg optimizer used.
Notation for the occupation measure and the intersection functional could be introduced once and used consistently to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment, and constructive suggestion. We address the single major comment below.

read point-by-point responses

Referee: The central convergence result rests on the compact LDP for unconditioned occupation measures holding in a topology compatible with spatial shifts and on the exponentially good approximation of the intersection functional against bounded measurable functions. These are stated to be established in the manuscript, but the precise topology (e.g., the precise form of the weak topology or the role of shifts in the rate function) is load-bearing for transferring the LDP to the conditioned setting; a short explicit verification that the approximation passes to the shifted measures would strengthen the argument.

Authors: We agree that an explicit verification of the compatibility with spatial shifts would improve clarity. The compact LDP is established in the topology of weak convergence of measures modulo translations (i.e., the quotient topology induced by the action of spatial shifts), with the rate function invariant under shifts, as required for the generalization of Mukherjee-Varadhan. The exponentially good approximation lemma is proved for all bounded measurable test functions; because a spatial shift of the occupation measure corresponds to a shift of the test function that preserves boundedness and measurability, the same exponential bound applies verbatim to shifted measures. In the revised version we will insert a short paragraph immediately after the statement of the approximation lemma making this extension explicit and confirming that it transfers directly to the conditioned LDP. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on external generalizations and independent tools

full rationale

The paper derives its central LDP for conditioned occupation measures and the weak convergence result by first establishing a compact LDP for unconditioned measures via generalization of the external Mukherjee-Varadhan result, then introducing an exponentially good approximation of the intersection functional against bounded measurable test functions as an independent tool. This approximation directly yields the LDP for the intersection measure itself, which is then used to transfer to the tilted and conditioned settings. No step reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation chain; the argument structure remains self-contained against external benchmarks with new approximation content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard results from stochastic processes and functional analysis without introducing new free parameters or postulated entities.

axioms (2)

standard math Brownian motion is a continuous Markov process with known occupation measure properties
Used to define the objects whose large deviations are studied.
standard math Existence of optimizers for the Gagliardo-Nirenberg inequality in appropriate function spaces
Invoked to identify the limiting density.

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discussion (0)

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Works this paper leans on

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Improving image generation with better captions.arXiv preprint arXiv:2310.07685, 2023

A. Adhikari and I. Okada. Moderate deviations for the capacity of the random walk range in dimension four, 2023. arXiv:2310.07685

work page arXiv 2023

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Adhikari and J

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work page arXiv 2025

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Asselah and B

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work page doi:10.1214/20-ejp560 2020

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work page doi:10.1007/s00440-021-01063-3 2021

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work page doi:10.1214/18-aop1288 2019

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work page doi:10.1214/08-aihp181 2009

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E. Bates and S. Chatterjee. The endpoint distribution of directed polymers.Ann. Probab., 48(2):817–871, 2020.doi: 10.1214/19-AOP1376

work page doi:10.1214/19-aop1376 2020

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work page doi:10.1002/cpa.21682 2017

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work page doi:10.1090/surv/157 2010

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A. Dembo and I. Okada. Capacity of the range of random walk: the law of the iterated logarithm.Ann. Probab., 52(5):1954– 1991, 2024.doi:10.1214/24-aop1692

work page doi:10.1214/24-aop1692 1954

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