Convergence of Brownian occupation measures with large intersections

Abstract: We prove that the occupation measures of Brownian motions conditioned to have large intersections converge weakly, up to spatial shifts, to a 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 on large self-intersections or mutual intersections. To this end, we develop a compact LDP for Brownian occupation measures, generalizing the work of Mukherjee and Varadhan. We also prove an LDP for Brownian occupation measures tilted by their intersections in the same topology. A key tool is an exponentially good approximation of the intersection measure tested against all bounded measurable functions, which may be of independent interest. As a consequence, we also obtain an LDP for the intersection measure of p independent Brownian motions.

• The paper establishes full large deviation principles (LDPs) for Brownian occupation measures conditioned on extreme self or mutual intersection local times.
• It employs a compactified topology and profile decomposition to identify limiting profiles linked to optimizers of the Gagliardo-Nirenberg inequality.
• Exponential approximations of singular intersection functionals enable rigorous conditional LDPs and connect probabilistic models with elliptic PDE extremal problems.

Convergence of Brownian Occupation Measures with Large Intersections: Technical Summary

Background and Problem Statement

The study addresses the asymptotics and structure of Brownian occupation measures under conditioning on having large (self-) or mutual intersection local times. For a Brownian motion $W_t$ in $\mathbb{R}^d$, the $q$-fold self-intersection local time $\beta([0, t]^q)$ and for $p$ independent Brownian motions $W^1,\dots,W^p$, the mutual intersection local time $\alpha([0, t]^p)$, quantify the extent to which the paths overlap or self-intersect. The tails of these functionals have been characterized variationally, with the large deviations rate given by minimizers of the Gagliardo-Nirenberg inequality, as established in the literature (e.g., Chen [2010]).

Despite this, the characterization of the conditional law of the occupation measure given these rare large intersection events remained incomplete. The functional dependence of the intersection local times on the occupation measure is highly singular (discontinuous). Prior analyses of rare events for Brownian occupation measures (e.g., under Coulomb or polaron tilts [Mukherjee, Varadhan 2016,2017]) relied on large deviation principles (LDPs) in canonically compactified measure spaces modulo translations, but did not directly address intersection local times.

Main Results

Large Deviation Principles for Brownian Occupation Measures

The author generalizes the compact topological framework for occupation measures—introduced by Mukherjee and Varadhan—by defining the space $$\widetilde{\mathcal{X}_{\le 1}^{\otimes p}(\mathbb{R}^d)}$$ of multisets of diagonal shift classes of tuples of sub-probability measures. This topology allows for profile decompositions and captures all possible limiting profiles modulo translations.

In this topology, the law of the $p$-tuple of occupation measures under independent Brownian motions admits a full LDP with a rate function

$$\mathcal{I}(\xi^{\otimes p}) = \begin{cases} \frac{1}{2} \sum_{i\in I} \sum_{j=1}^p \|\nabla \psi_i^j\|_2^2 & \psi_i^j = \sqrt{d\alpha_i^j/dx} \in H^1 \ +\infty & \text{otherwise}. \end{cases}$$

This rate characterizes the cost (in terms of large deviations) for the occupation measure to exhibit a prescribed empirical profile.

Conditional LDPs and Weak Convergence under Large Intersections

By combining the above LDP with an exponentially good approximation of intersection local times by continuous functionals on measure space, the author establishes conditional LDPs for the occupation measure, given that the intersection local time exceeds a rare large threshold (e.g., $\mathbb{R}^d$0).

The minimizers of the conditional rate functions correspond, up to translation, to probability measures with densities given by the square of optimizers of the Gagliardo-Nirenberg inequality:

• In the self-intersection case ($\mathbb{R}^d$1, $\mathbb{R}^d$2), the conditional occupation measure converges (mod translation) to one having density $\mathbb{R}^d$3, where $\mathbb{R}^d$4 uniquely solves the associated Gagliardo-Nirenberg extremal problem.
• In the mutual intersection case, the occupation tuples converge (again modulo diagonal shifts) to profiles whose entries are all copies of the same square of the optimizer $\mathbb{R}^d$5 for the corresponding variational problem.

Formally: $\mathbb{R}^d$6 where $\mathbb{R}^d$7 is the probability measure with density $\mathbb{R}^d$8.

LDPs and Weak Convergence under Gibbs Tilts Favoring Intersections

The same framework applies to Brownian occupation measures under Gibbs transforms that favor large intersection local times (i.e., measures tilted by exponentials of $\mathbb{R}^d$9 or $q$0). Here, the limiting occupation profile is determined by the unique solution to the associated variational inequality involving both the Dirichlet energy and intersection terms. For example, in the self-intersection Gibbs case, the limiting profile solves

$q$1

with optimality holding when $q$2 is below the critical threshold coming from Sobolev embedding.

Exponentially Good Approximations and Continuity

A central technical contribution is the establishment of exponentially good approximations for the intersection local times by smoothed (i.e., mollified) functionals, controlled in strong Orlicz (exponential) norms. Specifically, for the intersection measure $q$3 (used in the mutual intersection case), convolution with Gaussian kernels yields approximants that, when tested against bounded functions, converge exponentially fast (in the large deviation sense) to the singular intersection measure. These results extend to control the $q$4-norms relevant for the self-intersection case.

This continuity enables the application of contraction principles and the explicit computation of conditional/gibbsian rate functions on the compactified space.

Technical Innovations

• Full LDP in Translation-Compactified Product Measure Spaces: The joint occupation measures are analyzed in a topology that retains marginal information, circumventing issues faced in other approaches (e.g., [Mukherjee 2017], [Erhard-Poisat 2026]), where projections to marginals are not continuous.
• Profile Decomposition and Concentration-Compactness: The proof leverages profile decompositions (concentration-compactness), which classify the limiting behavior of sequenced measures (including tightness, vanishing, or dichotomy via decomposition into well-separated "profiles").
• Exponential Approximation of Singular Functionals: By refining and extending techniques pioneered for intersection local times and Wiener chaos (Le Gall, Chen, König, etc.), the author establishes exponential approximation for highly singular functionals of Brownian occupation, which enables rigorous LDP analysis via smoothing mollifiers.

Numerical and Structural Conclusions

• The unique minimizer of the conditional/gibbsian rate function is always the orbit of the optimizer of the Gagliardo-Nirenberg inequality, up to translation/diagonal translation.
• For fixed large intersection level, the occupation measure overwhelmingly concentrates near this profile as $q$5.
• The variational formulae for probabilities of large intersection events recover, and strictly strengthen, classical tail estimates for intersection local times.

Implications and Future Directions

Practically, the results provide a comprehensive variational and probabilistic description of the conditioned (or tilted) Brownian occupation profile under rare intersection events, with sharp asymptotics and identification of the most likely empirical path structure. The methodology is robust and extends to other functionals of Markov processes or random walks, such as occupation range, sausage volume, and more general path integrals.

On a theoretical level, the framework deepens understanding of the connection between large deviation minimizers and elliptic PDE extremals, suggesting avenues for studying rare event mechanisms in high-dimensional stochastic systems subject to singular functionals. The exponential approximation techniques developed are expected to be useful in a range of settings where occupation times interact through non-continuous functionals.

Anticipated future developments include extension to:

• Other stochastic processes (random walks, Levy flights, Markov jump processes),
• Singular path functionals not covered by intersection local times (capacity, extremes),
• Dynamical Gibbs models in statistical physics (polymers with self-interaction, etc.).

Conclusion

This work establishes large deviation conditional laws and weak limit theorems for Brownian occupation measures under large intersection constraints or Gibbs tilts, using an analytic and probabilistic framework based on refined compactification, profile decomposition, and exponential approximation techniques. The solutions connect to and are characterized by the extremals of the Gagliardo-Nirenberg inequality, and the methodology is broadly extensible within the domain of probabilistic analysis of singular stochastic path functionals.