Langevin dynamics along the zero set of real-analytic potentials

• URL: http://arxiv.org/abs/2608.09840v1
• Date: Mon, 10 Aug 2026 16:55:51 GMT
• Title: Langevin dynamics along the zero set of real-analytic potentials
• Abstract summary: We consider the Langevin diffusion $dX_t = - nabla V(X_t) dt + sqrt2 dB_t$ for a general nonnegative real-analytic potential $V$ and a large parameter $$.<n>We derive a candidate limiting evolution on the zero set.
• Score: 0.8947188600472255
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the Langevin diffusion $dX_t = - β\nabla V(X_t) dt + \sqrt{2} dB_t$ for a general nonnegative real-analytic potential $V$ and a large parameter $β$. In the large-$β$ limit the process is confined to the zero set of $V$, assuming that it starts there. We derive a candidate limiting evolution on the zero set. To do so, the zero set is partitioned into strata according to a measure of local codimension known as the local learning coefficient and its multiplicity. It is then shown that the Dirichlet form associated with $X$ converges in a certain sense to a hierarchy of Dirichlet forms corresponding to a stochastic evolution on the strata. This evolution is strongly biased toward higher-dimensional, or "more singular", strata. This result is motivated by a question from Watanabe's singular learning theory regarding the learning dynamics of overparameterized statistical models and the generalization puzzle in deep learning. The result suggests a mechanism for the observation that stochastic gradient methods tend to be biased toward singular solutions that generalize well.

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