Vanishing-noise asymptotics for Donsker-Varadhan rate functions on the circle
Abstract
We study the vanishing-noise limit of the rate function for the Donsker–Varadhan large deviation principle for one-dimensional diffusion processes on a circle. As is well known, the rate function can be represented either as the Legendre transform of the principal eigenvalue of the perturbed infinitesimal generator or by a variational formula. We analyze the asymptotic behavior of the principal eigenvalue as the parameter in front of the noise goes to zero and compare the Legendre transform of the limit with the expression obtained as the limit of the variational representation. We prove, in particular, that the resulting expressions do not always coincide, leading to continuity and discontinuity phenomena in infinite-dimensional functional spaces. Moreover, we use the previous analysis to pass to the limit in the LDP, using the notion of Γ-convergence.
Versions
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BibTeX
@article{koresski2026vanishingnoise,
title = {{Vanishing-noise asymptotics for Donsker-Varadhan rate functions on the circle}},
author = {Milan Koresski},
journal = {Electronic Journal of Differential Equations},
year = {2026},
month = sep,
doi = {10.58997/ejde.2026.71},
eprint = {2609.30113},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05763329v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Vanishing-noise asymptotics for Donsker-Varadhan rate functions on the circle
AU - Milan Koresski
PY - 2026
DA - 2026/09/01
JO - Electronic Journal of Differential Equations
DO - 10.58997/ejde.2026.71
UR - https://hal.science/hal-05763329v1
AB - We study the vanishing-noise limit of the rate function for the Donsker–Varadhan large deviation principle for one-dimensional diffusion processes on a circle. As is well known, the rate function can be represented either as the Legendre transform of the principal eigenvalue of the perturbed infinitesimal generator or by a variational formula. We analyze the asymptotic behavior of the principal eigenvalue as the parameter in front of the noise goes to zero and compare the Legendre transform of the limit with the expression obtained as the limit of the variational representation. We prove, in particular, that the resulting expressions do not always coincide, leading to continuity and discontinuity phenomena in infinite-dimensional functional spaces. Moreover, we use the previous analysis to pass to the limit in the LDP, using the notion of Γ-convergence.
SN - 1072-6691
ER -