Department of Mathematics
CY Cergy Paris Université
2 avenue Adolphe Chauvin
95302 Cergy-Pontoise, France
email: milankoresski@wanadoo.fr
Milan Koresski
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 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 \(\Gamma\)-convergence.
Submitted February 4, 2026. Published September 21, 2026.
Math Subject Classifications: 60F10, 60J60, 35B40.
Key Words: Second-order elliptic operators; principal eigenvalue; weak KAM theory;
Hamilton-Jacobi equations; Legendre transform; Gamma-convergence.
DOI: 10.58997/ejde.2026.71
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Milan Koresski Department of Mathematics CY Cergy Paris Université 2 avenue Adolphe Chauvin 95302 Cergy-Pontoise, France email: milankoresski@wanadoo.fr |
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