Brownian Representation of a Floquet Basis

P. McGill

2019, v.25, Issue 5, 797-820


Floquet's theorem sets out conditions for the circular equation
\psi'' = - \psi(\lambda + v)
to have a basis of multiplier solutions. Under white-noise potential,
and for $\lambda$ below the periodic groundstate eigenvalue, their logarithmic derivatives define
a pair of periodic Riccati processes. We establish absolute continuity with respect to
an intrinsic mixture of periodic Riccati bridges. The formula furnishes another perspective
on results discovered by S.\ Cambronero and H.~McKean (CPAM, 52, No.\thinspace 10 {pp.}\ 1277--1294 (1999)) and it
also makes sense for more general potential noises.

Keywords: Floquet theory, Brownian bridge, Riccati process, circular measures, absolute continuity


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