Mathematical Research Letters

Volume 18 (2011)

Number 5

Absolute Continuity of Brownian Bridges Under Certain Gauge Transformations

Pages: 875 – 887

DOI: http://dx.doi.org/10.4310/MRL.2011.v18.n5.a6

Authors

Andrea R. Nahmod (Department of Mathematics, University of Massachusetts, 710 N. Pleasant Street, Amherst MA 01003, USA)

Luc Rey-Bellet (Department of Mathematics, University of Massachusetts, 710 N. Pleasant Street, Amherst MA 01003, USA)

Scott Sheffield (Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA)

Gigliola Staffilani (Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA)

Abstract

We prove absolute continuity of Gaussian measures associated to complex Brownian bridges under certain gauge transformations. As an application we prove that the invariant measure for the periodic derivative nonlinear Schrödinger equation obtained by Nahmod, et al, in \cite{NORBS}, and with respect to which they proved almost surely global well-posedness, coincides with the weighted Wiener measure constructed by Thomann and Tzvetkov \cite{TTzv}. Thus, in particular we prove the invariance of the measure constructed in \cite{TTzv}.

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