Absolute Continuity of Brownian Bridges Under Certain Gauge Transformations

By Andrea R. Nahmod, Luc Rey-Bellet, Scott Sheffield, Gigliola Staffilani. 2011

54 citations recorded by the source corpus.

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 Schr¨ odinger equation obtained by Nahmod, Oh, Rey-Bellet and Staffilani in (20), and with respect to which they proved almost surely global well-posedness, coincides with the weighted Wiener measure con- structed by Thomann and Tzvetkov (24). Thus, in particular we prove the invariance of the measure constructed in (24).

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/16b72667c9ca119083771b385aa1f229ed1da8d3

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Citations in the corpus

Almost sure well-posedness for the periodic 3D quintic nonlinear Schr\"odinger equation below the energy space (2013, background): “ain [1, 2, 4] several works have appeared in which the well-posedness theory has been studied from a nondeterministic point of view relying on and adapting probabilistic ideas and tools as well (c.f. [11, 12, 34, 28, 29, 35, 26, 27, 32, 17, 25, 15, 18, 19] and references therein). It is by now well ”
An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schr\"odinger equation (2017, background): “ essential role in the remaining part of the paper, since all the estimates hold uniformly in t ∈ R. 14 T. OH, P. SOSOE, AND N. TZVETKOV When s= 1, Lemma 3.2 (i) basically follows from Theorem 3.1 in [30] which exploits the properties of the Brownian loop under conformal mappings. For general s>1”
Remarks on the Gibbs measures for nonlinear dispersive equations (2014, result): “This result may be seen as a consequence of a recent work by Nahmod, Oh, ReyBellet and Staffilani [33] and Nahmod, Ray-Bellet, Sheffield and Staffilani [34].”