Global dynamics for the stochastic KdV equation with white noise as initial data

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Abstract
We study the stochastic Korteweg-de Vries equation (SKdV) with an additive space-time white noise forcing, posed on the one-dimensional torus. In particular, we construct global-in-time solutions to SKdV with spatial white noise initial data. Due to the lack of an invariant measure, Bourgain’s invariant measure argument is not applicable to this problem. In order to overcome this difficulty, we implement a variant of Bourgain’s argument in the context of an evolution system of measures and construct global-in-time dynamics. Moreover, we show that the white noise measure with variance 1 + t 1+t is an evolution system of measures for SKdV with the white noise initial data.
Funding Information
  • H2020 European Research Council (637995)
  • H2020 European Research Council (864138)
  • H2020 European Research Council (482571)
  • H2020 European Research Council (2238423)
  • H2020 European Research Council (2154090)
  • Natural Sciences and Engineering Research Council of Canada (637995)
  • Natural Sciences and Engineering Research Council of Canada (864138)
  • Natural Sciences and Engineering Research Council of Canada (482571)
  • Natural Sciences and Engineering Research Council of Canada (2238423)
  • Natural Sciences and Engineering Research Council of Canada (2154090)
  • National Science Foundation (637995)
  • National Science Foundation (864138)
  • National Science Foundation (482571)
  • National Science Foundation (2238423)
  • National Science Foundation (2154090)

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