Potential Analysis

, Volume 25, Issue 4, pp 307–326 | Cite as

Young Integrals and SPDEs

  • Massimiliano Gubinelli
  • Antoine Lejay
  • Samy Tindel
Article

Abstract

In this note, we study the non-linear evolution problem
$$dY_t = -A Y_t dt + B(Y_t) dX_t,$$
where \(X\) is a \(\gamma\)-Hölder continuous function of the time parameter, with values in a distribution space, and \(-A\) the generator of an analytical semigroup. Then, we will give some sharp conditions on \(X\) in order to solve the above equation in a function space, first in the linear case (for any value of \(\gamma\) in \((0,1)\)), and then when \(B\) satisfies some Lipschitz type conditions (for \(\gamma>1/2\)). The solution of the evolution problem will be understood in the mild sense, and the integrals involved in that definition will be of Young type.

Mathematics Subject Classifications (2000)

60H15 60H05 

Keywords

stochastic heat equation fractional Brownian motion path-wise stochastic integration rough path theory 

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Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  • Massimiliano Gubinelli
    • 1
  • Antoine Lejay
    • 2
  • Samy Tindel
    • 3
  1. 1.Dip. di Matematica Applicata “U. Dini”Università di PisaPisaItaly
  2. 2.Project OMEGA, INRIA LorraineIECN, Campus ScientifiqueVandœuvre-lès-Nancy CedexFrance
  3. 3.Institut Elie CartanUniversité Henri Poincaré (Nancy)Vandœuvre-lès-Nancy CedexFrance

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