Solutions by Variational Method

  • Leszek GawareckiEmail author
  • Vidyadhar Mandrekar
Part of the Probability and Its Applications book series (PIA)


The variational method for solving stochastic partial differential equations (SPDE’s) of evolutionary type involves recasting them as SDE’s in a Gelfand triplet of Hilbert or Banach spaces VHV , where the embeddings are dense and continuous. We discuss only the case of separable Hilbert spaces. In order to construct a weak solution, we assume that the embeddings are compact, and use the “method of compact embedding” introduced in Chap. 3, together with the stochastic analogue of Lions’ theorem from Chap. 1. The solution is an H-valued stochastic process with continuous sample paths. Under the assumption of monotonicity, we obtain unique strong solution using pathwise uniqueness.

We also present the result on the existence of strong solutions, following the ideas in Prévôt and Röckner (A Concise Course on Stochastic Partial Differential Equations. LNM, vol. 1905. Springer, Berlin, 2007). Assuming that the coefficients are monotone suffices to produce a strong solution without the need for compactness of the embeddings in the Gelfand triplet. Using again the stochastic analogue of Lions’ theorem allows to put the solution in H and assure continuity of its sample paths. We also present results on Markov and strong Markov properties of strong variational solutions.


Weak Solution Strong Solution Stochastic Partial Differential Equation Lower Semicontinuous Function Coercivity Condition 
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  1. 64.
    C. Prévôt and M. Röckner. A Concise Course on Stochastic Partial Differential Equations, LNM 1905, Springer, Berlin (2007). Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of MathematicsKettering UniversityFlintUSA
  2. 2.Department of Statistics and ProbabilityMichigan State UniversityEast LansingUSA

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