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Hölder continuity of the solutions for a class of nonlinear SPDE’s arising from one dimensional superprocesses

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The Hölder continuity of the solution X t (x) to a nonlinear stochastic partial differential equation (see (1.2) below) arising from one dimensional superprocesses is obtained. It is proved that the Hölder exponent in time variable is arbitrarily close to 1/4, improving the result of 1/10 in Li et al. (to appear on Probab. Theory Relat. Fields.). The method is to use the Malliavin calculus. The Hölder continuity in spatial variable x of exponent 1/2 is also obtained by using this new approach. This Hölder continuity result is sharp since the corresponding linear heat equation has the same Hölder continuity.

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Author information

Correspondence to Fei Lu.

Additional information

Yaozhong Hu is partially supported by a grant from the Simons Foundation #209206 and David Nualart is supported by the NSF grant DMS0904538.

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Hu, Y., Lu, F. & Nualart, D. Hölder continuity of the solutions for a class of nonlinear SPDE’s arising from one dimensional superprocesses. Probab. Theory Relat. Fields 156, 27–49 (2013). https://doi.org/10.1007/s00440-012-0419-2

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  • Nonlinear stochastic partial differential equation
  • Stochastic heat kernel
  • Conditional transition probability density in a random environment
  • Malliavin calculus
  • Hölder continuity
  • Moment estimates

Mathematics Subject Classification

  • 60H07
  • 60H15
  • 60H30