Stochastic Heat and Wave Equations on a Lie Group

Abstract

We study nonlinear heat and wave equations on a Lie group. The noise is assumed to be a spatially homogeneous Wiener process. We give necessary and sufficient conditions for the existence of a function-valued solution in terms of the covariance kernel of the noise.

Keywords:

• Homogeneous Wiener process
• Stochastic evolution on a Lie group
• Stochastic heat and wave equations

Mathematics Subject Classification:

• Primary 60H15
• Secondary 60G60, 35K05, 35L05

The work of S.P. has been supported by EC FP6 Marie Curie ToK programme SPADE2.

The authors would like to thank Professors T. Coulhon, G. Folland, and P. Malliavin for very useful information on the analysis on Lie groups.