  We develop a theory of thermal fluctuations of spin density emerging in a
two-dimensional electron gas. The spin fluctuations probed at spatially
separated spots of the sample are correlated due to Brownian motion of
electrons and spin-obit coupling. We calculate the spatiotemporal correlation
functions of the spin density for both ballistic and diffusive transport of
electrons and analyze them for different types of spin-orbit interaction
including the isotropic Rashba model and persistent spin helix regime. The
measurement of spatial spin fluctuations provides direct access to the
parameters of spin-orbit coupling and spin transport in conditions close to the
thermal equilibrium.
