  We consider non-linear time-fractional stochastic heat type equation
$$\partial^\beta_tu_t(x)=-\nu(-\Delta)^{\alpha/2}
u_t(x)+I^{1-\beta}_t[\sigma(u)\stackrel{\cdot}{W}(t,x)]$$ in $(d+1)$
dimensions, where $\nu>0, \beta\in (0,1)$, $\alpha\in (0,2]$ and
$d<\min\{2,\beta^{-1}\}\a$, $\partial^\beta_t$ is the Caputo fractional
derivative, $-(-\Delta)^{\alpha/2} $ is the generator of an isotropic stable
process, $I^{1-\beta}_t$ is the fractional integral operator,
$\stackrel{\cdot}{W}(t,x)$ is space-time white noise, and
$\sigma:\RR{R}\to\RR{R}$ is Lipschitz continuous.
  Time fractional stochastic heat type equations might be used to model
phenomenon with random effects with thermal memory. We prove existence and
uniqueness of mild solutions to this equation and establish conditions under
which the solution is continuous. Our results extend the results in the case of
parabolic stochastic partial differential equations obtained in
\cite{foondun-khoshnevisan-09, walsh}. In sharp contrast to the stochastic
partial differential equations studied earlier in
\cite{foondun-khoshnevisan-09, khoshnevisan-cbms, walsh}, in some cases our
results give existence of random field solutions in spatial dimensions
$d=1,2,3$. Under faster than linear growth of $\sigma$, we show that time
fractional stochastic partial differential equation has no finite energy
solution. This extends the result of Foondun and Parshad \cite{foondun-parshad}
in the case of parabolic stochastic partial differential equations. We also
establish a connection of the time fractional stochastic partial differential
equations to higher order parabolic stochastic differential equations.
