  In this paper we study the linear stochastic heat equation, also known as
parabolic Anderson model, in multidimension driven by a Gaussian noise which is
white in time and it has a correlated spatial covariance. Examples of such
covariance include the Riesz kernel in any dimension and the covariance of the
fractional Brownian motion with Hurst parameter $H\in (\frac 14, \frac 12]$ in
dimension one. First we establish the existence of a unique mild solution and
we derive a Feynman-Kac formula for its moments using a family of independent
Brownian bridges and assuming a general integrability condition on the initial
data. In the second part of the paper we compute Lyapunov exponents, lower and
upper exponential growth indices in terms of a variational quantity. The last
part of the paper is devoted to study the phase transition property of the
Anderson model.
