  We study solutions of $\tfrac12 (-i\nabla - A(x))^2f=\lambda f$ on domains
$\Omega\subset \mathbb{R}^2$ with Dirichlet boundary conditions and prove
exponential decay estimates in terms of an Agmon type distance to a classically
allowed region. This metric depends only on the eigenvalue and associated
magnetic field. In fact the main quantity in the weight for this distance
function can be interpreted as an `average magnetic flux' of the magnetic field
along all domains whose boundary is the closed curved formed by joing the
`minimal path' to another random path. Our estimates are based on an analysis
of the associated heat kernel using the Feynman-Kac-It\'o formula.
