  We derive a general exact formula for the mean first passage time (MFPT) from
a fixed point inside a planar domain to an escape region on its boundary. The
underlying mixed Dirichlet-Neumann boundary value problem is conformally mapped
onto the unit disk, solved exactly, and mapped back. The resulting formula for
the MFPT is valid for an arbitrary space-dependent diffusion coefficient, while
the leading logarithmic term is explicit, simple, and remarkably universal. In
contrast to earlier works, we show that the natural small parameter of the
problem is the harmonic measure of the escape region, not its perimeter. The
conventional scaling of the MFPT with the area of the domain is altered when
diffusing particles are released near the escape region. These findings change
the current view of escape problems and related chemical or biochemical
kinetics in complex, multiscale, porous or fractal domains, while the
fundamental relation to the harmonic measure opens new ways of computing and
interpreting MFPTs.
