  The adjoint Fokker-Planck equation method is applied to study the runaway
probability function and the expected slowing-down time for highly relativistic
runaway electrons, including the loss of energy due to synchrotron radiation.
In direct correspondence to Monte Carlo simulation methods, the runaway
probability function has a smooth transition across the runaway separatrix,
which can be attributed to effect of the pitch angle scattering term in the
kinetic equation. However, for the same numerical accuracy, the adjoint method
is more efficient than the Monte Carlo method. The expected slowing-down time
gives a novel method to estimate the runaway current decay time in experiments.
A new result from this work is that the decay rate of high energyelectrons is
very slow when E is close to the critical electric field. This effect
contributes further to a hysteresis previously found in the runaway electron
population.
