  Here we present a multiscale method to calculate the saddle point associated
with the effective dynamics arising from a stochastic system which couples slow
deterministic drift and fast stochastic dynamics. This problem is motivated by
the transition states on free energy surfaces in chemical physics. Our method
is based on the gentlest ascent dynamics which couples the position variable
and the direction variable and has the local convergence to saddle points. The
dynamics of the direction vector is derived in terms of the covariance function
with respective to the equilibrium distribution of the fast stochastic process.
We apply the multiscale numerical methods to efficiently solve the obtained
multiscale gentlest ascent dynamics, {and discuss the acceleration techniques
based on the adaptive idea.} The examples of stochastic ordinary and partial
differential equations are presented.
