It is known that for a real valued stochastic process $X_t$ satisfying $$ d X_t = b(X_t) d t + \sigma d W_t $$ where $W$ is real valued Wiener process, the equation for the most probable path from a state $x$ at time $t$ to another state $y$ at time $T$, is of the form $$ \ddot x = b(x)b'(x)+ \frac{\sigma^2}{2}b''(x) $$ with $$ x(t) = x, \quad x(T) = y. $$ For instance, when $\sigma = 1$ and $b \equiv 0$, the most probable path connecting $x$ to $y$ between the instants $t$ and $T$ is $$ w(s) = \frac{1}{2}(x+y) + \frac{y-x}{T-t} \left ( s - \frac{T+t}{2}\right ) $$ (Not too bad for a process that is nowhere differentiable.)
Is there any reference concerning the equation for the most probable path for second order SDEs of the form $$ d X_t = F(X_t,Y_t) d t, \quad d Y_t = G(X_t,Y_t) d t + \sigma d W_t $$ I have in mind stochastic Hamiltonian systems in the sense of Talay, [Stochastic Hamiltonian dissipative systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Markov Processes and Related Fields 8(2), 163-198, 2002.]
