  We prove the integral Varadhan short-time formula for non-linear heat flow on
measured Finsler manifolds. To the best of the authors' knowledge, this is the
first result establishing a Varadhan-type formula for non-linear semigroups. We
do not assume the reversibility of the metric, and the distance function can be
asymmetric. Within this generality, we reveal that the probabilistic
interpretation is well-suited for our formula; the probability that a particle
starting from a set $A$ can be found in another set $B$ describes the distance
from $A$ to $B$. One side of the estimates (the upper bound of the probability)
is also established in the nonsmooth setting of infinitesimally strictly convex
metric measure spaces satisfying the local Sobolev-to-Lipschitz property.
