Integral Varadhan formula for non-linear heat flow

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, thus the distance function can be asymmetric. In 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.