  We present two approaches to the heat flow on a Finsler manifold $(M,F)$:
either as gradient flow on $L^2(M,m)$ for the energy; or as gradient flow on
the reverse $L^2$-Wasserstein space $\mathcal{P}_2(M)$ of probability measures
on $M$ for the relative entropy. Both approaches depend on the choice of a
measure $m$ on $M$ and then lead to the same nonlinear evolution semigroup. We
prove $\mathcal{C}^{1,\alpha}$-regularity for solutions to the (nonlinear) heat
equation on the Finsler space $(M,F,m)$. Typically, solutions to the heat
equation will not be $\mathcal{C}^2$. Moreover, we derive pointwise comparison
results a la Cheeger-Yau and integrated upper Gaussian estimates a la Davies.
