  We prove the existence of a viscosity solution of the following path
dependent nonlinear Kolmogorov equation: \[ \begin{cases}
\partial_{t}u(t,\phi)+\mathcal{L}u(t,\phi)+f(t,\phi,u(t,\phi),\partial_{x}u(t,\phi)
\sigma(t,\phi),(u(\cdot,\phi))_{t})=0,\;t\in[0,T),\;\phi\in\mathbb{\Lambda}\,
,u(T,\phi)=h(\phi),\;\phi\in\mathbb{\Lambda}, \end{cases} \] where
$\mathbb{\Lambda}=\mathcal{C}([0,T];\mathbb{R}^{d})$, $(u(\cdot
,\phi))_{t}:=(u(t+\theta,\phi))_{\theta\in[-\delta,0]}$ and \[
\mathcal{L}u(t,\phi):=\langle b(t,\phi),\partial_{x}u(t,\phi)\rangle+\dfrac
{1}{2}\mathrm{Tr}\big[\sigma(t,\phi)\sigma^{\ast}(t,\phi)\partial_{xx}
^{2}u(t,\phi)\big]. \] The result is obtained by a stochastic approach. In
particular we prove a new type of nonlinear Feynman-Kac representation formula
associated to a backward stochastic differential equation with time-delayed
generator which is of non-Markovian type.
  Applications to the large investor problem and risk measures via
$g$-expectations are also provided.
