In the book "Stochastic Differential Equations" by Bernt Øksendal, the Kolmogorov's backward equation is stated as following:
Let $X_t$ be an Ito diffusion and $A$ is the generator of $X_t$. Define $u(t,x)=E^x(f(X_t))$ where $E^x$ is the expectation with $X_0=x$. Then we have $$ \begin{array}{c}{\frac{\partial u}{\partial t}=A u, \quad t>0, x \in {R}^{n}} \\ {u(0, x)=f(x) ; \quad x \in {R}^{n}}\end{array}. $$ My question is why this equation is called "Backward"? Sicne now we have the initial condition $u(x,0)=f(x)$.
Besides, it seems that the definition of Kolmogorov's backward equation is in another form on Wiki: Kolmogorov backward equations (diffusion).
Although it is suggestive to think of forward and backward as the direction of time in the equation, this is slightly misleading. The idea to have in mind should be:
Backward $\leftrightarrow$ initial state $\leftrightarrow$ generator $A$ of the Markov process $\leftrightarrow$ Feynman-Kac.
Forward $\leftrightarrow$ terminal state $\leftrightarrow$ adjoint $A^*$ of the generator $\leftrightarrow$ Fokker-Planck.
Indeed, you can see that both Oksendal and Wikipedia the operator appearing is $A$. Indeed, if you let $T_{\cdot}$ be the semigroup generated by $A$, you see that:
$$ \mathbb{E}_{0,x}[f(X_t)] = T_t f(x), \qquad \mathbb{E}_{t,x}[f(X_T)] = T_{T-t}f(x)$$
so Wikipedia is just writing the equation for the time-reversed process presented by Oskendal. If you take the probability density of $X_t$ started in $x$ to be $p_t(x,y)$, the above tells you:
$$ \partial_t p_t(x,y) = A_x p_t(x, y), \qquad p_0(x,y)= \delta(x {-} y)$$
Instead, the forward equation tells you that:
$$\partial_t p_t(x,y) = A^*_y p_t(x,y), \qquad p_0(x,y) = \delta(x{-}y).$$