Understanding Einstein theory for Brownian motion. In wikipedia they describe Einstein theory as follows: Let $\Delta $ be the increment of a particle in time $\tau$ (it's a random variable) with density $\varphi (\Delta )$. Using Taylor at time $t+\tau$ for the density (number of particles per unit volume), we have $$\rho(x,t)+\tau\frac{\partial \rho (x,t)}{\partial t}+\mathcal O(t^2)=\rho(x,t+\tau)\underset{(*)}{=}\int_{\mathbb R}\rho(x-\Delta ,t)\varphi (\Delta )d\Delta =\rho(x,t)\int_{\mathbb R}\varphi (\Delta )d\Delta -\frac{\partial \rho}{\partial x}\int_{\mathbb R}\Delta \varphi (\Delta )d\Delta +\frac{\partial ^2\rho}{\partial x^2}\int_{\mathbb R}\frac{\Delta ^2}{2}\varphi (\Delta )d\Delta +O(\Delta ^3)=\rho(x,t)+\frac{\partial ^2\rho}{\partial x^2}\int_{\mathbb R}\frac{\Delta ^2}{2}\varphi (\Delta )d\Delta +O(\Delta ^3).$$ 
Q1) Where does equality $(*)$ come from? 
Q2) After they conclude that by setting $D=\int_{\mathbb R}\frac{\Delta ^2}{2\tau}\varphi (\Delta )d\Delta $, we get $$\frac{\partial \rho}{\partial t}=D\frac{\partial ^2\rho}{\partial x^2},$$ It's maybe obvious but I don't understand where this comes from.
 A: Q1) I think is like a continuos version of the marginal distribution of markov chains, both things are stochastic, in the discrete version you get something like
$$ \Pr(X_{n+1}=j) = \sum_{r \in S} p_{rj} \Pr(X_n=r)$$
so your probability of being at the state $j$ at the step $n+1$ depends on the all the possible states $r$ at the previous step $n$, similarty you have that
$$\rho(x,t+\tau)\underset{(*)}{=}\int_{\mathbb R}\rho(x-\Delta ,t)\varphi (\Delta )d\Delta$$
so your probability of being at the postion $x$ at the time $t+\tau$ depends on all the possible position $x-\Delta$ at the previous time $t$
Q2)Look at the first and last equation you wrote, and forget about terms $\mathcal{O(\tau^2)}$, $\mathcal{O(\Delta ^3)}$
$\rho(x,t)+\tau\frac{\partial \rho (x,t)}{\partial t}+\mathcal{O(\tau^2)}=\rho(x,t)+\frac{\partial ^2\rho}{\partial x^2}\int_{\mathbb R}\frac{\Delta ^2}{2}\varphi (\Delta )d\Delta +\mathcal{O(\Delta ^3)}$
$\rho(x,t)+\tau\frac{\partial \rho (x,t)}{\partial t}=\rho(x,t)+\frac{\partial ^2\rho}{\partial x^2}\int_{\mathbb R}\frac{\Delta ^2}{2}\varphi (\Delta )d\Delta$
$\tau\frac{\partial \rho (x,t)}{\partial t}=\frac{\partial ^2\rho}{\partial x^2}\int_{\mathbb R}\frac{\Delta ^2}{2}\varphi (\Delta )d\Delta$
$\frac{\partial \rho (x,t)}{\partial t}=\frac{\partial ^2\rho}{\partial x^2}\int_{\mathbb R}\frac{\Delta ^2}{2\tau}\varphi (\Delta )d\Delta$
$\frac{\partial \rho (x,t)}{\partial t}=D\frac{\partial ^2\rho}{\partial x^2}$
