Given the heat equation:


$$\partial\_{t}{\varPhi(x,t)}=k^2\partial\_{xx}{\varPhi(x,t)}$$


with the boundary conditions:


$$\Phi(x,0)=\Phi\_0$$


and a Neumann boundary condition of the kind:


$${\partial\_{x}}{\Phi(0,t)=\nu(t)+C}$$


where $\nu(t)$ is a stochastic variable with gaussian distribution ${\sigma=\sigma\_0,\mu=0}$ and $C$ a constant, what is the distribution of the $\Phi(L,t)$?


Thanks in advance



Stochastic Heat Equation
In this case $\Phi(x,t)$ is itself a stochastic process and this equation should be rewritten in a proper way. There is some literature as [this](https://www.math.lsu.edu/cosa/2-2-04%255B140%255D.pdf) and a more general theory of stochastic pde due to John Walsh (a tutorial can be found [here](http://www.math.utah.edu/~davar/ps-pdf-files/SPDE.pdf)). In this case, a general solution can be written down using the fundamental solution of the heat equation given by


$$\Delta(x,t)=\frac{1}{\sqrt{4\pi k^2 t}}e^{-\frac{x^2}{4k^2 t}}$$


and then one has


$$\Phi(x,t)=\int dx'\Phi\_0(x')\Delta(x-x',t)+k^2\int dt'[\nu(t')+C]\Delta(x,t-t')$$


and we can easily compute


$$\langle\Phi(x,t)\rangle = \int dx'\Phi\_0(x')\Delta(x-x',t)+k^2C\int dt'\Delta(x,t-t')$$


$$\langle\Phi(x,t)\Phi(y,s)\rangle=\int dx'\Phi\_0(x')\Delta(x-x',t)\int dy'\Phi\_0(y')\Delta(y-y',s)+$$
$$k^2C\int dx'\Phi\_0(x')\Delta(x-x',t)\int ds'\Delta(y,s-s')+$$
$$k^2C\int dt'\Delta(x,t-t')\int dy'\Phi\_0(y')\Delta(y-y',s)+$$
$$k^4C^2\int dt'\Delta(x,t-t')\int ds'\Delta(x,s-s')+k^4\sigma^2\_0\int dt'\Delta(x,t-t')\Delta(y,s-t')$$


where I used the fact that $\langle\nu(s)\nu(t)\rangle=\sigma^2\_0\delta(t-s)$. It is interesting to note the simplest case $\Phi\_0=0$ and $C=0$ producing immediately


$$\langle\Phi(x,t)\rangle = 0$$


and


$$\langle\Phi(x,t)\Phi(y,s)\rangle=k^4\sigma^2\_0\int dt'\Delta(x,t-t')\Delta(y,s-t').$$


So, all higher order even correlation functions are given by products of the fundamental solution of the heat equation properly integrate in intermediate times.


