Stochastic Heat Equation - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:07:29Z http://mathoverflow.net/feeds/question/87979 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87979/stochastic-heat-equation Stochastic Heat Equation Riccardo.Alestra 2012-02-09T10:34:55Z 2012-02-10T19:18:27Z <p>Given the heat equation:</p> <p>$$\partial_{t}{\varPhi(x,t)}=k^2\partial_{xx}{\varPhi(x,t)}$$</p> <p>with the boundary conditions:</p> <p>$$\Phi(x,0)=\Phi_0$$</p> <p>and a Neumann boundary condition of the kind:</p> <p>$${\partial_{x}}{\Phi(0,t)=\nu(t)+C}$$</p> <p>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 $\Phi(L,t)$?</p> <p>Thanks in advance</p> http://mathoverflow.net/questions/87979/stochastic-heat-equation/88131#88131 Answer by Jon for Stochastic Heat Equation Jon 2012-02-10T19:18:27Z 2012-02-10T19:18:27Z <p>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 <a href="https://www.math.lsu.edu/cosa/2-2-04%255B140%255D.pdf" rel="nofollow">this</a> and a more general theory of stochastic pde due to John Walsh (a tutorial can be found <a href="http://www.math.utah.edu/~davar/ps-pdf-files/SPDE.pdf" rel="nofollow">here</a>). In this case, a general solution can be written down using the fundamental solution of the heat equation given by</p> <p>$$\Delta(x,t)=\frac{1}{\sqrt{4\pi k^2 t}}e^{-\frac{x^2}{4k^2 t}}$$</p> <p>and then one has</p> <p>$$\Phi(x,t)=\int dx'\Phi_0(x')\Delta(x-x',t)+k^2\int dt'[\nu(t')+C]\Delta(x,t-t')$$</p> <p>and we can easily compute</p> <p>$$\langle\Phi(x,t)\rangle = \int dx'\Phi_0(x')\Delta(x-x',t)+k^2C\int dt'\Delta(x,t-t')$$</p> <p>$$\langle\Phi(x,t)\Phi(y,s)\rangle=\int dx'\Phi_0(x')\Delta(x-x',t)\int dy'\Phi_0(y')\Delta(y-y',s)+$$ $$k^2C\int dx'\Phi_0(x')\Delta(x-x',t)\int ds'\Delta(y,s-s')+$$ $$k^2C\int dt'\Delta(x,t-t')\int dy'\Phi_0(y')\Delta(y-y',s)+$$ $$k^4C^2\int dt'\Delta(x,t-t')\int ds'\Delta(x,s-s')+k^4\sigma^2_0\int dt'\Delta(x,t-t')\Delta(y,s-t')$$</p> <p>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</p> <p>$$\langle\Phi(x,t)\rangle = 0$$</p> <p>and</p> <p>$$\langle\Phi(x,t)\Phi(y,s)\rangle=k^4\sigma^2_0\int dt'\Delta(x,t-t')\Delta(y,s-t').$$</p> <p>So, all higher order even correlation functions are given by products of the fundamental solution of the heat equation properly integrate in intermediate times.</p>