Consider $\partial_{t}u=\partial_{xx}u$ with Neumann boundary condition $u_{x}(0,t)=u_{x}(1,t)=0$ and initial condition $u(x,0)=f(x)\geqslant0$. Then up to time $T$, the maximal value of $u$ should be at the ``boundary''. So how about the stochastic equation $\partial_{t}u=\partial_{xx}u+\dot{W}(t,x)$? Here $\dot{W}$ is intepreted by Walsh's approach. Does it still hold? Or we can make a coefficient before $\dot{W}$ to make the noise under certain control, and then make it possible to find the maximal value on the boundary? Or we can at least find some property like this for other variant heat equations? Any comments?
