I got this advice for calculating a partial differential equation with drift, that said I should integrate the approximation of the previous time step against the heat kernel.
Now I wonder, what does it mean to integrate a function against the heat kernel, as I originally thought integrating $f$ against $g$ means $\int f dg$ which does not make sense to me in this case.
The heat kernel is $$\phi(x,y,t)=\frac{1}{\sqrt{2\pi\sigma^2t}}\exp\bigg\{-\frac{(x-y)^2}{2\sigma^2t}\bigg\}$$ It is the solution of the PDE $$\frac{\partial \phi}{\partial t}=\frac{\sigma^2}{2}\frac{\partial^2 \phi}{\partial x^2}$$ at time $t>0$ with initial condition $\phi(x,y,0)=\delta(x-y)$. In my experience, integrating $f(x)$ against the heat kernel means computing $$\int_\mathbb{R}\phi(x,y,t)f(y)dy$$ But to be safe, you should compare this with your context.