I am just working my way through a talk from an Italian researcher and I don´t get one of his points. Let´s look at these two slides.
I don´t understand why he says on slide 2 that $u_t$ is a time dependent probability measure i.e density. Because in my opinion if $u_t$ were a probability density then the total derivate can´t include a source term i.e
$$ \frac{d u}{dt} = \frac{\partial u}{\partial t} - \frac {1}{2} \triangle u $$
Otherwise $\int_{\Omega} u(t,x)dx = 1$ is not fulfilled. But as I assume the researcher is right there must be any mistake in my considerations. Can someone help?
By time dependent density he just means the density u(t,x) is the probability to start from a particular initial point (t0, x0) and to arrive later at the ('uncertain') state (t,x). I don't know the exact context of the physical problem from your slides, but I don't see anything wrong in it. Your statement: $ \int_\Omega{u(t,x)dx}=1$ is correct but your statement for total derivate: $$\frac{du}{dt}=\frac{du}{dt}-\frac{1}{2}\delta u$$ seems wrong and is different than in the slide. The PDE from your slide is a Fisher-KPP type of PDE: $$\frac{du}{dt}=\frac{1}{2}\Delta u+\lambda u$$
Cheers.