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Like this: Using the fact that $u(t,x)=\int f(y) p(y,t,x)dy$, for bounded $f$ and due to $p$ having bounded derivatives, I can use dominated convergence to interchange differentiation and integration. As $p$ solves the heat equation I get a zero integral and thus $\partial_t u-1/2\Delta u=0$?
My question is "Where can I read more about ...", so it would be great if you could give some references for learning and understanding what you just wrote :-)
Thank you for this comprehensive answer! I still fail to grasp how to use dominated convergence to show that $u$ satisfies the heat equation and would really appreciate more details or a reference. Thanks!
Interesting! While I think I found the statements about the smoothing properties of the semigroup $P_t$ in Cerrai, I cannot find anything about $P_tf$ being the solution to a PDE if $f\in B_b$. Is it in Cerrai?
Thanks for bearing with me. I marked my proof as wrong and added a new one. The remaining question is: How exactly do you show the $L^2$ convergence for the $\sigma$ integral (see also my new proof)? Do you think the sufficient convergence in probability could be easier to prove, i.e. without $L^2$ convergence?
