this is a very nice way to prove it! I don't know why I was expecting something quite involved, but this is a straightforward consequence of the integration by parts. By the way is this a set of notes of some course? If yes from which university? I also live in Italy and I didn't know Pratelli

@AbdelmalekAbdesselam Thank you so much for the references! Actually I am more used to work on $S'(\mathbb R)$, this was just a curiosity that I had while reading Da Prato's book. I wanted to know if both approaches were somehow equivalent. Thanks again!

Why do we end up integrating wrt to $t$ alone, what about the $\vec{x}$? Sorry if this is a stupid question, I know very little about this stuff.

Ey Willie sorry for bothering you again, in the last step, how can I apply the Gronwall inequality? I am not familiar with the application to PDEs.

This may be a matter of notation but is there a proper meaning for $:x^{\otimes n}:$? Is it the same as $x\diamond \cdots \diamond x$? But this doesn't make sense since $x$ is not a random variable, it's actually a tempered distribution.

Hey Martin, I recently came back to this question. Sorry for not giving you any feedback, I've been pretty busy with courses and trying to figure out what my thesis will be about haha. I find your answer quite interesting, but I still feel dubious. In particular regarding the concept of Wick tensor. Namely when I write $\langle :x^{\otimes n}:, f^{\otimes n}\rangle$ I know that this is equivalent to write $I(f)\diamond \cdots \diamond I(f)$, where the stochastic integrals (random variables) are evaluated at the fixed $x$ (chance parameter).

thank you so much Willie! This is really helpful

@WillieWong Yes, the $A_n$ are actually diagonal, and depend only on $t$. The equations are coupled just through the function $\mathbf b$.

@WillieWong Still, there's a way in which I can get rid of the linear part, and we may assume as well that $c$ will depend on $u$ and $x_1,\cdots,x_n$ and it's bounded and continuously differentiable as many times as needed. Can I do something in that case?

@WillieWong what a shame! I hope Kato's result could still be applied. Thanks a lot for the feedback!

@WillieWong more in particular $b_i(\cdots)= c_i(\mathbf u)+\sum_{n=1}^K \xi_n(t)x_n$ where $c_i$ is continuous differentiable as many times as needed, and $\xi_n$ is the $n$-th Hermite function

@WillieWong so we can assume that $\mathbf b=(b_1,\cdots,b_N)$ (each component) is continuous differentiable w.r.t $\mathbf u$ as many times as needed with bounded first derivatives; it's linear in $x_i, i=1,\cdots, K$; it's a rapidly decreasing function in $t$.

@WillieWong thank you so much for the comments, I asked that because I've received an answer (on a related question) saying that if the equations on the system were coupled just through the non-homegeneity $b$ (as in my question above) then the method of characteristics could be applied. Tomorrow I'll give you more information regarding the function $b$. Thanks again!