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In the context of functional inequalities for Markov semigroups $(\mathcal P_t)_{t\ge0}$, what is one denoting by $\nabla\mathcal P_tf$? For example, I've found the following assumption in this paper:

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Above, $(\mathcal P_t)_{t\ge0}$ is the semigroup given by $$\mathcal P_t(x,B):=\operatorname P\left[\Phi_t(\;\cdot\;,x)\in B\right]$$ and $\phi$ is a Lipschitz continuous Fréchet differentiable function. However, there is no assumption ensuring that $\mathcal P_t$ preserves Fréchet differentiability. So, how does one understand the differential ${\rm D}\mathcal P_t$?

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    $\begingroup$ A probabilistic way to understand $D \mathcal{P}_t$ is using a Bismut-Elworthy-Li formula; see, e.g., Theorem 10.3 of hairer.org/notes/Imperial.pdf Note that this formula involves the underlying process (10.1) and its derivative (10.2). $\endgroup$ Jun 8, 2020 at 5:49
  • $\begingroup$ @NawafBou-Rabee Maybe I don't understand the result, but isn't Theorem 10.3 an identity for ${\rm D}\mathcal P_t$ for the particular flow given by the solution of $(10.1)$? My question is more basic: What kind of differential is ${\rm D}\mathcal P_t$? Is it the ordinary Fréchet derivative? I mean, as indicated in the paper, $x\mapsto\Phi_t(\omega,x)$ is Fréchet differentiable for all $(\omega,t)$. Does this imply that $\mathcal P_tf$ is Fréchet differentiable for all Fréchet differentiable $f$? $\endgroup$
    – 0xbadf00d
    Jun 8, 2020 at 6:15
  • $\begingroup$ @NawafBou-Rabee Or do we need to understand it in terms of the "modulus of gradient" (local Lipschitz constant)? $\endgroup$
    – 0xbadf00d
    Jun 8, 2020 at 6:22
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    $\begingroup$ $D \mathcal{P}_t$ is a Fréchet derivative defined by directional derivatives of $\mathcal{P}_t \phi$. $\endgroup$ Jun 8, 2020 at 6:44
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    $\begingroup$ A nice reference is the following book by Cerrai: springer.com/gp/book/9783540421368. The idea is to use a stochastic representation of $\mathcal{P}_t f(x)$ in terms of an expectation involving the underlying stochastic process, show the process is mean-square differentiable with respect to the initial point, differentiate the stochastic representation and then interchange the derivative with the expectation as in math.stackexchange.com/questions/217702/… $\endgroup$ Jun 8, 2020 at 8:07

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