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:
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 ${\rm D}\mathcal P_t$?