Let $(p_t)_{t >0}$ be the Gaussian heat kernel on $\mathbb R^d$ and $(P_t)_{t >0}$ its induced semi-group, i.e., $$ \begin{align} p_t (x) &:= (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}}, \quad t>0, x \in \mathbb R^d, \\ P_t f (x) &:= \int_{\mathbb R^d} p_t ( x- y) f(y) \, \mathrm d y. \end{align} $$

Let $\nu$ be a Borel probability measure on $\mathbb R^d$ with finite first moment. We denote by $\ell_\nu$ the probability density function of $\nu$. For $R>0$, let $B_R$ be the open ball centerted at $0$ with radius $R$. Let $B_R^c := \mathbb R^d \setminus B_R$ and $$ I_R := \int_{B_R^c} |x| (P_t \ell_\nu) (x) \, \mathrm d x, \quad R >0. $$

Is there an upper bound of $I_R$ in terms of $R, \nu, t$?

Thank you so much for your elaboration!