A calculation of dynamic polarizabilities of rovibrational states with vibrational quantum number $v=0-7$ and rotational quantum number $J=0,1$ in the 1s$sigma_g$ ground-state potential of HD$^+$ is presented. Polarizability contributions by transitions involving other 1s$sigma_g$ rovibrational states are explicitly calculated, whereas contributions by electronic transitions are treated quasi-statically and partially derived from existing data [R.E. Moss and L. Valenzano, textit{Molec. Phys.}, 2002, textbf{100}, 1527]. Our model is valid for wavelengths $>4~mu$m and is used to to assess level shifts due to the blackbody radiation (BBR) electric field encountered in experimental high-resolution laser spectroscopy of trapped HD$^+$ ions. Polarizabilities of 1s$sigma_g$ rovibrational states obtained here agree with available existing accurate textit{ab initio} results. It is shown that the Stark effect due to BBR is dynamic and cannot be treated quasi-statically, as is often done in the case of atomic ions. Furthermore it is pointed out that the dynamic Stark shifts have tensorial character and depend strongly on the polarization state of the electric field. Numerical results of BBR-induced Stark shifts are presented, showing that Lamb-Dicke spectroscopy of narrow vibrational optical lines ($sim 10$ Hz natural linewidth) in HD$^+$ will become affected by BBR shifts only at the $10^{-16}$ level.
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