A path-integral formulation of the two-dimensional Hubbard model is used in which scattering of electrons across the magnetic Brillouin zone by spin fluctuations (umklapp processes) is included. With this formulation, we have calculated the spin-wave velocity ${\mathit{c}}_{\mathit{s}}$ and the specific heat ${\mathit{C}}_{\mathit{v}}$ for the half-filled-band case. For the quadratic form of the Hubbard model due to Schrieffer, we obtain ${\mathit{c}}_{\mathit{s}}$=1.5${\mathit{c}}_{0}$ in the large-U limit (U is the intrasite Coulomb repulsion, ${\mathit{c}}_{0}$= \ensuremath{\surd}2 J is the spin-wave velocity in linear-spin-wave theory, J=4${\mathit{t}}^{2}$/U is the superexchange interaction, and t is the hopping integral for nearest neighbors). Our result is in good agreement with various numerical calculations based on the Heisenberg model, e.g., ${\mathit{c}}_{\mathit{s}}$=1.22${\mathit{c}}_{0}$ by Liu and Manousakis [Phys. Rev. B 40, 11 437 (1989)], with use of the variational Monte Carlo technique. Our present calculation differs from previous path-integral calculations, which lead to ${\mathit{c}}_{\mathit{s}}$\ensuremath{\sim}t in the large-U limit. A general free-energy formula, which includes all kinds of fluctuation, is obtained. At low temperature, the specific heat in the large-U limit is given by ${\mathit{C}}_{\mathit{v}}$\ensuremath{\simeq}0.51(T/J${)}^{2}$. The present calculation can also be applied to the Hubbard model written in other quadratic forms, in one of which the saddle-point approximation leads to the Hartree-Fock solution and ${\mathit{c}}_{\mathit{s}}$=${\mathit{c}}_{0}$ and ${\mathit{C}}_{\mathit{v}}$=1.15(T/J${)}^{2}$ in the large-U limit.