Consider the heat kernel on the 2D-torus
$$ K(t,z-z_1):= \sum_{m \in \mathbb{Z}^2} \exp\Big(-(1-4\pi^2|m|^2)t +2\pi i m \cdot (z-z_1)\Big) $$
It satisfies the semigroup property $$ K(t_1+t_2,z_1-z_2)=\int_{\mathbb{T}^2} K(t_1,z_1-z) K(t_2,z_2-z) dz $$ Let now $\rho_\epsilon$ a mollifier and consider the regularized kernel $K * \rho_\epsilon$.
I'm expecting this new kernel to satisfy the semigroup property as well, but doing the calculation I have $$ I:=\int_{\mathbb{T}^2} \int \int K(t_1-\tau_1,z_1-z-x_1) \rho_\epsilon(\tau_1,x_1) K(t_2-\tau_2,z_2-z-x_2) \rho_\epsilon(\tau_2,x_2) d\tau_1 dx_1 d\tau_2 dx_2 dz $$ By using the semigroup property of $K$ and Fubini we have
$$ I=\int \int K(t_1+t_2-\tau_1-\tau_2,z_1-x_1-z_2+x_2) \rho_\epsilon(\tau_1, x_1) \rho_\epsilon(\tau_2,x_2) d\tau_1 dx_1 d\tau_2 dx_2=K *\rho_\epsilon *\rho_\epsilon(t_1+t_2,z_1-z_2) $$ This is strange as $K * \rho_\epsilon$ should be the kernel associated to the regularized stochastic heat equation i.e $$ \partial u^\epsilon=\Delta u^\epsilon + \xi^\epsilon $$ where $\xi^\epsilon := \xi * \rho_\epsilon$ is the regularization of the white noise and so it should satisfy the semigroup property. Did i have done some mistakes in my calculation?