Let \[ LU=\frac{\partial u(t,x)}{\partial t}+Hu(t,x),\quad t\in [0,1), x\in \mathbb R^d, \] where \(Hu(t,x)= a^{ij}(t,x)\partial_{x_i}\partial_{x_j}u(t,x)+ b_i(t,x)\partial_{x_i}u(t,x).\) The authors formulate rather general conditions on functions \(a^{ij}(t,x), b_i(t,x)\) that allow to prove the existence of a family \(\mu_t\) of probability measures on \(\mathbb R^d\) satisfying in the weak sense the parabolic equation \(L^*\mu=0\) and the condition \(\mu_0=\nu\) for a probability measure \(\nu\) under an additional assumption about existence of a suitable Lyapunov function.