Consider the problem of finding a family of probability measures $(\mu_t)_{t \geq 0}$ on $(\mathbb{R}, \mathscr{B}( \mathbb{R}))$ such that $$ \int_{\mathbb{R}} \varphi(x) \mu_t(dx) = \int_{\mathbb{R}} \varphi(x) \mu_0(dx) + \int_0^t ds \int_{ \mathbb{R} } (A_s \varphi)(x) \mu_s(dx) $$ for all real-valued smooth functions $\varphi$ with compact support in $\mathbb{R}$. Here, $$ (A_s \varphi)(x) := a(s,x) \varphi'(x) + b ( s, x) \varphi''(x) $$ for sufficiently nice functions $a$ and $b$.
I have come across the following reasoning on why the solution to the equation is not unique:
If $A$ has an invariant probability measure $\mu$ then any family of probability measures $(\mu_t)_{t \geq 0}$ with $\mu_0 = \mu$ solves the equation above.
What does it mean for the differential operator $A$ to have an invariant probability measure $\mu$? Is it always possible to find such a measure?