Elliptic principal eigenfunction analysis for Langevin dynamics with a varying source term

Question

Consider the Kolmogorov forward equation for a Langevin dynamic: $$\DeclareMathOperator{\Div}{div} \begin{cases} \dfrac{\partial}{\partial t} f = \Delta f + \Div(f\nabla V)\\ \\ \displaystyle\int_{\mathbb{R}^d} f(x) dx = 1, \end{cases}$$ it's well-known that this system admits an invariant density function $\nu(x) = Ce^{-V(x)}$, which is exactly the solution to the equation $$\Delta \nu + \Div(\nu \nabla V) = 0.$$

Now if I add a source term in the evolution equation, and assume the source term is monotonic function of the original potential: $$ \frac{\partial}{\partial t} f = \Delta f + \Div(f\nabla V) + m(V) f, $$ where $m: \mathbb{R} \to \mathbb{R}$ is monotonically decreasing. This system is generally not a evolution of probabilistic density, or to say, $$ \frac{\partial}{\partial t} \int_{\mathbb{R}^d} f(x,t) dx \neq 0. $$ BTW, we can still define a probabilistic density $p$ proportional to $f$: $p(x,t) = \frac{f(x,t)}{\int_{\mathbb{R}^d} f(y,t) dy}$. It can be shown that the limit distribution $\lim_{t \to \infty}p(t,x) = \mu(x)$ is exactly the principal eigenfunction to the elliptic eigenvalue problem: $$ \Delta \mu + \Div(\mu\nabla V) + m(V) \mu = \lambda \mu. $$ Here raise a question: is $\mu$ sharper than $\nu$? Or to say, whether there exists a monotonically decreasing function $\gamma : \mathbb{R} \to \mathbb{R}^+$ such that $\frac{\mu(x)}{\nu(x)} = \gamma(V(x))$ or not?

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Why to ask this question, let's take $\nu$ as the initial condition to the parabolic equation $$\frac{\partial}{\partial t} \int_{\mathbb{R}^d} f(x,t) dx \neq 0$$ and then will arrive at $$\frac{\partial}{\partial t} f(x,0) = m(V) f(x,0)$$ take $t = \epsilon$, $$f(x,\epsilon) \approx (1 + \epsilon m(V))\nu(x).$$ Thus I wonder if this property will preserve in a long time, as well as the limit situation.