TITLE: Variant of parabolic maximum principle: $\partial_t u\ge \Delta_x u + b\cdot\nabla_x u-u \;\Rightarrow\; u\ge e^{-t}\inf_y u(0,y)$
QUESTION [5 upvotes]: Let $b\colon[0,\infty)\times\Bbb R^d\to\Bbb R^d$ some smooth bounded function and $u\colon[0,\infty)\times\Bbb R^d\to\Bbb R$ a smooth function with
$$ \partial_t u\ge \Delta_x u + b\cdot\nabla_x u-u. \label{1}\tag{1}$$
A paper I've been reading recently then quotes the maximum principle as
$$ u(t,x)\ge e^{-t}\inf_{y\in\Bbb R^d}u(0,y) \quad \text{for all $(t,x)\in[0,\infty)\times\Bbb R^d$.} \label{2}\tag{2}$$
No reference is provided.
I thought I was familiar with the parabolic maximum principle, but I have never seen this variant before. Can anyone give me a hint where I could find it in the literature? Would it need an additional growth condition on $u$?
I'm a bit puzzled by the fact that \eqref{2} is similar to what Gronwall's inequality would imply if one omitted the terms in \eqref{1} with derivatives with respect to $x$.

REPLY [2 votes]: Here is a probabilistic solution. Let $L(t)$ be the operator which sends a function $u$ to $ \Delta_x u+ b(t,\cdot ) \cdot \nabla_x u$. This family of operators generates the diffusion process $$dX_t = dB_t + b(t,X_t)dt,$$
which is inhomogeneous Markov. Let $p_{s,t}(x,y)$ be the associated transition densities, i.e., they are defined by the relation $p_{s,t}(x,y) dy = P(X_t \in dy|X_s=x)$, and they solve the adjoint equation $\partial_t p_{s,t}(x,y) = \Delta_y p_{s,t}(x,y) -\text{div}_y\big(b(t,y)  p_{s,t}(x,y)\big)$ with initial data $\delta_x$ (a more abstract way to think about $p_{s,t}$ can be found in the comments below).
Then the solution of the equation $\partial_tv = L(t)v $ can be written as $v(t,x) = \int_{\Bbb R^d} v(0,y)p_{0,t} (x,y)dy$. Thus the solution of the equation $\partial_t w = L(t) w - w$ is given by $w(t,x)=e^{-t}\int_{\Bbb R^d} w(0,y)p_{0,t} (x,y)dy$.
Now consider any $u$ satisfying $\partial_t u \geq L(t) u -u$. Define $f(x,t) := \partial_t u - L(t)u+u\ge 0.$ Then $\partial_t u = L(t)u-u+f$, so by the Duhamel principle we can write $$u(t,x) = e^{-t}\int_{\Bbb R^d} u(0,y) p_{0,t}(x,y)dy+\int_0^t\int_{\Bbb R^d} e^{-(t-s)}p_{s,t}(x,y) f(s,y)\;dy\; ds.$$ The second integral is non-negative because $f$ is non-negative. The first integral is bounded below by $e^{-t} \inf_{\Bbb R^d} u(0,\cdot)$ because $p_{0,t}$ is a probability measure, completing the proof.
As Arctic Char points out in the comments above, $u(0,\cdot)$ should be reasonable enough so that these integrals are well defined, i.e. at-worst exponential growth will suffice to show that it can be integrated against $p_{0,t}$.
Sorry if this is unclear, I can try to update later.