= Weighted Neumann heat kernel with constant drift
{title2=$K_\beta(x,y,t)$}
On $(0,L)$, the <differential operator> $\mathcal L=\partial_x^2+\beta\partial_x=e^{-\beta x}\partial_x(e^{\beta x}\partial_x)$ with homogeneous <Neumann boundary conditions> is <self-adjoint> in the <weighted inner product> with weight $e^{\beta x}$. Its <eigenfunctions> and nonnegative decay rates are
$$
\phi_0=1,\quad\lambda_0=0,\qquad
\phi_k=e^{-\beta x/2}\left[\cos(q_kx)+\frac\beta{2q_k}\sin(q_kx)\right],\quad
\lambda_k=q_k^2+\frac{\beta^2}4,\quad q_k=\frac{k\pi}L.
$$
Their squared <norms> are $N_0=(e^{\beta L}-1)/\beta$ and $N_k=L[1+\beta^2/(4q_k^2)]/2$. The <Sturm-Liouville eigenfunction expansion> gives
$$
K_\beta(x,y,t)=\sum_{k=0}^\infty\frac{e^{-\lambda_kt}\phi_k(x)\phi_k(y)}{N_k}.
$$
This kernel acts against the measure $e^{\beta y}dy$, not Lebesgue measure alone. It is symmetric in $x,y$; the transition density against Lebesgue measure is $e^{\beta y}K_\beta(x,y,t)$.
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