Weighted Neumann heat kernel with constant drift ($K_{\beta }(x,y,t)$)

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 areTheir 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 givesThis 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 $e^{\beta y}{K}_{\beta }(x,y,t)$.

For $L>0$, constant $\beta >0$ and $\mathrm{Re}p>0$, on $(0,L)$ put $b=\beta /2$, $\kappa =\sqrt{p+b^2}$ with the principal square root, andThese satisfy $(p-{\partial }_x^2-\beta {\partial }_x)u_p=(p-{\partial }_x^2-\beta {\partial }_x)v_p=0$ and the left and right homogeneous Neumann boundary conditions, respectively. The weighted Wronskian givesThe first derivative jumps by $-e^{-\beta y}$, so the resolvent integrates against $e^{\beta y}dy$. Its time Laplace transform interpretation is ${\mathcal{R}}_p={\int }_0^{\infty }{e}^{-pt}{K}_{\beta }(t)dt$. The factor $p$ records the zero eigenvalue and constant stationary eigenfunction.

For $u_t=u_{xx}+\beta u_x$ with $u_x(0,t)=g(t)$ and $u_x(L,t)=h(t)$, two applications of integration by parts against a Neumann eigenfunction give the boundary forcing $e^{\beta L}{\varphi }_k(L)h-{\varphi }_k(0)g$. Solving the resulting modal linear differential equation givesThe endpoint derivatives are interior limits: the short-time singularity prevents evaluating each homogeneous boundary derivative before the time integral and infinite sum. The constant eigenfunction gives the weighted-mass law ${\partial }_t{\int }_0^L{e}^{\beta x}udx=e^{\beta L}h-g$.