This question is tangentially related to Probability a Brownian particle with an exponentially distributed lifetime hits a sphere before vanishing.

I have an infinitely long cylinder of some radius $r_c$ in some infinite three-dimensional volume. Inside this cylinder, Brownian particles come into existence at a rate $J \space \frac{particles}{meter^3}$, and have individual lifetimes given by an exponential distribution with rate parameter $\lambda$. Particles are also free to diffuse outside of the cylindrical volume unhindered.

Let $\omega(d)$ represent the particle density a Euclidean distance $d$ from the closest point $p_i$ along the long-axis of the cylinder. (This is not the primary question, is there a simple analytic expression for $\omega(d)$?).

Now, I create infinite plane, which acts as a reflecting boundary for the Brownian particles created inside the cylinder, and that intersects the cylinder somewhere s.t. the long-axis of the cylinder has an angle $\alpha$ to the plane. If $\alpha = \frac{\pi}{2}$, making the long-axis of the cylinder normal to the plane, is our function for particle density vs. displacement from the long-axis of the cylinder, $\omega(d)$, the same as before? In other words, are there cylinder-plane orientations where the density distribution of the particles (now executing reflected Brownian motion) is indistinguishable from the density distribution without the reflecting barrier being present?

If not, can we say anything meaningful regarding what happens near the plane?


A similar question might be the following:

Imagine one infinitely long cylinder of radius $r_2$ enclosing another (parallel) infinite cylinder of radius $r_1$ in $R^3$. Hold the outer cylinder's surface at a temperature $T_2$ and the inner cylinder's surface at a temperature $T_1 \neq T_2$. Let $H(t,\theta,q)$, $r_1 \leq q \leq r_2$ be a heat kernel for the volume between the surfaces of the two cylinders.

How does $H(t,\theta,q)$ change if we insert an infinitely thin perfectly insulating plane that bisects the two cylinders s.t. the angle $\alpha$ between the plane and the long-axes of either cylinder is $\alpha = \frac{\pi}{2}$? Do we now need a $z$ component, relative to the distance from the plane, i.e. $H(t,\theta,q,z)$, in order to properly characterize the heat kernel?


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