Consider the piecewise-deterministic Markov process on $\mathbf{R}$ which

- moves according to the vector field $\phi (x) = 1$,
- experiences events at rate $\lambda(x) = 1$, and
- at events, jumps according to the transition kernel

\begin{align} Q (x \to dy) = \frac{\exp(- y - \exp(-y))}{\exp( - \exp (-x))} \, \mathbf{I} [y \leqslant x] dy \end{align}

i.e. the location is resampled according to a Gumbel distribution, conditioned to be to the left of $x$.

It can be shown that this process admits the Gumbel distribution as its stationary measure, that is, the measure with density $G(x) = \exp(- x - \exp(-x))$. Moreover, its generator is given by

\begin{align} \mathcal{L} f (x) = f'(x) + \int_{-\infty}^x [f(y) - f(x)] Q (x \to dy). \end{align}

It is worth remarking that the process is non-reversible, and hence its generator is not self-adjoint.

For reference, the generator's adjoint in $L^2 (dG)$ is given by

\begin{align} \mathcal{L}^* f (x) = - f'(x) + \exp (-x) \int_x^{\infty} [f(y) - f(x)] Q^* (x \to dy) \end{align}

where

\begin{align} Q^* (x \to dy) = \exp(- (y - x)) \mathbf{I} [y \geqslant x] dy, \end{align}

which can be derived by computing the time-reversal of the original process.

Now, I'm interested in studying the eigenfunctions of the generator, i.e. solutions to

\begin{align} \mathcal{L} f (x) = \lambda f (x) \end{align}

After some calculations, I can simplify this equation to

\begin{align} f''(x) + (\exp ( -x) - 1 ) f'(x) = \lambda \left\{ \exp (-x) f(x) + f' (x) \right\} \end{align}

which can be put into `quasi-Sturm-Liouville' form as

\begin{align} \frac{d}{dx} \left( G(x) f'(x) \right) &= \lambda G(x) \left\{ f' (x) + \exp (-x) f(x)\right\}, \end{align}

and even into the form

\begin{align} \frac{d}{dx} \left( G(x) f'(x) \right) &= \lambda \exp (-x) \frac{d}{dx} \left( \exp ( - \exp (-x)) f(x) \right), \end{align}

which could conceivably be useful.

Anyways, although this all looks very reminiscent of Sturm-Liouville theory, it's not *quite* there, in as much as the term with the eigenvalue $\lambda$ don't just involve $f$, but also its derivatives.

**My main questions are thus as follows:**

Is there a name for this type of system? i.e. S-L-type structure, but where the $\lambda$-term depends on the derivatives of $f$ as well. If so, I'd appreciate relevant references.

Are there any other techniques which might be of use in trying to solve this eigensystem? Ideally there will be solutions of the form

\begin{align} f(x; \lambda_n) = \{ \text{polynomial of degree } n \} \times \text{fixed function}, \end{align}

but this might be a bit (/very) optimistic on my part.

A related consideration (which I'm in the process of doing the calculations for) is to carry out a spectral decomposition of $\mathcal{L} \mathcal{L}^*$ and $\mathcal{L}^* \mathcal{L}$. Given that these will be self-adjoint, some of the theory might make life a little easier; on the other hand, both operators involve double integrals, so it may be time-consuming. I'll update with details once I've had a go of that. Any advice on that would also certainly be welcome.