For a (real valued, finite variance, centered) stationary process $X_t$ on $\mathbb R$, the auto-correlation function $k(\tau) = \mathbb E(X_{t+\tau}-X_t)^2$ and its inverse Fourier transform $\rho$, the spectral measure of $X$ ($k(\tau)= \int e^{is\tau}\rho(ds)$) could be any (even) positive definite function $k$ and (even) bounded positive measure $\rho$, as exemplified by Gaussian processes.

I wonder which of these are obtainable as auto-correlations and/or spectral measures for this special class of Feller processes: $X_t$ is Markov, $dX_t/dt=c$ a.e. ($c>0$) and $Prob(X_{t+\Delta t}\in dy |X_t=x) = \nu(x,dy)\Delta t + o(\Delta t)$, where the jump rate $\nu(x,dy)$ vanishes on $[x,\infty)$. In other words, the infinitesimal generator $$Af(x):=lim_{\Delta t \to 0}\mathbb E([f(X_{t+\Delta t})-f(X_t)]/\Delta t|X_t=x)$$ has the form $$Af(x)=cf'(x)+\int_{-\infty}^x \nu(x,dy)(f(y)-f(x))$$and satisfies what is needed so that $X_t$ has finite variance and mean $0$.

In particular, is it possible that $\rho$ has a density, $\rho(d\lambda) = r(\lambda) d\lambda$, satisfying $r(\lambda) \propto|\lambda|^m $ for small $\lambda$, for any value of the exponent $m>-1$ ? I think a solution should use the resolvant $R(\lambda)=(A-\lambda I)^{-1}$, acting on the identity function $j(x)=x$ near $\lambda=0$, but I don't know how to relate relevant properties of the resolvant with the particular form of $A$.

This question is related to finding self-similar statistical solutions of (inviscid) Burgers equation $u_t+(u^2/2)_x=0$. We know the Markov (in $x$) property is preserved, and the question is whether the initial $u_0$ can be any self-similar Gaussian (generalized) process, either derivative of fractional Brownian, or even more irregular.