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For a (real valued, finite variance) stationary process $X_t$ on $\mathbb R$ with $\mathbb EX_t=m$, the auto-correlation function $k(\tau) = \mathbb E[(X_{t+\tau}-m)(X_t-m)]$ 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 $$Af(x):=lim_{t \to 0^+}t^{-1}\mathbb E^x[f(X_t)-f(x)]$$ 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.

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 $R(\lambda)=(\lambda I-A)^{-1}$, 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.

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