We have a centered Gaussian process $X_{t}$ where we have exact equality $$E[X_{t}X_{s}]=a_{1}-a_{2}|t-s|$$ for $|t-s|<\epsilon_{0}\ll \frac{a_{1}}{a_{2}}$ and $a_{i}>0$.

Q: I am curious if there is any other

concreteGaussian process $(Y_{s})_{s\in [0,\epsilon_{0}]}$ out there with the same exact covariance when $|t-s|<\epsilon_{0}$ for some $\epsilon_{0}>0$ (not asymptotical behaviour with error term, but exact equality).

It will be interesting if $Y_{t}$ is in terms of some known process like a functional of Brownian motion or a stationary solution of some SDE.

We are not concerned with $Y_{t}$ having different distribution than $X_{t}$(even though they do looked as Gaussian processes over $t\in [0,\epsilon']$ for $\epsilon'$ small enough). Our main concern is if such covariances have been studied in the literature or if we can devise one.

Some idea: start from $Y_{t}=\int_{0}^{t}f(r,t)dW_{r}$ and try to find a deterministic $f(r,t)$ with the desired covariance: by Ito isometry $\int_{0}^{s}f(r,s+h)f(r,s)ds=a_{1}-a_{2}h$.

**Our process**

Let $X_{\epsilon}(x)\sim N(0,\ln\frac{1}{\epsilon})$ with covariance:

For simplicity above we suppressed the $\epsilon$ and just let $X_{t}:=X_{\epsilon}(t)$.

Our particular process. Consider the hyperbolic measure $\lambda:=\frac{1}{y^{2}}dx dy$ in the upper half-plane and a White noise process W indexed by Borel sets of finite hyperbolic area:

$$\{A\subset \mathbb{H}: \lambda(A)<\infty; \sup_{(x,y),(x',y')\in A}|x-x'|<\infty\}$$

with covariance:

$$E[W(A_{1})W(A_{2})]:=\lambda(A_{1}\cap A_{2}).$$

Then let $X_{t}=W(V_{\epsilon}+t)$ for

$$V_{\epsilon}:=\{(x,y)\in \mathbb{H}: x\in [-1/4,1/4]\text{ and }max(2|x|,\epsilon)\leq y<1/2\}.$$