I'm given the bounded Markovian stochastic process $-n\leq X(t)\leq n,\, \langle X(t)\rangle=0$ and try to find an upper bound for the probability that the difference between $X(t)$ and the inital value $X(0)$ up to time $T\to\infty$ is zero, i.e. the value remained unchanged until $T\to\infty$, ideally in terms of just the expected value of $X(t)$: $$\operatorname{P}\left(\lim_{T\to\infty}\frac{1}{T}\int_{0}^Tdt\,\bigl(X(t)-X(0)\bigr)^2=0\right)\leq F\bigl(\langle X(t)\rangle\bigr).$$ I tried to apply well-known inequalities like Markov's or Chebychev's, but non are really valid in my case. Alternatively the probability can equivalently expressed as $$\operatorname{P}\left(\lim_{T\to\infty}\frac{1}{T}\int_{0}^Tdt\,\langle X(0)X(t)\rangle=1\right)\leq F\bigl(\langle X(0)X(t)\rangle\bigr)$$ so stated instead in terms of the autocorrelation.
Additional thoughts: In some sense this probablility seems similar to the probability of fixation from theoretical biology $$\operatorname{P}\left(X=1,t\to\infty\mid X=X_0,t=0\right)$$ which let's me question if there is an underlying Fokker-Planck equation to the problem
More additional stuff: One obvious statement about the probability is that it's upper bounded by the argument itself: $$\phi=\operatorname{P}\left(\lim_{T\to\infty}\frac{1}{T}\int_{0}^Tdt\,\langle X(0)X(t)\rangle=1\right)\leq \lim_{t\to\infty}\langle X(0)X(t)\rangle$$ because we can write the RHS as $$\lim_{t\to\infty}\langle X(0)X(t)\rangle=\phi+(1-\phi)A,$$ where $A$ is zero if $X(t)$ becomes decorrelated or some finite number if not. But this bound is rather loose and I wondered if there's a better one.