If you look at a noise-driven damped harmonic oscillator, the autocorrelation of “oscillator state at time t1” and “oscillator state at time t2” cycles positive and negative at the oscillator frequency [if it’s not overdamped], but with an envelope that gradually decays to zero when t1 and t2 get far enough apart from each other.
This whole thing is time-symmetric—knowing the state at time 0 is unhelpful for guessing the state at very positive timestamps AND unhelpful for guessing the state at very negative timestamps.
But the OP question was about fixing an initial state and talking about later times, so I was talking in those terms, which is more intuitive anyway. I.e., as time moves forward, the influence of the initial state gradually decays to zero (because the wave is damped), while meanwhile the accumulated influence of the noise driver gradually increases.
If you look at a noise-driven damped harmonic oscillator, the autocorrelation of “oscillator state at time t1” and “oscillator state at time t2” cycles positive and negative at the oscillator frequency [if it’s not overdamped], but with an envelope that gradually decays to zero when t1 and t2 get far enough apart from each other.
This whole thing is time-symmetric—knowing the state at time 0 is unhelpful for guessing the state at very positive timestamps AND unhelpful for guessing the state at very negative timestamps.
But the OP question was about fixing an initial state and talking about later times, so I was talking in those terms, which is more intuitive anyway. I.e., as time moves forward, the influence of the initial state gradually decays to zero (because the wave is damped), while meanwhile the accumulated influence of the noise driver gradually increases.