Manuscript Psy02-B@2 · Psy02-B-v2.tex · proposition · cfd06e2e

Proposition: Windowed carrier variance

Track A3: concentration and temporal fluctuations › Bath covariances and filters · lines 729–753 · proof included or linked

Psy02-B@2#prop:codevariance

For the named Markov process,

Var (\int_{0}^{T} k (t) X_{t} d t) = \int_{0}^{T} \int_{0}^{T} k (t) k (s) C_{X} (t, s) d t d s \leq \norm k_{L^{1}}^{2} .

The displayed covariance is generally nonstationary.

Proof. For $t \geq s$ , the adjoint evolution multiplies the compensated $X$ by $e^{- r (t - s)}$ . The symmetrized regression expression becomes

Tr [X Λ_{R} (t - s) \frac{X ρ _{R} ( s ) + ρ _{R} ( s ) X}{2}] = e^{- r (t - s)} Tr (P ρ_{R} (s)),

because $X^{2} = P$ . Integrate the centered covariance. The dilation observable has norm at most one, giving the bound.

Displayed formulas in this passage

unlabeled · line 732
$Var (\int_{0}^{T} k (t) X_{t} d t) = \int_{0}^{T} \int_{0}^{T} k (t) k (s) C_{X} (t, s) d t d s \leq \norm k_{L^{1}}^{2} .$
unlabeled · line 744
$Tr [X Λ_{R} (t - s) \frac{X ρ _{R} ( s ) + ρ _{R} ( s ) X}{2}] = e^{- r (t - s)} Tr (P ρ_{R} (s)),$