= Ornstein-Uhlenbeck multiplicative amplification
{c}
{title2=$\gamma_n=\kappa\tau^2n^2/2$}
Let $d\sigma=-\sigma\,dt/\tau+\sqrt\kappa\,dW$, $\sigma_0=0$, and $\dot B=\sigma B$, $B_0>0$. The logarithm $\log(B_t/B_0)=\int_0^t\sigma_sds$ is a centered <Gaussian random variable> of <variance> $V(t)=\kappa\tau^2[t-2\tau(1-e^{-t/\tau})+\tau(1-e^{-2t/\tau})/2]$. Squaring the stochastic kernel $\sqrt\kappa\tau[1-e^{-(t-u)/\tau}]$ and integrating proves this formula. The <exponential moment of a Gaussian linear functional> gives $\mathbb E B_t^n=B_0^ne^{n^2V(t)/2}$ and the displayed long-time growth exponent. Positive moment growth with zero mean strain reflects the breadth of the <log-normal distribution>; it does not imply positive growth of the mean logarithm.
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