Ornstein-Uhlenbeck multiplicative amplification (${\gamma }_n=\kappa {\tau }^2{n}^2/2$)

Let $d\sigma =-\sigma dt/\tau +\sqrt{\kappa }dW$, ${\sigma }_0=0$, and $\dot{B}=\sigma B$, $B_0>0$. The logarithm $\log (B_t/B_0)={\int }_0^t{\sigma }_{s}ds$ 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 $\mathbb{E}{B}_t^n=B_0^n{e}^{n^{2}V(t)/2}$ 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.

The weighted strain density in Ornstein-Uhlenbeck multiplicative amplification evolves under ${\mathcal{L}}_{n}p=(\kappa /2)p^{\prime \prime }+{\tau }^{-1}(\sigma p{)}^{\prime }+n\sigma p$. Setting $p=e^{-{\sigma }^2/(2\kappa \tau )}\psi$ eliminates its first derivative. Shifting and scaling to $x=(\sigma -\kappa n{\tau }^2)/\sqrt{\kappa \tau }$ gives ${\psi }_{xx}+[1+\kappa n^2{\tau }^3-2\gamma \tau -x^2]\psi =0$. The quantum harmonic oscillator levels yield the displayed eigenvalues. Its ground state is the dominant positive mode; higher sign-changing modes may occur in a transient expansion.