Unveiling out-of-time-order correlators from stochastic operator variance

We consider the dynamics generated by a fluctuating Hamiltonian. We introduce the concept of stochastic variance of operators and find their equation of motion. We show that the stochastic operator variance (SOV) is related to the out-of-time-order correlator (OTOC) introduced in the theory of quantum chaos to define a quantum Lyapunov exponent $\lambda$. Our findings are illustrated in a stochastic Lipkin-Meshkov-Glick (sLMG) Hamiltonian undergoing energy dephasing, where the action of noise changes the stability region compared to the noiseless LMG, as demonstrated from the SOV-OTOC relation.