Consider a sample-continuous stochastic process $\left\{ X_t \right\}_{t \in T}$ s.t. each $X_t$ is real-valued and $$\int_\Omega | X_t(\omega) | ^p \, \mathrm{d} P(\omega)< \infty$$ for all $1 \leq p < \infty$. We denote $\mathbb{E}[X_t]=\mu_t$.

Assume that $$\mathbb{E}\left[\, \left|X_t - \mu_t \right|^2 \, \middle| \, \,|X_0-\mu_0|^2 > \delta \right] > \delta t,$$ i.e. that the paths with an offset in the beginning are expected to "spread out". Is there a nice lower bound for $\text{Var}[X_t]$ in terms of $\text{Var}[X_0]$?