The mean-variance model in portfolio theory does suffer from being static. This is why it has been extended to dynamic portfolio optimization, which captures investors' behavior of rebalancing their portfolios periodically. They rebalance because, as time varies, individual assets returns and volatilities change and diverge from their measurements they had during the initial allocation.
It is not normal to rebalance every minute though. Investors hold their portfolios for much longer horizons so that they rebalance monthly, quarterly, or semi-annually. After all, assets hardly change over the course of one minute, but can change dramatically by the end of a month's time.
Since it is hard to tell what the portfolio will be like in the future, allocations that should be optimal the next quarter, for example, have to be estimated from current and historical data. Upon reaching each rebalancing period, you repeat this process using preceding historical data that has accumulated up to that specific point in time in the form of rolling windows. This is the typical design of forecasting models, where volatility is typically the basis for re-weighting the portfolio, mainly due to volatility being much more easy to forecast than expected returns.
In summary, in dynamic portfolio optimization, the static model is just repeated (re-estimated) for each new rebalancing period as just described using new historical data.