Lower confirmation time schemes: any requirement between confirmation time and the probability to find a block during that confirmation time? Bitcoin, the first blockchain-based cryptocurrency, set a 10 minute confirmation interval time; LiteCoin set a 2 minute confirmation interval time; DogeCoin set it at 1 minute; QuarkCoin set it at 30 seconds; Ethereum set it at 12 seconds.

Is there any serious analysis about the stability of the blockchain when the confirmation time is reduced?

To be more specific: any requirement between confirmation time and the probability to find a block during that confirmation time ( specially when confirmation time is going to zero) ?
The lower confirmation time schemes for mining a block will have a pronounced effect on orphaned blocks, with an inversely proportional relationship prevailing.

Generally this can be explained quite simply; because shorter confirmation times for blocks will mean numerous simultaneous competing chains lengths. Of these competing chains, one will eventually be integrated into the ledger when the block height is greater than the competing chains' heights. However, the chains which are rendered obsolete (ie the shorter chains not integrated into the ledger) will mean orphaned blocks are much more likely as confirmation time decreases. 

Take an example where it's 1 minute confirmation times. If there's 4 competing chains each 5 blocks in height/length then all it takes is for one of the chains' blocks to be confirmed in, say, 10 seconds, followed by the next confirmation in 10 seconds (which is not a rare event) which will make the block height 7. Therefore, the other chains of height 5 (or perhaps 6 if a block was solved) will be rendered orphaned, and all Txns therein will be orphaned.

There is a balance that needs to be struck between confirmation time and orphaned blocks, and from my experience with altcoins, the 2 minute mark (as employed by Litecoin) is the functional limit. Ethereum has a much, much shorter confirmation time because it's a totally different kind of altcoin. However for POW coins like LTC/BTC, a ~20s confirmation time is infeasible.