Question: With early 2013 cameras, it's generally accepted that using contrast detection autofocus on SLRs (ie live view on *most* SLRs) is something which is really suitable only for static, or close to static, subjects due to the slow focusing speed. On the other hand, the current best of breed mirrorless cameras (the Olympus OM-D E-M5 often being quoted here) have autofocus systems which are significantly quicker at achieving a focus lock, if not quite being up to the performance of phase detection autofocus systems.

My understanding is that both systems are using the same technology, so why is it the case that mirrorless cameras have much quicker autofocus systems than SLRs in Live View mode? Is it the case that the *lenses* for mirrorless systems are optimized for quick CDAF performance, and if so, what are those optimizations?

Edit: in response to one of the answers, I'm not thinking about how cameras like the Nikon 1 series or the Canon EOS M have quicker autofocus due to the use of phase detection elements in the sensor; I understand how using an entirely different technology will improve things - what I'm interested in here is how some manufacturers have made contrast detection autofocus much quicker than is apparently possible in SLRs. Similar reasoning applies to the Sony SLT series as that's again using PDAF rather than CDAF.

Answer: 
Major reason is that the **DSLR lenses are optimized for Phase Detection**. Every component of the lens is tailored towards quick movement and stopping the glass in precisely picked moment. Contrast detection on the other hand works best with stepper motors capable of quickly switching directions so that you can move lenses inside back and forth looking for highest contrast on the image. 

Phase detection knows straight away where the focus is and how much should the lenses move to achieve perfect focus. Contrast detection needs to "find it". This forces different engineering solutions in lenses manufacturing.

Also the DSLRs are usually made with **live view as an afterthought**. Most of the manufacturers think of it as an manual focus assist. They don't try to create fast AF for video as they know that pro videophotographers usually relay on manual focusing (not that they have any choice with such performance...) while photographers usually use viewfinders anyway. Hence they rarely have dedicated processors for contrast detection, and if the main CPU is occupied with focusing - it doesn't perform as well as dedicated unit.

Also the statement in your question **isn't entirely true**. Sony's **SLT** DSLRs have AF that's much faster in live view than Mirrorless, as SLT design basically allows camera to fully utilize it's PDAF (Phase Detection Auto Focus) sensors all the time during live view. So you get DSLR-quality AF with Live view at the same time. Also the older generation of Sony DSLRs offered **Quick AF Live View** - which never flipped the main mirror up for Live View - instead it used secondary sensor in viewfinder allowing DSLR-quality AF for a price of additional delay before shooting photo (mirror had to flip up in order to capture the photograph).
It is simply a case of optimization. Sensors used on those cameras have been optimized to perform efficient autofocus. Some use contrast-detect while others even have phase-detect sensors.

In the case of the OM-D E-M5, it uses Contrast-Detect which is basically a loop that measures local contrast, moves the lens and repeats until the maximum contrast is found. This latest generation of sensors performs this iteration at 240 Hz with processing to analyze the data correspondingly fast. Olympus is so confident in their ability to make this faster than DSLRs that they are not considering on-chip phase-detection.

Nikon on the other hand choose to use Phase-Detection which is why they can focus so fast. That system requires much fewer iterations - like on a DSLR using the OVF - because the data collected informs the camera of the direction and amount of misfocus. This is not precise enough to do it in a single bound but gets there rather quickly. One thing to note is that the splitters for on-chip phase-detection tiny which is why these systems do not work as quickly as those used by DSLRs in low-light. Nikon 1 cameras switch to Contrast-Detect AF when light is low.

As you mention, lenses have an influence on autofocus speed. The key is in the loop described above. With contrast-detect, the camera continuously moves the lens in tiny increments. In contrast (no pun intended), phase detection does the majority of focusing with larger movement. Motors, controls and feedback from the lens have to be tuned for each particular case. In the case of modern DSLR lenses, the ultra-sonic motor used in many lenses works against them.

For example, Canon introduced lenses with linear motors (STM) with the EOS M while their high-end lenses feature ultra-sonic ones (USM). Along the release, the explicitly stated that the new motors are designed to work better with autofocus of the EOS M which uses contrast-detect to fine-tune AF after phase-detection places focus in the ballpark area.