Does your car stop running after a short time, and start again after you wait a few minutes?

No fuel supply check fuel filter and fuel pump

Well, lets put it this way. First, you didn't bother to mention YEAR, MAKE, MODEL, or ENGINE SIZE.
This is kinda nice to know if you want answers that actually apply to your vehicle.

Second, I would not put it past the manufacturers to eventually come up with such a thing as an electric oil pump. They will most likely figure out a way to control and monitor it electronically. That way, they can have the excitement and pleasure of adding an entire array of fault codes to the allready mind-boggling list of possible OBD codes. Of course, they will claim it is for fuel efficiency, but it will really be to make it so you have to spend an entire years' salary to diagnose and fix the thing (and the engine that will have been destroyed) when it quits working.

However, I have not yet seen one of these animals.

Hi, The Escape is a great vehicle, compact and great on fuel mileage. As with all vehicles, it has a few concerns that are more common specific to this vehicle. I've seen engine misfire concerns, usually caused by tiny fractures/cracks in the ignition coil housings or due to the rubber boot part of the coil deteriorating and allowing moisture to short it out. A scan tool can be used to retrieve codes that would determine which cylinder is affected.

Another concern is misfires caused by a break in the wiring between the computer (PCM) and the coils or fuel injectors. The harness constantly flexes when the vehicle accelerates and decelerates.

Another concern is the lower control arm bushing rubber deteriorating. When this happens the alignment will change and can cause tire wear.

The Escape has a short wheel base and is prone to tire wear. It's important to rotate tires regularly (7500 - 10,000) miles. If the tires wear (cup) enough the vehicle will exibit an audible roaring noise while driving.

If maintained properly, the Escape is a great vehicle.

Note the type of conditions that this happens in like high humidity,rain completely dry or does any of that make a difference? I would suspect a faulty crankshaft position sensor that is possibly cracked or corroded wires to it. If there is oil leaking onto the plug occassionally this can disrupt the signal as well.