Even with the current DSOs it is no problem to utilize your PC for more processing and analysis. Even the cheap Chinese DSOs these days have USB interfaces, so you can get the data to the PC.
Mid-range and upwards DSO vendors have PC interfaces since ages. I regularly import data from my DSO into GNU octave for further processing. High-End vendors even deliver their own Matlab oscilloscope interfaces, e.g. Tek and LeCroy. AFAIR Agilent even has some instruments with a build-in PC board and Matlab running inside the instrument.
(where is only analog part
Only? ONLY? The analog part is the hard part! The (digital) processing stuff is child's play compared to building a good analog frontend.
And "good" means not only 200 MHz or 500 MHz bandwidth for your 2 GHz sampler (good luck with that), but also 300 V CAT II safety (as opposite to a 5 V input range and a mayor fire incident at 10 V), linearity, defined input impedance, temperature stability, low aging, etc.
What people just don't understand when looking at the cost for a DSO is that you don't pay for the parts. It is not the (color) LCD screen, the knobs on the front plate or the CPUs and FPGAs inside that make them expensive. Oscilloscope makers can get that stuff rather cheap (not what you would have to pay for qty 1). It is the effort that went into designing it.
Every now and then you see some company coming up with a DIY DSO kit. All I have seen were junk as a DSO. Nice if you want to get some soldering practice, but utter crap as a measurement instrument. Not enough bandwidth, a cheap OpAmp as analog front-end, non or barely non input protection.
Oh, and good luck with calibrating your 2 GHz sampler. How many k$ (kilo-Dollars) are you willing to invest in measurement and calibration equipment?