I need circuit able to detect the minimum value of a positive signal
If you can deal with offsets, then you can use a modified LP filter.



Link to simulation.


The left graph = square wave with 1% pulses, -peak = 0 V, +peak = 5 V.
The middle graph = min(the square wave) + offset
The right graph = max(the square wave) - offset
The 1 Ω resistor is there for simulation purposes.


The offset is non-linear, but fairly linear in the region that it might be used. In this particular setup, the offset is about 500 mV, which stems from the forward voltages of the diodes. These two offsets can easily be removed in software with a simple addition and subtraction. 

The input impedance is pretty low, so you might want to use an op-amp acting as a buffer right before it goes into the diodes. Otherwise you will load the noise, so what you measure will show a less value than if your measurement hadn't been there. 



If you, for some reason cannot perform addition and subtraction in software, or if the non-linearity from the forward voltage of the diodes are either unknown, messy, or just too non-linear, then you can use this setup instead:



Link to simulation.


The graph's are the same as above
The 1 Ω resistors are there for simulation purposes.


Here you are getting high impedance for free and proper max and min values. If you are going to go with this setup, then make sure the op-amp's output is in the range of the expected noise voltage + offset. So if you are going to use an LM358 as the op-amp (not something I recommend, because it's slow), then the negative voltage source for the op-amp would be at least -1 V for the negative output. Because the diode will have a forward voltage of maximum 1 V, and the most negative part of your noise will be 0 V I presume. And the positive voltage source would have to be 5 V (maximum noise) + 1 V (diode) + 1.5 V (LM358 isn't rail-to-rail) = 7.5 V. 



Edit: Here's a link for the first one with proper noise, and here's the link for the second one with proper noise. 

Here's an image of them two side by side.



As you can see, they both have pros and cons. Op-amp = messier to set up, no op-amp = offsets.