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To properly answer this question, you need to show us a real circuit with a capacitor across the supply.
There are various reasons a capacitor across a power supply might make sense. Protecting the supply from thermal noise created by a resistive load is way way down the list of likely explanations.
Probably the most common reason for a cap between power and ground is for it to act like a local high-speed reservoir. Put another way, the cap lowers the impedance of the supply locally at high frequencies. Ideal supplies have 0 impedance. Real ones don't of course. Also, the feed lines from the supply to where the power is consumed always add some impedance. Between these two issues, sudden load changes can cause voltage fluctuations in the local supply. Capacitance across the local supply reduces these fluctuations.
In the frequency-space view, a local bypass cap is the shunt for the high frequency current generated by the load. By shunting this high frequency current locally, it doesn't generate a voltage across the inevitable supply impedance.
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Is C2 here to protect the voltage source from thermal noise?
There's absolutely no chance that any level of thermal noise produced by a resistor will be sufficient to cause any reasonably designed power supply (voltage source) any problems whatsoever. Even badly designed power supplies will cope with this easily.
For instance if R1+R2 was 1 MΩ and they were heated up to 200 °C, the open circuit noise they would produce is 161 μV. That's a piddling amount of noise and power.
Also, consider this; a power supply is a practical voltage source and, will almost certainly have several, to hundreds, or thousands of micro-farads of capacitance already across its output terminals so, what do you think the presence of another capacitor is going to bring to the party?
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