Since the salt is molten, the only ions are $\ce{Ag+}$ and $\ce{NO3-}$, and both of them can only do a reduction reaction. Who goes to the anode? Also, there is no acid in the solution — only the salt — therefore the following reaction cannot occur, right?

$$\ce{ NO3- + 4H+ + 3e- -> NO + H2O}$$

So, what happens? The only possible reaction is the reduction of the $\ce{Ag+}$ to $\ce{Ag (s)}$. There is literally nothing else.

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    $\begingroup$ You can't have reduction without oxidation. Then again, who said they can only do a reduction reaction? $\endgroup$ – Ivan Neretin Feb 7 '19 at 9:22
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    $\begingroup$ Why oxygen shouldn't evolve? $\endgroup$ – Alchimista Feb 7 '19 at 9:58
  • $\begingroup$ @IvanNeretin the nitrogen and the silver already have their highest oxidation numbers. $\endgroup$ – angelo Feb 7 '19 at 10:49
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    $\begingroup$ True, but are they the only elements here? $\endgroup$ – Ivan Neretin Feb 7 '19 at 11:00
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    $\begingroup$ Well, if it were spontaneous, then AgNO3 would not exist in the first place, would it? $\endgroup$ – Ivan Neretin Feb 7 '19 at 12:50

In principle, a nitrate reduction in the presence or silver ions only would have to give silver oxide; but this decomposes between 200 and 300°C. So, perforce, you would ultimately get elemental silver from electrolyzing the nitrate melt.

However, that does not mean it's best. With a little additional heating, to 440°C, you get decomposition to elemental silver just with the heating.

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