That depends on how you define Steady state. If your derivatives are never actually zero, you will never have a fully perfect steady state, so your problem is impossible if the goal is a perfect steady state. Considering that x and y are spatial coordinates, one way of thinking of this is to find which position will give you the smallest possible variance, which, since your noise is zero mean gaussian, will actually be in the place where:
-ax +by^2 = 0
-cy+dx+w = 0
That is the same solution you would become had you solved the problem completely ignoring the noise. This is related to the Maximum likelihood estimation and it basically says "Considering that my data has zero mean Gaussian noise, which is the most likely solution between all solutions?", and this is the one where your mean "error" is zero and all that's left are the noise fluctuations.