Hi everyone,
Do you know how to check the existence of solution for non-linear equation set, especially for the case with random coefficients?
For example, we have the following equations:
f1(A1, x) = 0
f2(A2, x) = 0
where f1(.) and f2(.) are non-linear functions.
Moreover, if A1 and A2 are continuous random variables, and x is a scalar (one dimension; or more generally, the dimensionality of x is less than the number of equations), can we claim that "with probability 1, the solution of these equation set does not exist"? What type of techniques can we use?
The intuition we believe it should be true is that if (under certain condition) each function f_i(A_i, x) restricts the variable x to a curve (the degree of freedom is reduced). If A1 and A2 are continuous random variables, then the intersection of these curves is empty with probability 1.
But it seems to be difficult to describe it in a rigorous manner. It may be related to "noise prevent singularity".
Thank you very much!
You can look up the book by Hasminskii from Wayne State University
Try this Stochastic Stability of Differential Equations, but there are also some others by other authors. You need something on stability of stochastic DEs. I think there are some basic theorems on stability of linear and linear with parametric noise systems, but in general nonlinear case you may not find anything. Usually it is required to coefficient to be Lipshiz and some conditions for noise intensity to be fulfilled,
I would suggest this: if you can prove existence of a solution for constant values, then it is really a question of what values for A1 and A2 do not allow a solution. If you know a range of values, then you would simply prove existence for each edge of the segment. If you are concerned for poles that may lie within the range of values, then you will need to find those critical points and analyze around them.
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