Suppose I have found a (finite) solution $x$ to an ill-posed problem, e.g. \begin{equation} b = A x \end{equation} where $A$ is a $N\times N$ matrix with a large condition number, $b$ is a vector collecting $N$ observations. I was wondering if it could be analytically proven why adding a random noise to $b$ of the form \begin{equation} b_i \rightarrow \tilde b_i = b_i + \varepsilon \eta_i\,, \quad \varepsilon \ll \vert b_i \vert\,, \quad \langle \eta_i \rangle = 0\,, \quad \langle \eta_i \eta_j \rangle = \delta(i-j) \end{equation} causes the reconstructed solution $\tilde x$ of the problem \begin{equation} \tilde b = A \tilde x \end{equation} to be divergent or substantially different than $x$.
Suppose now that a regulation to the matrix $A$ of the form \begin{equation} A \rightarrow A + \lambda \mathbb{1} \end{equation} is applied. How and why does the above argument change?