On convergence of solutions of stochastic diffusion equations with unlimited increasing drift coefficients on finite segments (Q2726267)

The following equation is studied: NEWLINE\[NEWLINEdX_{n}^{x}(t)=n1_{[a,b]}(X_{n}^{x}(t)) dt+dw(t), \quad t>0,NEWLINE\]NEWLINE where \(1_{[a,b]}(x)\) is the indicator function of the segment \([a,b],\) \(w(t)\) is a Wiener process defined on the probability space \((\Omega, F, P)\), and \(X_{n}^{x}(0)=x.\) The limit behaviour of solutions of the equation is investigated as \(n\to +\infty.\) Weak convergence (in the uniform metric) of solutions \(X_{n}^{x}(t)\) to a Wiener process with reflecting barrier at the point \(0\) is proved for \(x\geq b.\) Weak convergence in the \(M_{1}\) metric of solutions \(X_{n}^{x}(t)\) to certain processes with discontinuous trajectories is proved in the case of \(a\leq x< b\) and in the case of \(x< a.\)NEWLINENEWLINEFor the entire collection see [Zbl 0956.00022].