Spiking and collapsing in large noise limits of SDEs (Q6103973)

The paper under review explores the limit of one-dimensional diffusion processes \(X^\gamma\) as \(\gamma \rightarrow +\infty\). These processes satisfy the stochastic differential equation: \[ dX_t^\gamma=b(X_t^\gamma)dt+\sigma(X_t^\gamma)dW_t. \] Here, \(b\) represents a Lipschitz continuous drift term, \(\sigma\) represents a Lipschitz continuous diffusion coefficient, and \(W\) denotes a standard Wiener process. The main results of the paper demonstrate that as the noise intensity increases, the solutions of the stochastic differential equations display a collapsing behaviour. This can be interpreted as convergence to pure jump processes, reminiscent of a metastability phenomenon. Additionally, the limiting jump process is accompanied by a spike process.