A limit theorem for singular stochastic differential equations

TL;DR

This paper investigates the weak limits of solutions to a class of stochastic differential equations with singular drift terms, revealing that the limits can be Bessel, skew Bessel, or mixed Bessel processes.

Contribution

It establishes a limit theorem for SDEs with singular drift involving delta functions, characterizing possible limit processes.

Findings

Limits include Bessel and skew Bessel processes

Convergence depends on the parameters of the drift

Provides a framework for understanding singular SDEs

Abstract

We study the weak limits of solutions to SDEs \[dX_n(t)=a_n\bigl(X_n(t)\bigr)\,dt+dW(t),\] where the sequence $\{a_n\}$ converges in some sense to $(c_- 1\mkern-4.5mu\mathrm{l}_{x<0}+c_+ 1\mkern-4.5mu\mathrm{l}_{x>0})/x+\gamma\delta_0$. Here $\delta_0$ is the Dirac delta function concentrated at zero. A limit of $\{X_n\}$ may be a Bessel process, a skew Bessel process, or a mixture of Bessel processes.