Coupling methods for multidimensional diffusion processes. (English) Zbl 0686.60083

Chen, Mufa; Li, Shaofu

Ann. Probab. 17, No. 1, 151-177 (1989).

Probability measures \(P^{x,y}\), \(x,y\in R^ d\), on \(C([0,\infty);R^{2d})\) are considered, such that the canonical process \(Z(t)=(X(t),Y(t))\), \(t\geq 0\), is the \(P^{x,y}\)-diffusion process with the matrix of diffusion coefficients \(a(x,y)=\left( \begin{matrix} a(x)\\ c(x,y)^*\end{matrix} \begin{matrix} c(x,y)\\ a(y)\end{matrix} \right)\) and the vector of drift coefficients \(b(x,y)=(b(x)\quad \quad b(y))'\). Criteria are found for the success of coupling, i.e. \[ P^{x,y}\{T<\infty \}=1\quad and\quad P^{x,y}\{X(t)=Y(t),\quad t\geq T\}=1, \] where \(T=\inf \{t\geq 0:\) \(X(t)=Y(t)\}\). Some examples and applications are also studied.

Reviewer: B.Grigelionis

MSC:

60J60	Diffusion processes
60J65	Brownian motion
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60J45	Probabilistic potential theory
60J70	Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)

Keywords:

multidimensional diffusion processes; coupling