Coupling methods for multidimensional diffusion processes (Q1262623)

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.