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

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


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.)
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