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On reflecting diffusion processes and Skorokhod decompositions. (English) Zbl 0767.60074

Summary: Let \(G\) be a \(d\)-dimensional bounded Euclidean domain, \(H^ 1(G)\) the set of \(f\) in \(L^ 2(G)\) such that \(\nabla f\) (defined in the distribution sense) is in \(L^ 2(G)\). Reflecting diffusion processes associated with the Dirichlet spaces \((H^ 1(G),{\mathcal E})\) on \(L^ 2(G,\rho dx)\) are considered, where \[ {\mathcal E}(f,g)={1\over 2}\int_ G\sum^ d_{i,j=1}a^{ij}{\partial f\over\partial x_ i}{\partial g\over\partial x_ j}\rho dx,\quad\text{for }f,g\in H^ 1(G), \] \({\mathcal A}=(a^{ij})\) is a symmetric, bounded, uniformly elliptic \(d\times d\) matrix-valued function such that \(a^{ij}\in H^ 1(G)\) for each \(i\), \(j\), and \(\rho\in H^ 1(G)\) is a positive bounded function on \(G\) which is bounded away from zero. A Skorokhod decomposition is derived for the continuous reflecting Markov processes associated with \((H^ 1(G),{\mathcal E})\) having starting points in \(G\) under a mild condition which is satisfied when \(\partial G\) has finite \((d-1)\)-dimensional lower Minkowski content.

MSC:

60J60 Diffusion processes
60J65 Brownian motion
60J55 Local time and additive functionals
60J35 Transition functions, generators and resolvents
31C25 Dirichlet forms
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