Sankhya: The Indian Journal of Statistics

1996, Volume 58, Series A, Pt. 3, pp. 347--381

TRANSITION DENSITIES OF REFLECTING DIFFUSIONS

By

S. RAMASUBRAMANIAN, Indian Statistical Institute

SUMMARY.  Using a parametrix method, localization procedure and probabilistic arguments we construct the transition density function for nondegenerate diffusion process in a smooth domain, with oblique reflection at the boundary and with "killing" terms in the interior and on the boundary; (our method obtains the density also on the boundary). In the case of half space, a Gaussian type upper bound and minimality are established for the transition density. We also prove some auxiliary results concerning Green function and Poisson kernel for a second order parabolic operator with the mixed boundary conditions in certain domains.

AMS (1991) subject classification.  Primary 60J60, secondary 35K15, 35K20.

Key words and phrases. Fundamental solution for parabolic equations, oblique reflection, parametrix method, Green function, Poisson kernel, exit time, strong Markov property, boundary local time.

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