Elementary properties of the Dirichlet heat kernel for symmetric Markov processes and their applications
DOI:
https://doi.org/10.19136/jobs.a11n32.6567Keywords:
Dirichlet heat kernel, symmetric strong Markov process, Dirichlet transition density, exit time, killed process, mild solution, blow upAbstract
In this diffusion article some basic properties of the Dirichlet heat kernel are demonstrated. In probability theory this mathematical object is the transition density of a killed Markov process. In this work, symmetric strong Markov processes that could be discontinuous are considered. Among the basic properties demonstrated are: continuity, symmetry, and the Chapman-Kolmogorov equation. An important application to the theory of non-autonomous semilinear reaction-diffusion equations with Dirichlet boundary conditions is also presented. Diffusion in this case is the generator of the associated Markov process, which is known to be a non-local integro-differential operator.
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