Geodesic
Parametrix, heat kernel asymptotics, and regularized trace of the diffusion semigroup
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and topology. II, Tome 271 (2010), pp. 241-258

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We establish a relationship between a path integral representation of the heat kernel and the construction of a fundamental solution to a diffusion-type equation by the parametrix method; this relationship is used to find the coefficients of a short-time asymptotic expansion of the heat kernel. We extend the approach proposed to the case of diffusion with drift and obtain two-sided estimates for the regularized trace of the corresponding evolution semigroup. 
@article{TM_2010_271_a16,
     author = {S. A. Stepin},
     title = {Parametrix, heat kernel asymptotics, and regularized trace of the diffusion semigroup},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {241--258},
     publisher = {mathdoc},
     volume = {271},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2010_271_a16/}
}
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S. A. Stepin. Parametrix, heat kernel asymptotics, and regularized trace of the diffusion semigroup. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and topology. II, Tome 271 (2010), pp. 241-258. http://geodesic.mathdoc.fr/item/TM_2010_271_a16/