A~probabilistic approach tо a~nonlinear differential equation on a~Riemannian manifold
Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 2, pp. 336-341
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We investigate the minimal solution of the problem \begin{gather*} Lu=u^\alpha в D,
u=f на O, \end{gather*}
where $1\le\alpha\le2$, $D$ is an open subset f a Riemannian manifold, O is a regular relatively, open subset of $\partial D$, and $f$ is a mapping from $\partial D$ to $[0,\infty]$ which is continuous on $O$ and vanishes on $\partial D\setminus O$. An explicit formula for such a solution is given in terms of the $(L,\alpha)$-superdiffusion.
Keywords:
nonlinear differential equations, Riemannian manifolds, Markov process, minimal positive solution.
Mots-clés : $L$-diffusion
Mots-clés : $L$-diffusion
@article{TVP_1997_42_2_a6,
author = {E. B. Dynkin},
title = {A~probabilistic approach t{\cyro} a~nonlinear differential equation on {a~Riemannian} manifold},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {336--341},
publisher = {mathdoc},
volume = {42},
number = {2},
year = {1997},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TVP_1997_42_2_a6/}
}
TY - JOUR AU - E. B. Dynkin TI - A~probabilistic approach tо a~nonlinear differential equation on a~Riemannian manifold JO - Teoriâ veroâtnostej i ee primeneniâ PY - 1997 SP - 336 EP - 341 VL - 42 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TVP_1997_42_2_a6/ LA - en ID - TVP_1997_42_2_a6 ER -
E. B. Dynkin. A~probabilistic approach tо a~nonlinear differential equation on a~Riemannian manifold. Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 2, pp. 336-341. http://geodesic.mathdoc.fr/item/TVP_1997_42_2_a6/