Stochastic calculus and degenerate boundary value problems

Patrick Cattiaux

Annales de l'institut Fourier (1992)

  • Volume: 42, Issue: 3, page 541-624
  • ISSN: 0373-0956

Abstract

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Consider the boundary value problem (L.P): ( h - A ) u = f in D , ( v - Γ ) u = g on ∂ D where A is written as A = 1 / 2 ∑ i = 1 m Y i 2 + Y 0 , and Γ is a general Venttsel’s condition (including the oblique derivative condition). We prove existence, uniqueness and smoothness of the solution of (L.P) under the Hörmander’s condition on the Lie brackets of the vector fields Y i ( 0 ≤ i ≤ m ), for regular open sets D with a non-characteristic boundary.Our study lies on the stochastic representation of u and uses the stochastic calculus of variations for the ( A , Γ ) -diffusion process in D ‾ . Applications to the decomposition of C ∞ ( D ‾ ) , and to invariant measures are also discussed.

How to cite

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Cattiaux, Patrick. "Stochastic calculus and degenerate boundary value problems." Annales de l'institut Fourier 42.3 (1992): 541-624. <http://eudml.org/doc/74966>.

@article{Cattiaux1992,
abstract = {Consider the boundary value problem (L.P):$(h-A)u=f$ in $D$, $(v- \Gamma )u=g$ on $\partial D$ where $A$ is written as $A= \{1/ 2\} \sum ^ m_\{i=1\} Y^ 2_ i+Y_ 0$, and $\Gamma $ is a general Venttsel’s condition (including the oblique derivative condition). We prove existence, uniqueness and smoothness of the solution of (L.P) under the Hörmander’s condition on the Lie brackets of the vector fields $Y_ i$ ($0\le i\le m$), for regular open sets $D$ with a non-characteristic boundary.Our study lies on the stochastic representation of $u$ and uses the stochastic calculus of variations for the $(A,\Gamma )$-diffusion process in $\overline\{D\}$. Applications to the decomposition of $C^ \infty (\overline\{D\})$, and to invariant measures are also discussed.},
author = {Cattiaux, Patrick},
journal = {Annales de l'institut Fourier},
keywords = {diffusions with a boundary condition; hypoellipticity; resolvent; diffusion process; Venttsel's condition; oblique derivative; existence; uniqueness; smoothness; stochastic representation; stochastic calculus of variations; invariant measures},
language = {eng},
number = {3},
pages = {541-624},
publisher = {Association des Annales de l'Institut Fourier},
title = {Stochastic calculus and degenerate boundary value problems},
url = {http://eudml.org/doc/74966},
volume = {42},
year = {1992},
}

TY - JOUR
AU - Cattiaux, Patrick
TI - Stochastic calculus and degenerate boundary value problems
JO - Annales de l'institut Fourier
PY - 1992
PB - Association des Annales de l'Institut Fourier
VL - 42
IS - 3
SP - 541
EP - 624
AB - Consider the boundary value problem (L.P):$(h-A)u=f$ in $D$, $(v- \Gamma )u=g$ on $\partial D$ where $A$ is written as $A= {1/ 2} \sum ^ m_{i=1} Y^ 2_ i+Y_ 0$, and $\Gamma $ is a general Venttsel’s condition (including the oblique derivative condition). We prove existence, uniqueness and smoothness of the solution of (L.P) under the Hörmander’s condition on the Lie brackets of the vector fields $Y_ i$ ($0\le i\le m$), for regular open sets $D$ with a non-characteristic boundary.Our study lies on the stochastic representation of $u$ and uses the stochastic calculus of variations for the $(A,\Gamma )$-diffusion process in $\overline{D}$. Applications to the decomposition of $C^ \infty (\overline{D})$, and to invariant measures are also discussed.
LA - eng
KW - diffusions with a boundary condition; hypoellipticity; resolvent; diffusion process; Venttsel's condition; oblique derivative; existence; uniqueness; smoothness; stochastic representation; stochastic calculus of variations; invariant measures
UR - http://eudml.org/doc/74966
ER -

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