Stochastic calculus and degenerate boundary value problems
type: Item
instance of: scholarly article
title: Stochastic calculus and degenerate boundary value problems
zbMATH Open document ID: 0780.35023
author: Patrick Cattiaux
publication date: 1993-02-08T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/74966
review text: 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 \(A= {1\over 2} \sum^ m_{i=1} Y^ 2_ i+Y_ 0\), 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 \(Y_ i\) (\(0\leq i\leq m\)), 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.
Mathematics Subject Classification ID: 35H10
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 118277
zbMATH Keywords: diffusions with a boundary condition
zbMATH Keywords: hypoellipticity
zbMATH Keywords: resolvent
zbMATH Keywords: diffusion process
zbMATH Keywords: Venttsel's condition
zbMATH Keywords: oblique derivative
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: smoothness
zbMATH Keywords: stochastic representation
zbMATH Keywords: stochastic calculus of variations
zbMATH Keywords: invariant measures
MaRDI profile type: Publication
OpenAlex ID: W2030378797
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DOI: 10.5802/AIF.1302
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 118277