MOUVEMENTS BROWNIENS ASYMÉTRTQUES MODIFIÉS EN
DIMENSION FINIE ET OPÉRATEURS DIFFÉRENTIELS À COEFFICIENTS
DISCONTINUS
Michèle Mastrangelo
Mouloud Talbi
Abstract: We consider a partial differential equation of parabolic type on
(x,t) | = Lu(x,t), x E,t R+, | (1)
|
u(x,0) | = f(x), u(.,t)/ E = 0 | | |
where
![V](files/11.1/HTML/11.1.4.abs4x.png)
and
![W](files/11.1/HTML/11.1.4.abs5x.png)
being two subdomains of
![E](files/11.1/HTML/11.1.4.abs6x.png)
such that
![E = V U W U S,](files/11.1/HTML/11.1.4.abs7x.png)
![V /~\ W /= Ø](files/11.1/HTML/11.1.4.abs8x.png)
and
![S](files/11.1/HTML/11.1.4.abs9x.png)
being a
![2
C](files/11.1/HTML/11.1.4.abs10x.png)
-variety. The functions
![C](files/11.1/HTML/11.1.4.abs11x.png)
and
![D](files/11.1/HTML/11.1.4.abs12x.png)
are
![2
C](files/11.1/HTML/11.1.4.abs13x.png)
on
![E](files/11.1/HTML/11.1.4.abs14x.png)
,
![dy](files/11.1/HTML/11.1.4.abs15x.png)
is the surface-vector-measure on
![S](files/11.1/HTML/11.1.4.abs16x.png)
,
![A](files/11.1/HTML/11.1.4.abs17x.png)
is a function defined on
![S](files/11.1/HTML/11.1.4.abs18x.png)
which
will be precised later on,
![dSA](files/11.1/HTML/11.1.4.abs19x.png)
is a generalized drift,
![\~/](files/11.1/HTML/11.1.4.abs20x.png)
[resp.
![/_\](files/11.1/HTML/11.1.4.abs21x.png)
] is the classical gradient
[resp. Laplacian operator] on
![d
R](files/11.1/HTML/11.1.4.abs22x.png)
.
We give, via a modified skew Brownian motion, a stochastic resolution of (1) -
being
considered as a generalized infinitesimal generator - and we study the continuity properties of
the transition probability densities and of their derivatives at the neighbourhood of
2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;
Key words and phrases: -