IV: 08, 76-76, LNM 124 (1970)
DELLACHERIE, Claude
Un lemme de théorie de la mesure (
Measure theory)
A lemma used by Erdös, Kesterman and Rogers (
Coll. Math., XI, 1963) is reduced to the fact that a sequence of bounded r.v.'s contains a weakly convergent subsequence
Keywords: Convergence in norm,
SubsequencesNature: Original proofs
Retrieve article from Numdam
VII: 17, 172-179, LNM 321 (1973)
MEYER, Paul-André
Application de l'exposé précédent aux processus de Markov (
Markov processes)
This paper is devoted to results of J.F. Mertens on optimal stopping and strongly supermedian functions for a right Markov process (
Zeit. für W-theorie, 26, 1973), which are shown to be closely related to those of the preceding paper
716 on general gambling houses. An interesting result of Mokobodzki is included, showing that the extreme points of the convex set of all balayées of a given measure $\lambda$ are the balayées of $\lambda$ on sets
Comment: See related papers by Mertens in
Zeit. für W-theorie, 22, 1972 and
Invent. Math.,
23, 1974. The original result of Mokobodzki appeared in the
Sémin. Théorie du Potentiel, 1969-70
Keywords: Excessive functions,
Supermedian functions,
RéduiteNature: Exposition,
Original proofs Retrieve article from Numdam
VII: 21, 210-216, LNM 321 (1973)
MEYER, Paul-André
Note sur l'interprétation des mesures d'équilibre (
Markov processes)
Let $(X_t)$ be a transient Markov process (we omit the detailed assumptions) with a potential density $u(x,y)$. Let $\mu$ be the measure whose potential is the equilibrium potential of a set $A$. Then the distribution of the process at the last exit time from $A$ is given by $$E_x[f\circ X_{L-}, 0<L<\infty]=\int u(x,y)\,f(y)\,\mu(dy)$$ This formula, due to Chung, is deduced under minimal duality hypotheses from a general formula of Azéma, and a well-known theorem on Revuz measures
Keywords: Equilibrium potentials,
Last exit time,
Revuz measuresNature: Exposition,
Original proofs Retrieve article from Numdam
XIII: 31, 371-377, LNM 721 (1979)
DELLACHERIE, Claude
Inégalités de convexité pour les processus croissants et les sousmartingales (
Martingale theory)
Several inequalities concerning general convex functions are classical in martingale theory (e.g. generalizations of Doob's inequality) and the general theory of processes (e.g. estimates on dual projections of increasing processes). The proof of such inequalities given here is slightly more natural than those in Dellacherie-Meyer,
Probabilités et Potentiels B, Chapter VI
Keywords: Martingale inequalities,
Convex functionsNature: Exposition,
Original proofs Retrieve article from Numdam
XIX: 28, 314-331, LNM 1123 (1985)
LE GALL, Jean-François
Sur le temps local d'intersection du mouvement brownien plan et la méthode de renormalisation de Varadhan (
Brownian motion)
The normalized self-intersection local time of planar Brownian motion was shown to exist by Varadhan (Appendix to
Euclidean quantum field theory, by K.~Symanzik, in
Local Quantum Theory, Academic Press, 1969). This is established anew here by a completely different method, using the intersection local time of two independent planar Brownian motions (whose existence was established by Geman, Horowitz and Rosen,
Ann. Prob. 12, 1984) and a sequence of dyadic decompositions of the triangle $\{0<s<t\le1\}$
Comment: Later, Dynkin, Rosen, Le Gall and others have shown existence of a renormalized local time for the multiple self-intersection of arbitrary order $n$ of planar Brownian motion. A good reference is Le Gall,
École d'Été de Saint-Flour XX, Springer LNM 1527
Keywords: Brownian motion,
Local times,
Self-intersectionNature: Original proofs Retrieve article from Numdam
XIX: 29, 332-349, LNM 1123 (1985)
YOR, Marc
Compléments aux formules de Tanaka-Rosen (
Brownian motion)
Several variants of Rosen's works (
Comm. Math. Phys. 88 (1983),
Ann. Proba. 13 (1985),
Ann. Proba. 14 (1986)) are presented. They yield Tanaka-type formulae for the self-intersection local times of Brownian motion in dimension 2 and beyond, establishing again Varadhan's normalization result (Appendix to
Euclidean quantum field theory, by K.~Symanzik, in
Local Quantum Theory, Academic Press, 1969). The methods involve stochastic calculus, which was not needed in
1928Comment: Examples of further work on this subject, using stochastic calculus or not, are Werner,
Ann. I.H.P. 29 (1993) who gives many references, Khoshnevisan-Bass,
Ann. I.H.P. 29 (1993), Rosen-Yor
Ann. Proba. 19 (1991)
Keywords: Brownian motion,
Local times,
Self-intersectionNature: Original proofs Retrieve article from Numdam
XX: 35, 543-552, LNM 1204 (1986)
YOR, Marc
Sur la représentation comme intégrales stochastiques des temps d'occupation du mouvement brownien dans ${\bf R}^d$ (
Brownian motion)
Varadhan's renormalization result (Appendix to
Euclidean quantum field theory, by K.~Symanzik, in
Local Quantum Theory consists in centering certain sequences of Brownian functionals and showing $L^2$-convergence. The same results are obtained here by writing these centered functionals as stochastic integrals
Comment: One of mny applications of stochastic calculus to the existence and regularity of self-intersection local times. See Rosen's papers on this topic in general, and page 196 of Le Gall,
École d'Été de Saint-Flour XX, Springer LNM 1527
Keywords: Local times,
Self-intersection,
Previsible representationNature: Original proofs Retrieve article from Numdam
