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IV: 19, 240-282, LNM 124 (1970)

**DELLACHERIE, Claude**; **DOLÉANS-DADE, Catherine**; **LETTA, Giorgio**; **MEYER, Paul-André**

Diffusions à coefficients continus, d'après Stroock et Varadhan (Markov processes, Diffusion theory)

This paper consists of four seminar talks on a celebrated paper of Stroock-Varadhan (*Comm. Pure Appl. Math.*, **22**, 1969), which constructs by a probability method a unique semigroup whose generator is an elliptic second order operator with continuous coefficients (the analytic approach either deals with operators in divergence form, or requires some Hölder condition). The contribution of G.~Letta nicely simplified the proof

Comment: The results were so definitive that apparently the subject attracted no further work. See Stroock-Varadhan,*Multidimensional Diffusion Processes,* Springer 1979

Keywords: Elliptic differential operators, Uniqueness in law

Nature: Exposition

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VII: 11, 95-117, LNM 321 (1973)

**EL KAROUI, Nicole**; **REINHARD, Hervé**

Processus de diffusion dans ${\bf R}^n$ (Diffusion theory)

This paper concerns diffusions (without boundaries) whose generators have Borel bounded coefficients. It consists of two parts. The first one is devoted to the equivalence between the existence and uniqueness of the diffusion semigroup and the uniqueness in law of the solution of the corresponding Ito stochastic differential equation. This allows the authors to use in the elliptic case the deep results of Krylov on s.d.e.'s. The second part concerns mostly the Lipschitz case, and discusses several properties of the diffusion process in itself: the representation of additive functional martingales; the relations between the number of martingales necessary for the representation and the rank of the generator (locally); the existence of a dual diffusion; the support and absolute continuity properties of the semi-group

Comment: This paper is in part an improved version of a paper on degenerate diffusions by Bonami, El-Karoui, Reinhard and Roynette (*Ann. Inst. H. Poincaré,* **7**, 1971)

Keywords: Construction of diffusions, Diffusions with measurable coefficients, Degenerate diffusions

Nature: Original

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VIII: 18, 316-328, LNM 381 (1974)

**PRIOURET, Pierre**

Construction de processus de Markov sur ${\bf R}^n$ (Markov processes, Diffusion theory)

The problem is to construct a Markov process which satisfies a martingale problem relative to a generator involving a diffusion part and a jump part. The method used is analytic

Comment: To be completed

Keywords: Processes with jumps

Nature: Original

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IX: 35, 534-554, LNM 465 (1975)

**EL KAROUI, Nicole**

Processus de réflexion dans ${\bf R}^n$ (Diffusion theory)

In the line of the seminar on diffusions 419 this talk presents the theory of diffusions in a half space with continuous coefficients and a boundary condition on the boundary hyperplane involving a reflexion part, but more general than the pure reflexion case considered by Stroock-Varadhan (*Comm. Pure Appl. Math.*, **24**, 1971). The point of view is that of martingale problems

Comment: This talk is a late publication of work done by the author in 1971

Keywords: Boundary reflection, Local times

Nature: Original

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X: 16, 240-244, LNM 511 (1976)

**YAMADA, Toshio**

On the uniqueness of solutions of stochastic differential equations with reflecting barrier conditions (Stochastic calculus, Diffusion theory)

A stochastic differential equation is considered on the positive half-line, driven by Brownian motion, with time-dependent coefficients and a reflecting barrier condition at $0$ (Skorohod style). Skorohod proved pathwise uniqueness under Lipschitz condition, and this is extended here to moduli of continuity satisfying integral conditions

Comment: This extends to the reflecting barrier case the now classical result in the ``free'' case due to Yamada-Watanabe,*J. Math. Kyoto Univ.*, **11**, 1971. Many of these theorems have now simpler proofs using local times, in the spirit of Revuz-Yor, *Continuous Martingales and Brownian Motion,* Chapter IX

Keywords: Stochastic differential equations, Boundary reflection

Nature: Original

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XII: 32, 446-456, LNM 649 (1978)

**TAYLOR, John C.**

Some remarks on Malliavin's comparison lemma and related topics (Diffusion theory)

The comparison lemma considered here gives estimates for the hitting probabilities of a several dimensional diffusion in terms of the hitting probabilities of a half line for suitably constructed one-dimensional diffusions. A self-contained proof is given

Keywords: Hitting probabilities

Nature: Original

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XIII: 45, 521-532, LNM 721 (1979)

**JEULIN, Thierry**

Un théorème de J.W. Pitman (Brownian motion, Diffusion theory)

This paper contains an appendix by M. Yor. Let $(B_t)$ and $(Z_t)$ be a Brownian motion and a Bes$_3$ process both starting at $0$. Put $S_t=\sup_{s\le t} B_t$ and $J_t=\inf_{s\ge t}Z_t$. Then Pitman's theorem asserts that, in law, $2S-B=Z$ and $2J-Z=B$ (both statements being in fact equivalent). A complete proof of the theorem is given, using techniques from the general theory of processes. The appendix shows that, granted that $2S-B$ is Markov, it is easy to see that it is a Bes$_3$

Keywords: Bessel processes

Nature: New proof of known results

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XIII: 48, 557-569, LNM 721 (1979)

**CARMONA, René**

Processus de diffusion gouverné par la forme de Dirichlet de l'opérateur de Schrödinger (Diffusion theory)

Standard conditions on the potential $V$ imply that the Schrödinger operator $-(1/2)ėlta+V$ (when suitably interpreted) is essentially self-adjoint on $L^2(**R**^n,dx)$. Assume it has a ground state $\psi$. Then transferring everything on the Hilbert space $L^2(\mu)$ where $\mu$ has the density $\psi^2$ the operator becomes formally $Df=(-1/2)ėlta f + \nabla h.\nabla f$ where $h=-log\psi$. A problem which has aroused some excitement ( due in part to Nelson's ``stochastic mechanics'') was to construct true diffusions governed by this generator, whose meaning is not even clearly defined unless $\psi$ satisfies regularity conditions, unnatural in this problem. Here a reasonable positive answer is given

Comment: This problem, though difficult, is but the simplest case in Nelson's theory. In this seminar, see 1901, 1902, 2019. Seemingly definitive results on this subject are due to E.~Carlen,*Comm. Math. Phys.*, **94**, 1984. A recent reference is Aebi, *Schrödinger Diffusion Processes,* Birkhäuser 1995

Keywords: Nelson's stochastic mechanics, Schrödinger operators

Nature: Original

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XIV: 39, 347-356, LNM 784 (1980)

**CHUNG, Kai Lai**

On stopped Feynman-Kac functionals (Markov processes, Diffusion theory)

Let $(X_t)$ be a strong Markov process with continuous paths on the line, and let $\tau_b$ be the hitting time of the point $b$. It is assumed that $\tau_b$ is $P_a$-a.s. finite for all $a,b$. The purpose of the paper is to study the quantities $u(a,b)=E_a[\,\exp(\int_0^{\tau_b} q(X_s)\,ds)\,]$ where $q$ is bounded. Then (among other results) if $u(a,b)<\infty$ for all $a<b$, we have $u(a,b)\,u(b,a)\le 1$ for all $a,b$

Keywords: Hitting probabilities

Nature: Original

Retrieve article from Numdam

Diffusions à coefficients continus, d'après Stroock et Varadhan (Markov processes, Diffusion theory)

This paper consists of four seminar talks on a celebrated paper of Stroock-Varadhan (

Comment: The results were so definitive that apparently the subject attracted no further work. See Stroock-Varadhan,

Keywords: Elliptic differential operators, Uniqueness in law

Nature: Exposition

Retrieve article from Numdam

VII: 11, 95-117, LNM 321 (1973)

Processus de diffusion dans ${\bf R}^n$ (Diffusion theory)

This paper concerns diffusions (without boundaries) whose generators have Borel bounded coefficients. It consists of two parts. The first one is devoted to the equivalence between the existence and uniqueness of the diffusion semigroup and the uniqueness in law of the solution of the corresponding Ito stochastic differential equation. This allows the authors to use in the elliptic case the deep results of Krylov on s.d.e.'s. The second part concerns mostly the Lipschitz case, and discusses several properties of the diffusion process in itself: the representation of additive functional martingales; the relations between the number of martingales necessary for the representation and the rank of the generator (locally); the existence of a dual diffusion; the support and absolute continuity properties of the semi-group

Comment: This paper is in part an improved version of a paper on degenerate diffusions by Bonami, El-Karoui, Reinhard and Roynette (

Keywords: Construction of diffusions, Diffusions with measurable coefficients, Degenerate diffusions

Nature: Original

Retrieve article from Numdam

VIII: 18, 316-328, LNM 381 (1974)

Construction de processus de Markov sur ${\bf R}^n$ (Markov processes, Diffusion theory)

The problem is to construct a Markov process which satisfies a martingale problem relative to a generator involving a diffusion part and a jump part. The method used is analytic

Comment: To be completed

Keywords: Processes with jumps

Nature: Original

Retrieve article from Numdam

IX: 35, 534-554, LNM 465 (1975)

Processus de réflexion dans ${\bf R}^n$ (Diffusion theory)

In the line of the seminar on diffusions 419 this talk presents the theory of diffusions in a half space with continuous coefficients and a boundary condition on the boundary hyperplane involving a reflexion part, but more general than the pure reflexion case considered by Stroock-Varadhan (

Comment: This talk is a late publication of work done by the author in 1971

Keywords: Boundary reflection, Local times

Nature: Original

Retrieve article from Numdam

X: 16, 240-244, LNM 511 (1976)

On the uniqueness of solutions of stochastic differential equations with reflecting barrier conditions (Stochastic calculus, Diffusion theory)

A stochastic differential equation is considered on the positive half-line, driven by Brownian motion, with time-dependent coefficients and a reflecting barrier condition at $0$ (Skorohod style). Skorohod proved pathwise uniqueness under Lipschitz condition, and this is extended here to moduli of continuity satisfying integral conditions

Comment: This extends to the reflecting barrier case the now classical result in the ``free'' case due to Yamada-Watanabe,

Keywords: Stochastic differential equations, Boundary reflection

Nature: Original

Retrieve article from Numdam

XII: 32, 446-456, LNM 649 (1978)

Some remarks on Malliavin's comparison lemma and related topics (Diffusion theory)

The comparison lemma considered here gives estimates for the hitting probabilities of a several dimensional diffusion in terms of the hitting probabilities of a half line for suitably constructed one-dimensional diffusions. A self-contained proof is given

Keywords: Hitting probabilities

Nature: Original

Retrieve article from Numdam

XIII: 45, 521-532, LNM 721 (1979)

Un théorème de J.W. Pitman (Brownian motion, Diffusion theory)

This paper contains an appendix by M. Yor. Let $(B_t)$ and $(Z_t)$ be a Brownian motion and a Bes$_3$ process both starting at $0$. Put $S_t=\sup_{s\le t} B_t$ and $J_t=\inf_{s\ge t}Z_t$. Then Pitman's theorem asserts that, in law, $2S-B=Z$ and $2J-Z=B$ (both statements being in fact equivalent). A complete proof of the theorem is given, using techniques from the general theory of processes. The appendix shows that, granted that $2S-B$ is Markov, it is easy to see that it is a Bes$_3$

Keywords: Bessel processes

Nature: New proof of known results

Retrieve article from Numdam

XIII: 48, 557-569, LNM 721 (1979)

Processus de diffusion gouverné par la forme de Dirichlet de l'opérateur de Schrödinger (Diffusion theory)

Standard conditions on the potential $V$ imply that the Schrödinger operator $-(1/2)ėlta+V$ (when suitably interpreted) is essentially self-adjoint on $L^2(

Comment: This problem, though difficult, is but the simplest case in Nelson's theory. In this seminar, see 1901, 1902, 2019. Seemingly definitive results on this subject are due to E.~Carlen,

Keywords: Nelson's stochastic mechanics, Schrödinger operators

Nature: Original

Retrieve article from Numdam

XIV: 39, 347-356, LNM 784 (1980)

On stopped Feynman-Kac functionals (Markov processes, Diffusion theory)

Let $(X_t)$ be a strong Markov process with continuous paths on the line, and let $\tau_b$ be the hitting time of the point $b$. It is assumed that $\tau_b$ is $P_a$-a.s. finite for all $a,b$. The purpose of the paper is to study the quantities $u(a,b)=E_a[\,\exp(\int_0^{\tau_b} q(X_s)\,ds)\,]$ where $q$ is bounded. Then (among other results) if $u(a,b)<\infty$ for all $a<b$, we have $u(a,b)\,u(b,a)\le 1$ for all $a,b$

Keywords: Hitting probabilities

Nature: Original

Retrieve article from Numdam