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IV: 13, 151-161, LNM 124 (1970)

**MAISONNEUVE, Bernard**; **MORANDO, Philippe**

Temps locaux pour les ensembles régénératifs (Markov processes)

This paper uses the results of the preceding one 412 to define and study the local time of a perfect regenerative set with empty interior (e.g. the set of zeros of Brownian motion), a continuous adapted increasing process whose set of points of increase is exactly the given set

Comment: Same references as the preceding paper 412

Keywords: Renewal theory, Regenerative sets, Local times

Nature: Original

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V: 15, 147-169, LNM 191 (1971)

**MAISONNEUVE, Bernard**

Ensembles régénératifs, temps locaux et subordinateurs (General theory of processes, Renewal theory)

New approach to the theory of regenerative sets (Kingman; Krylov-Yushkevic 1965, Hoffmann-Jørgensen,*Math. Scand.*, **24**, 1969), including a general definition of local time of a random set

Comment: See Meyer 412, Morando-Maisonneuve 413, later work of Maisonneuve in 813 and later

Keywords: Local times, Subordinators, Renewal theory

Nature: Original

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V: 17, 177-190, LNM 191 (1971)

**MEYER, Paul-André**

Processus de Poisson ponctuels d'après K. Ito (Markov processes, Point processes)

Presents (a preliminary form of) the celebrated paper of Ito (*Proc. Sixth Berkeley Symposium,* **3**, 1972) on excursion theory, with an extension (the use of possibly unbounded entrance laws instead of initial measures) which has become part of the now classical theory

Comment: A slip in the definition of Poisson point processes is corrected in vol. VI p.253. The material has appeared repeatedly in book form

Keywords: Poisson point processes, Excursions, Local times

Nature: Exposition, Original additions

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V: 20, 209-210, LNM 191 (1971)

**MEYER, Paul-André**

Un théorème sur la répartition des temps locaux (Markov processes)

Kesten discovered that the value at a terminal time $T$ of the local time $L$ of a Markov process $X$ at a single point has an exponential distribution, and that $X_T$ and $L_T$ are independent. A short proof is given

Comment: The result can be deduced from excursion theory

Keywords: Local times

Nature: New exposition of known results

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VIII: 10, 134-149, LNM 381 (1974)

**KNIGHT, Frank B.**

Existence of small oscillations at zeros of brownian motion (Brownian motion)

The one-dimensional Brownian motion path is shown to have an abnormal behaviour (an ``iterated logarithm'' upper limit smaller than one) at uncountably many times on his set of zeros

Comment: This result may be compared to Kahane,*C.R. Acad. Sci.* **248**, 1974

Keywords: Law of the iterated logarithm, Local times

Nature: Original

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VIII: 13, 172-261, LNM 381 (1974)

**MAISONNEUVE, Bernard**; **MEYER, Paul-André**

Ensembles aléatoires markoviens homogènes (5 talks) (Markov processes)

This long exposition is a development of original work by the first author. Its purpose is the study of processes which possess a strong Markov property, not at all stopping times, but only at those which belong to a given homogeneous random set $M$---a point of view introduced earlier in renewal theory (Kingman, Krylov-Yushkevich, Hoffmann-Jörgensen, see 412). The first part is devoted to technical results: the description of (closed) optional random sets in the general theory of processes, and of the operations of balayage of random measures; homogeneous processes, random sets and additive functionals; right Markov processes and the perfection of additive functionals. This last section is very technical (a general problem with this paper).\par Chapter II starts with the classification of the starting points of excursions (``left endpoints'' below) from a random set, and the fact that the projection (optional and previsible) of a raw AF still is an AF. The main theorem then computes the $p$-balayage on $M$ of an additive functional of the form $A_t=\int_0^th\circ X_s ds$. All these balayages have densities with respect to a suitable local time of $M$, which can be regularized to yield a resolvent and then a semigroup. Then the result is translated into the language of homogeneous random measures carried by the set of left endpoints and describing the following excursion. This section is an enlarged exposition of results due to Getoor-Sharpe (*Ann. Prob.* **1**, 1973; *Indiana Math. J.* **23**, 1973). The basic and earlier paper of Dynkin on the same subject (* Teor. Ver. Prim.* **16**, 1971) was not known to the authors.\par Chapter III is devoted to the original work of Maisonneuve on incursions. Roughly, the incursion at time $t$ is trivial if $t\in M$, and if $t\notin M$ it consists of the post-$t$ part of the excursion straddling $t$. Thus the incursion process is a path valued, non adapted process. It is only adapted to the filtration ${\cal F}_{D_t}$ where $D_t$ is the first hitting time of $M$ after $t$. Contrary to the Ito theory of excursions, no change of time using a local time is performed. The main result is the fact that, if a suitable regeneration property is assumed only on the set $M$ then, in a suitable topology on the space of paths, this process is a right-continuous strong Markov process. Considerable effort is devoted to proving that it is even a right process (the technique is heavy and many errors have crept in, some of them corrected in 932-933).\par Chapter IV makes the connection between II and III: the main results of Chapter II are proved anew (without balayage or Laplace transforms): they amount to computing the Lévy system of the incursion process. Finally, Chapter V consists of applications, among which a short discussion of the boundary theory for Markov chains

Comment: This paper is a piece of a large literature. Some earlier papers have been mentioned above. Maisonneuve published as*Systèmes Régénératifs,* *Astérisque,* **15**, 1974, a much simpler version of his own results, and discovered important improvements later on (some of which are included in Dellacherie-Maisonneuve-Meyer, *Probabilités et Potentiel,* Chapter XX, 1992). Along the slightly different line of Dynkin, see El~Karoui-Reinhard, *Compactification et balayage de processus droits,* *Astérisque 21,* 1975. A recent book on excursion theory is Blumenthal, *Excursions of Markov Processes,* Birkhäuser 1992

Keywords: Regenerative systems, Regenerative sets, Renewal theory, Local times, Excursions, Markov chains, Incursions

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: 14, 216-234, LNM 511 (1976)

**WILLIAMS, David**

The Q-matrix problem (Markov processes)

This paper completely solves the Q-matrix problem (find necessary and sufficient conditions for an infinite matrix $q_{ij}$ to be the pointwise derivative at $0$ of a transition matrix) in the case when all states are instantaneous. Though the statement of the problem and the two conditions given are elementary and simple, the proof uses sophisticated ``modern'' methods. The necessity of the conditions is proved using the Ray-Knight compactification method, the converse is a clever construction which is merely sketched

Comment: This paper crowns nearly 20 years of investigations of this problem by the English school. It contains a promise of a detailed proof which apparently was never published. See the section of Markov chains in Rogers-Williams*Diffusions, Markov Processes and Martingales,* vol. 1 (second edition), Wiley 1994. See also 1024

Keywords: Markov chains, Ray compactification, Local times, Excursions

Nature: Original

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X: 15, 235-239, LNM 511 (1976)

**WILLIAMS, David**

On a stopped Brownian motion formula of H.M.~Taylor (Brownian motion)

This formula gives the joint distribution of $X_T$ and $T$, where $X$ is standard Brownian motion and $T$ is the first time $M_T-X_T=a$, $M_t$ denoting the supremum of $X$ up to time $t$. Two different new proofs are given

Comment: For the original proof of Taylor see*Ann. Prob.* **3**, 1975. For modern references, we should ask Yor

Keywords: Stopping times, Local times, Ray-Knight theorems, Cameron-Martin formula

Nature: Original

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XIII: 09, 126-131, LNM 721 (1979)

**PRATELLI, Maurizio**

Le support exact du temps local d'une martingale continue (Martingale theory)

It is well known in the Brownian case that the zero set and the support of the local time are the same. For a continuous local martingale $(X_t)$ with zero set $H$ and local time $(L_t)$, it is shown that the support of $dL$ is exactly the perfect kernel of the boundary of $H$

Keywords: Local times

Nature: Original

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XIII: 37, 441-442, LNM 721 (1979)

**CHOU, Ching Sung**

Démonstration simple d'un résultat sur le temps local (Stochastic calculus)

It follows from Ito's formula that the positive parts of those jumps of a semimartingale $X$ that originate below $0$ are summable. A direct proof is given of this fact

Comment: Though the idea is essentially correct, an embarrassing mistake is corrected as 1429

Keywords: Local times, Semimartingales, Jumps

Nature: Original

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XIII: 38, 443-452, LNM 721 (1979)

**EL KAROUI, Nicole**

Temps local et balayage des semimartingales (General theory of processes)

This paper is the first one in a series of reports on the balayage of semimartingales, and the following description is common to all of them. \par Let $H$ be a right-closed optional set, and let $g_t=\sup\{s<t, s\in H\}$ and $D_t=\inf\{s>t,s\in H\}$. Put $L=g_{\infty}$. Let also $G$ be the set of all left-endpoints of intervals contiguous to $H$, i.e., of all points $g_t$ for $t\notin H$. For simplicity we assume here that $D_0=0$ and that $H=\{X=0\}$, where $X$ is a semimartingale with decomposition $X=M+V$, though for a few results (including the balayage formula itself) it is sufficient that $X=0$ on $H$. \par One of the starting points of this paper is the*balayage formula * (see Azéma-Yor, introduction to *Temps Locaux *, *Astérisque *, **52-53**): if $Z$ is a locally bounded previsible process, then $$Z_{g_t}X_t=\int_0^t Z_{g_s}dX_s$$ and therefore $Y_t=Z_{g_t}X_t$ is a semimartingale. The main problem of the series of reports is: what can be said if $Z$ is not previsible, but optional, or even progressive?\par This particular paper is devoted to the study of the non-adapted process $$K_t=\sum_{g\in G,g\le t } (M_{D_g}-M_g)$$ which turns out to have finite variation

Comment: This paper is completed by 1357

Keywords: Local times, Balayage, Balayage formula

Nature: Original

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XIII: 39, 453-471, LNM 721 (1979)

**YOR, Marc**

Sur le balayage des semi-martingales continues (General theory of processes)

For the general notation, see 1338. This paper is independent from the preceding one 1338, and some overlap occurs. The balayage formula is extended to processes $Z$ which are not locally bounded, and the local time of the semimartingale $Y$ is computed. The class of continuous semimartingales $X$ with canonical decomposition $X=M+V$ such that $dV$ is carried by $H=\{X=0\}$ is introduced and studied. It turns out to be an important class, closely related to ``relative martingales'' (Azéma, Meyer and Yor 2623). A number of results are given, too technical to be stated here. Stopping previsible, optional and progressive processes at the last exit time $L$ from $H$ leads to three $\sigma$-fields, ${\cal F}_L^p$, ${\cal F}_L^o$, ${\cal F}_L^{\pi}$, and it was considered surprising that the last two could be different (see 1240). Here it is shown that if $X$ is a continuous uniformly integrable martingale with $X_0=0$, $E[X_{\infty}|{\cal F}_L^o]=0\neq E[X_{\infty}|{\cal F}_L^{\pi}]$

Comment: See 1357

Keywords: Local times, Balayage, Balayage formula

Nature: Original

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XIV: 08, 76-101, LNM 784 (1980)

**SHARPE, Michael J.**

Local times and singularities of continuous local martingales (Martingale theory)

This paper studies continuous local martingales $(M_t)$ in the open interval $]0,\infty[$. After recalling a few useful results on local martingales, the author proves that the sample paths a.s., either have a limit (possibly $\pm\infty$) at $t=0$, or oscillate over the whole interval $]-\infty,\infty[$ (this is due to Walsh 1133, but the proof here does not use conformal martingales). Then the quadratic variation and local time of $M$ are defined as random measures which may explode near $0$, and it is shown that non-explosion of the quadratic variation (of the local time) measure characterizes the sample paths which have a finite limit (a limit) at $0$. The results are extended in part to local martingale increment processes, which are shown to be stochastic integrals with respect to true local martingales, of previsible processes which are not integrable near $0$

Comment: See Calais-Genin 1717

Keywords: Local times, Local martingales, Semimartingales in an open interval

Nature: Original

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XIV: 29, 254-254, LNM 784 (1980)

**YOEURP, Chantha**

Rectificatif à l'exposé de C.S. Chou (Stochastic calculus)

A mistake in the proof of 1337 is corrected, the result remaining true without additional assumptions

Keywords: Local times, Semimartingales, Jumps

Nature: Correction

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XV: 12, 189-190, LNM 850 (1981)

**BARLOW, Martin T.**

On Brownian local time (Brownian motion)

Let $(L^a_t)$ be the standard (jointly continuous) version of Brownian local times. Perkins has shown that for fixed $t$ $a\mapsto L^a_t$ is a semimartingale relative to the excusrion fields. An example is given of a stopping time $T$ for which $a\mapsto L^a_T$ is not a semimartingale

Keywords: Local times

Nature: Original

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XV: 13, 191-205, LNM 850 (1981)

**MAISONNEUVE, Bernard**

On Lévy's downcrossing theorem and various extensions (Excursion theory)

Lévy's downcrossing theorem describes the local time $L_t$ of Brownian motion at $0$ as the limit of $\epsilon D_t(\epsilon)$, where $D_t$ denotes the number of downcrossings of the interval $(0,\epsilon)$ up to time $t$. To give a simple proof of this result from excursion theory, easy to generalize, the paper uses the weaker definition of regenerative systems described in 1137. A gap in the related author's paper*Zeit. für W-Theorie,* **52**, 1980 is repaired at the end of the paper

Keywords: Excursions, Lévy's downcrossing theorem, Local times, Regenerative systems

Nature: Original

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XV: 14, 206-209, LNM 850 (1981)

**McGILL, Paul**

A direct proof of the Ray-Knight theorem (Brownian motion)

The (first) Ray-Knight theorem describes the law of the process $(L_T^{1-a})_{0\le a\le 1}$ where $(L^a_t)$ is the family of local times of Brownian motion starting from $0$ and $T$ is the hitting time of $1$. A direct proof is given indeed. It is reproduced in Revuz-Yor,*Continuous Martingales and Brownian Motion,* Chapter XI, exercice (2.7)

Keywords: Local times, Ray-Knight theorems, Bessel processes

Nature: New proof of known results

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XVI: 15, 209-211, LNM 920 (1982)

**BARLOW, Martin T.**

$L(B_t,t)$ is not a semimartingale (Brownian motion)

The following question had been open for some time: given a jointly continuous version $L(a,t)$ of the local times of Brownian motion, is $Y_t=L(B_t,t)$ a semimartingale? It is proved here that $Y$ fails to be Hölder continuous of order 1/4, and therefore cannot be a semimartingale

Keywords: Local times, Semimartingales

Nature: Original

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XVI: 17, 213-218, LNM 920 (1982)

**FALKNER, Neil**; **STRICKER, Christophe**; **YOR, Marc**

Temps d'arrêt riches et applications (General theory of processes)

This paper starts from the existence of increasing left-continuous processes $(A_t)$ which generate the previsible $\sigma$-field, i.e., every previsible process can be represented as $f(X_t)$ for some Borel function $f$ (see 1123), to prove the existence (discovered by the first named author) of ``rich'' stopping times $T$, i.e., previsible stopping times which encode the whole past up to time $T$: $\sigma(T)={\cal F}_{T-}$ (a few details are omitted here). This result leads to counterexamples: a non-reversible semimartingale (see the preceding paper 1616) and a stopping time $T$ for Brownian motion such that $L^a_T$ is not a semimartingale in its space variable $a$

Keywords: Stopping times, Local times, Semimartingales, Previsible processes

Nature: Original

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XVI: 20, 234-237, LNM 920 (1982)

**YOEURP, Chantha**

Une décomposition multiplicative de la valeur absolue d'un mouvement brownien (Brownian motion, Stochastic calculus)

A positive submartingale like $X_t=|B_t|$ vanishes too often to be represented as a product of a local martingale and an increasing process. Still, one may look for a kind of additive decomposition of $\log X$, from which the required multiplicative decomposition would follow by taking exponentials. Here the (Ito-Tanaka) additive decomposition of $\log(X\lor\epsilon)$ is studied, as well as its limiting behaviour as $\epsilon\rightarrow0$

Comment: See 1023, 1321

Keywords: Multiplicative decomposition, Change of variable formula, Local times

Nature: Original

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XVI: 21, 238-247, LNM 920 (1982)

**YOR, Marc**

Sur la transformée de Hilbert des temps locaux browniens et une extension de la formule d'Itô (Brownian motion)

This paper is about the application to the function $(x-a)\log|x-a|-(x-a)$ (whose second derivative is $1/x-a$) of the Ito-Tanaka formula; the last term then involves a formal Hilbert transform $\tilde L^a_t$ of the local time process $L^a_t$. Such processes had been defined by Ito and McKean, and studied by Yamada as examples of Fukushima's ``additive functionals of zero energy''. Here it is proved, as a consequence of a general theorem, that this process has a jointly continuous version---more precisely, Hölder continuous of all orders $<1/2$ in $a$ and in $t$

Comment: For a modern version with references see Yor,*Some Aspects of Brownian Motion II*, Birkhäuser 1997

Keywords: Local times, Hilbert transform, Ito formula

Nature: Original

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XVII: 26, 227-239, LNM 986 (1983)

**VALLOIS, Pierre**

Le problème de Skorokhod sur $\bf R$ : une approche avec le temps local (Brownian motion)

A solution to Skorohod's embedding problem is given, using the first hiting time of a set by the 2-dimensional process which consists of Brownian motion and its local time at zero. The author's aim is to ``correct'' the asymmetry inherent to the Azéma-Yor construction

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding, Local times

Nature: Original

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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-intersection

Nature: Original proofs

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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 1928

Comment: 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-intersection

Nature: Original proofs

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XIX: 30, 350-365, LNM 1123 (1985)

**YOR, Marc**

Renormalisation et convergence en loi pour des temps locaux d'intersection du mouvement brownien dans ${\bf R}^3$ (Brownian motion)

It is shown that no renormalization à la Varadhan occurs for the self-intersection local times of 3-dimensional Brownian motion; but a weaker result is established: when the point $y\in**R**^3$ tends to $0$, the self-intersection local time at $y$, on the triangle $\{0<s<u\le t\},\ t\ge0$, centered and divided by $(-\log|y|)^{1/2}$, converges in law to a Brownian motion. Several variants of this theorem are established

Comment: This result was used by Le Gall in his work on fluctuations of the Wiener sausage:*Ann. Prob.* **16** (1988). Many results by Rosen have the same flavour

Keywords: Brownian motion, Local times, Self-intersection

Nature: Original

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XX: 33, 515-531, LNM 1204 (1986)

**ROSEN, Jay S.**

A renormalized local time for multiple intersections of planar Brownian motion (Brownian motion)

Using Fourier techniques, the existence of a renormalized local time for $n$-fold self-intersections of planar Brownian motion is obtained, thus extending the case $n=2$, obtained in the pioneering work of Varadhan (Appendix to*Euclidean quantum field theory*, by K.~Symanzik, in *Local Quantum Theory*, Academic Press, 1969)

Comment: Closely related to 2036. A general reference is Le Gall,*École d'Été de Saint-Flour XX*, Springer LNM 1527

Keywords: Local times, Self-intersection

Nature: Original

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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 representation

Nature: Original proofs

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XXV: 33, 407-424, LNM 1485 (1991)

**ROSEN, Jay S.**

Second order limit laws for the local times of stable processes (Limit theorems)

Using the method of moments, a central limit theorem is established for the increments $L^x_t-L^0_t$ of the local times of a symmetric $\beta$-stable process ($\beta>1$). The limit law is that of a fractional Brownian sheet, with Hurst index $\beta-1$, time-changed via $L_t^0$ in its time variable

Comment: Another proof due to Eisenbaum 3120 uses Dynkin's isomorphism. Ray-Knight theorems for these local times can be found in Eisenbaum-Kaspi-Marcus-Rosen-Shi*Ann. Prob.* **28** (2000). A good reference on this subject is Marcus-Rosen, *Markov Processes, Gaussian Processes, and Local Times*, Cambridge University Press (2006)

Keywords: Local times, Stable processes, Method of moments, Fractional Brownian motion, Brownian sheet

Nature: Original

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XXIX: 26, 266-289, LNM 1613 (1995)

**EISENBAUM, Nathalie**

Une version sans conditionnement du théorème d'isomorphisme de Dynkin (Limit theorems)

After establishing an unconditional version of Dynkin's isomorphism theorem, the author applies this theorem to give a new proof of Ray-Knight theorems for Brownian local times, and also to give another proof to limit theorems due to Rosen 2533 concerning the increments of the local times of a symmetric $\beta$-stable process for $\beta>1$. Some results by Marcus-Rosen (*Proc. Conf. Probability in Banach Spaces~8*, Birkhäuser 1992) on Laplace transforms of the increments of local time are extended

Comment: A general reference on the subject is Marcus-Rosen,*Markov Processes, Gaussian Processes, and Local Times*, Cambridge University Press (2006)

Keywords: Stable processes, Local times, Central limit theorem, Dynkin isomorphism, Fractional Brownian motion, Brownian sheet

Nature: Original

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XXXI: 20, 216-224, LNM 1655 (1997)

**EISENBAUM, Nathalie**

Théorèmes limites pour les temps locaux d'un processus stable symétrique (Limit theorems)

Using Dynkin's isomorphism, a central-limit type theorem is derived for the local times of a stable symmetric process of index $\beta$ at a finite number $n$ of levels. The limiting process is expressed in terms of a fractional, $n$-dimensional Brownian sheet with Hurst index $\beta-1$. The case when $n=1$ is due to Rosen 2533, and, for Brownian local times, to Yor 1709

Comment: This kind of result is now understood as a weak form of theorems à la Ray-Knight, describing the local times of a stable symmetric process: see Eisenbaum-Kaspi-Marcus-Rosen-Shi*Ann. Prob.* **28** (2000) for a Ray-Knight theorem involving fractional Brownian motion. Marcus-Rosen, *Markov Processes, Gaussian Processes, and Local Times*, Cambridge University Press (2006) is a general reference on the subject

Keywords: Stable processes, Local times, Central limit theorem, Dynkin isomorphism, Fractional Brownian motion, Brownian sheet

Nature: Original

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XXXIII: 20, 388-394, LNM 1709 (1999)

**PITMAN, James W.**

The distribution of local times of a Brownian bridge (Brownian motion)

Several useful identities for the one-dimensional marginals of local times of Brownian bridges are derived. This is a variation and extension on the well-known joint law of the maximum and the value of Brownian motion at a given time

Comment: Useful references are Borodin,*Russian Math. Surveys* (1989) and the book *Brownian motion and stochastic calculus* by Karatzas-Shrieve (Springer, 1991)

Keywords: Local times, Brownian bridge

Nature: Original

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XLII: 12, 331-363, LNM 1978 (2009)

**DEBS, Pierre**

Penalisation of the standard random walk by a function of the one-sided maximum of the local time or of the duration of the excursions (Theory of stochastic processes)

Keywords: Penalisation, Excursions, Local times

Nature: Original

XLIII: 03, 95-104, LNM 2006 (2011)

**ROSEN, Jay**

A stochastic calculus proof of the CLT for the $L^{2}$ modulus of continuity of local time (Theory of Brownian motion)

Keywords: Central Limit Theorem, Moduli of continuity, Local times, Brownian motion

Nature: Original

XLIII: 04, 105-126, LNM 2006 (2011)

**MATSUMOTO, Ayako**; **YANO, Kouji**

On a zero-one law for the norm process of transient random walk (Theory of random walks)

Keywords: Zero-one law, Random walk, Local times, Jeulin's lemma

Nature: Original

XLIII: 09, 221-239, LNM 2006 (2011)

**MAROUBY, Matthieu**

Simulation of a Local Time Fractional Stable Motion (Theory of processes)

Keywords: Stable processes, Self-similar processes, Shot noise series, Local times, Fractional Brownian motion, Simulation

Nature: Original

XLIV: 08, 167-190, LNM 2046 (2012)

**HAJRI, Hatem**

Discrete approximation to solution flows of Tanaka's SDE related to Walsh Brownian motion (Stochastic calculus, Limit theorems)

Keywords: Walsh's Brownian motion, Tanaka's SDE, Local times

Nature: Original

Temps locaux pour les ensembles régénératifs (Markov processes)

This paper uses the results of the preceding one 412 to define and study the local time of a perfect regenerative set with empty interior (e.g. the set of zeros of Brownian motion), a continuous adapted increasing process whose set of points of increase is exactly the given set

Comment: Same references as the preceding paper 412

Keywords: Renewal theory, Regenerative sets, Local times

Nature: Original

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V: 15, 147-169, LNM 191 (1971)

Ensembles régénératifs, temps locaux et subordinateurs (General theory of processes, Renewal theory)

New approach to the theory of regenerative sets (Kingman; Krylov-Yushkevic 1965, Hoffmann-Jørgensen,

Comment: See Meyer 412, Morando-Maisonneuve 413, later work of Maisonneuve in 813 and later

Keywords: Local times, Subordinators, Renewal theory

Nature: Original

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V: 17, 177-190, LNM 191 (1971)

Processus de Poisson ponctuels d'après K. Ito (Markov processes, Point processes)

Presents (a preliminary form of) the celebrated paper of Ito (

Comment: A slip in the definition of Poisson point processes is corrected in vol. VI p.253. The material has appeared repeatedly in book form

Keywords: Poisson point processes, Excursions, Local times

Nature: Exposition, Original additions

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V: 20, 209-210, LNM 191 (1971)

Un théorème sur la répartition des temps locaux (Markov processes)

Kesten discovered that the value at a terminal time $T$ of the local time $L$ of a Markov process $X$ at a single point has an exponential distribution, and that $X_T$ and $L_T$ are independent. A short proof is given

Comment: The result can be deduced from excursion theory

Keywords: Local times

Nature: New exposition of known results

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VIII: 10, 134-149, LNM 381 (1974)

Existence of small oscillations at zeros of brownian motion (Brownian motion)

The one-dimensional Brownian motion path is shown to have an abnormal behaviour (an ``iterated logarithm'' upper limit smaller than one) at uncountably many times on his set of zeros

Comment: This result may be compared to Kahane,

Keywords: Law of the iterated logarithm, Local times

Nature: Original

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VIII: 13, 172-261, LNM 381 (1974)

Ensembles aléatoires markoviens homogènes (5 talks) (Markov processes)

This long exposition is a development of original work by the first author. Its purpose is the study of processes which possess a strong Markov property, not at all stopping times, but only at those which belong to a given homogeneous random set $M$---a point of view introduced earlier in renewal theory (Kingman, Krylov-Yushkevich, Hoffmann-Jörgensen, see 412). The first part is devoted to technical results: the description of (closed) optional random sets in the general theory of processes, and of the operations of balayage of random measures; homogeneous processes, random sets and additive functionals; right Markov processes and the perfection of additive functionals. This last section is very technical (a general problem with this paper).\par Chapter II starts with the classification of the starting points of excursions (``left endpoints'' below) from a random set, and the fact that the projection (optional and previsible) of a raw AF still is an AF. The main theorem then computes the $p$-balayage on $M$ of an additive functional of the form $A_t=\int_0^th\circ X_s ds$. All these balayages have densities with respect to a suitable local time of $M$, which can be regularized to yield a resolvent and then a semigroup. Then the result is translated into the language of homogeneous random measures carried by the set of left endpoints and describing the following excursion. This section is an enlarged exposition of results due to Getoor-Sharpe (

Comment: This paper is a piece of a large literature. Some earlier papers have been mentioned above. Maisonneuve published as

Keywords: Regenerative systems, Regenerative sets, Renewal theory, Local times, Excursions, Markov chains, Incursions

Nature: Original

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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

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X: 14, 216-234, LNM 511 (1976)

The Q-matrix problem (Markov processes)

This paper completely solves the Q-matrix problem (find necessary and sufficient conditions for an infinite matrix $q_{ij}$ to be the pointwise derivative at $0$ of a transition matrix) in the case when all states are instantaneous. Though the statement of the problem and the two conditions given are elementary and simple, the proof uses sophisticated ``modern'' methods. The necessity of the conditions is proved using the Ray-Knight compactification method, the converse is a clever construction which is merely sketched

Comment: This paper crowns nearly 20 years of investigations of this problem by the English school. It contains a promise of a detailed proof which apparently was never published. See the section of Markov chains in Rogers-Williams

Keywords: Markov chains, Ray compactification, Local times, Excursions

Nature: Original

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X: 15, 235-239, LNM 511 (1976)

On a stopped Brownian motion formula of H.M.~Taylor (Brownian motion)

This formula gives the joint distribution of $X_T$ and $T$, where $X$ is standard Brownian motion and $T$ is the first time $M_T-X_T=a$, $M_t$ denoting the supremum of $X$ up to time $t$. Two different new proofs are given

Comment: For the original proof of Taylor see

Keywords: Stopping times, Local times, Ray-Knight theorems, Cameron-Martin formula

Nature: Original

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XIII: 09, 126-131, LNM 721 (1979)

Le support exact du temps local d'une martingale continue (Martingale theory)

It is well known in the Brownian case that the zero set and the support of the local time are the same. For a continuous local martingale $(X_t)$ with zero set $H$ and local time $(L_t)$, it is shown that the support of $dL$ is exactly the perfect kernel of the boundary of $H$

Keywords: Local times

Nature: Original

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XIII: 37, 441-442, LNM 721 (1979)

Démonstration simple d'un résultat sur le temps local (Stochastic calculus)

It follows from Ito's formula that the positive parts of those jumps of a semimartingale $X$ that originate below $0$ are summable. A direct proof is given of this fact

Comment: Though the idea is essentially correct, an embarrassing mistake is corrected as 1429

Keywords: Local times, Semimartingales, Jumps

Nature: Original

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XIII: 38, 443-452, LNM 721 (1979)

Temps local et balayage des semimartingales (General theory of processes)

This paper is the first one in a series of reports on the balayage of semimartingales, and the following description is common to all of them. \par Let $H$ be a right-closed optional set, and let $g_t=\sup\{s<t, s\in H\}$ and $D_t=\inf\{s>t,s\in H\}$. Put $L=g_{\infty}$. Let also $G$ be the set of all left-endpoints of intervals contiguous to $H$, i.e., of all points $g_t$ for $t\notin H$. For simplicity we assume here that $D_0=0$ and that $H=\{X=0\}$, where $X$ is a semimartingale with decomposition $X=M+V$, though for a few results (including the balayage formula itself) it is sufficient that $X=0$ on $H$. \par One of the starting points of this paper is the

Comment: This paper is completed by 1357

Keywords: Local times, Balayage, Balayage formula

Nature: Original

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XIII: 39, 453-471, LNM 721 (1979)

Sur le balayage des semi-martingales continues (General theory of processes)

For the general notation, see 1338. This paper is independent from the preceding one 1338, and some overlap occurs. The balayage formula is extended to processes $Z$ which are not locally bounded, and the local time of the semimartingale $Y$ is computed. The class of continuous semimartingales $X$ with canonical decomposition $X=M+V$ such that $dV$ is carried by $H=\{X=0\}$ is introduced and studied. It turns out to be an important class, closely related to ``relative martingales'' (Azéma, Meyer and Yor 2623). A number of results are given, too technical to be stated here. Stopping previsible, optional and progressive processes at the last exit time $L$ from $H$ leads to three $\sigma$-fields, ${\cal F}_L^p$, ${\cal F}_L^o$, ${\cal F}_L^{\pi}$, and it was considered surprising that the last two could be different (see 1240). Here it is shown that if $X$ is a continuous uniformly integrable martingale with $X_0=0$, $E[X_{\infty}|{\cal F}_L^o]=0\neq E[X_{\infty}|{\cal F}_L^{\pi}]$

Comment: See 1357

Keywords: Local times, Balayage, Balayage formula

Nature: Original

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XIV: 08, 76-101, LNM 784 (1980)

Local times and singularities of continuous local martingales (Martingale theory)

This paper studies continuous local martingales $(M_t)$ in the open interval $]0,\infty[$. After recalling a few useful results on local martingales, the author proves that the sample paths a.s., either have a limit (possibly $\pm\infty$) at $t=0$, or oscillate over the whole interval $]-\infty,\infty[$ (this is due to Walsh 1133, but the proof here does not use conformal martingales). Then the quadratic variation and local time of $M$ are defined as random measures which may explode near $0$, and it is shown that non-explosion of the quadratic variation (of the local time) measure characterizes the sample paths which have a finite limit (a limit) at $0$. The results are extended in part to local martingale increment processes, which are shown to be stochastic integrals with respect to true local martingales, of previsible processes which are not integrable near $0$

Comment: See Calais-Genin 1717

Keywords: Local times, Local martingales, Semimartingales in an open interval

Nature: Original

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XIV: 29, 254-254, LNM 784 (1980)

Rectificatif à l'exposé de C.S. Chou (Stochastic calculus)

A mistake in the proof of 1337 is corrected, the result remaining true without additional assumptions

Keywords: Local times, Semimartingales, Jumps

Nature: Correction

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XV: 12, 189-190, LNM 850 (1981)

On Brownian local time (Brownian motion)

Let $(L^a_t)$ be the standard (jointly continuous) version of Brownian local times. Perkins has shown that for fixed $t$ $a\mapsto L^a_t$ is a semimartingale relative to the excusrion fields. An example is given of a stopping time $T$ for which $a\mapsto L^a_T$ is not a semimartingale

Keywords: Local times

Nature: Original

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XV: 13, 191-205, LNM 850 (1981)

On Lévy's downcrossing theorem and various extensions (Excursion theory)

Lévy's downcrossing theorem describes the local time $L_t$ of Brownian motion at $0$ as the limit of $\epsilon D_t(\epsilon)$, where $D_t$ denotes the number of downcrossings of the interval $(0,\epsilon)$ up to time $t$. To give a simple proof of this result from excursion theory, easy to generalize, the paper uses the weaker definition of regenerative systems described in 1137. A gap in the related author's paper

Keywords: Excursions, Lévy's downcrossing theorem, Local times, Regenerative systems

Nature: Original

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XV: 14, 206-209, LNM 850 (1981)

A direct proof of the Ray-Knight theorem (Brownian motion)

The (first) Ray-Knight theorem describes the law of the process $(L_T^{1-a})_{0\le a\le 1}$ where $(L^a_t)$ is the family of local times of Brownian motion starting from $0$ and $T$ is the hitting time of $1$. A direct proof is given indeed. It is reproduced in Revuz-Yor,

Keywords: Local times, Ray-Knight theorems, Bessel processes

Nature: New proof of known results

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XVI: 15, 209-211, LNM 920 (1982)

$L(B_t,t)$ is not a semimartingale (Brownian motion)

The following question had been open for some time: given a jointly continuous version $L(a,t)$ of the local times of Brownian motion, is $Y_t=L(B_t,t)$ a semimartingale? It is proved here that $Y$ fails to be Hölder continuous of order 1/4, and therefore cannot be a semimartingale

Keywords: Local times, Semimartingales

Nature: Original

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XVI: 17, 213-218, LNM 920 (1982)

Temps d'arrêt riches et applications (General theory of processes)

This paper starts from the existence of increasing left-continuous processes $(A_t)$ which generate the previsible $\sigma$-field, i.e., every previsible process can be represented as $f(X_t)$ for some Borel function $f$ (see 1123), to prove the existence (discovered by the first named author) of ``rich'' stopping times $T$, i.e., previsible stopping times which encode the whole past up to time $T$: $\sigma(T)={\cal F}_{T-}$ (a few details are omitted here). This result leads to counterexamples: a non-reversible semimartingale (see the preceding paper 1616) and a stopping time $T$ for Brownian motion such that $L^a_T$ is not a semimartingale in its space variable $a$

Keywords: Stopping times, Local times, Semimartingales, Previsible processes

Nature: Original

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XVI: 20, 234-237, LNM 920 (1982)

Une décomposition multiplicative de la valeur absolue d'un mouvement brownien (Brownian motion, Stochastic calculus)

A positive submartingale like $X_t=|B_t|$ vanishes too often to be represented as a product of a local martingale and an increasing process. Still, one may look for a kind of additive decomposition of $\log X$, from which the required multiplicative decomposition would follow by taking exponentials. Here the (Ito-Tanaka) additive decomposition of $\log(X\lor\epsilon)$ is studied, as well as its limiting behaviour as $\epsilon\rightarrow0$

Comment: See 1023, 1321

Keywords: Multiplicative decomposition, Change of variable formula, Local times

Nature: Original

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XVI: 21, 238-247, LNM 920 (1982)

Sur la transformée de Hilbert des temps locaux browniens et une extension de la formule d'Itô (Brownian motion)

This paper is about the application to the function $(x-a)\log|x-a|-(x-a)$ (whose second derivative is $1/x-a$) of the Ito-Tanaka formula; the last term then involves a formal Hilbert transform $\tilde L^a_t$ of the local time process $L^a_t$. Such processes had been defined by Ito and McKean, and studied by Yamada as examples of Fukushima's ``additive functionals of zero energy''. Here it is proved, as a consequence of a general theorem, that this process has a jointly continuous version---more precisely, Hölder continuous of all orders $<1/2$ in $a$ and in $t$

Comment: For a modern version with references see Yor,

Keywords: Local times, Hilbert transform, Ito formula

Nature: Original

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XVII: 26, 227-239, LNM 986 (1983)

Le problème de Skorokhod sur $\bf R$ : une approche avec le temps local (Brownian motion)

A solution to Skorohod's embedding problem is given, using the first hiting time of a set by the 2-dimensional process which consists of Brownian motion and its local time at zero. The author's aim is to ``correct'' the asymmetry inherent to the Azéma-Yor construction

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding, Local times

Nature: Original

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XIX: 28, 314-331, LNM 1123 (1985)

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

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,

Keywords: Brownian motion, Local times, Self-intersection

Nature: Original proofs

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XIX: 29, 332-349, LNM 1123 (1985)

Compléments aux formules de Tanaka-Rosen (Brownian motion)

Several variants of Rosen's works (

Comment: Examples of further work on this subject, using stochastic calculus or not, are Werner,

Keywords: Brownian motion, Local times, Self-intersection

Nature: Original proofs

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XIX: 30, 350-365, LNM 1123 (1985)

Renormalisation et convergence en loi pour des temps locaux d'intersection du mouvement brownien dans ${\bf R}^3$ (Brownian motion)

It is shown that no renormalization à la Varadhan occurs for the self-intersection local times of 3-dimensional Brownian motion; but a weaker result is established: when the point $y\in

Comment: This result was used by Le Gall in his work on fluctuations of the Wiener sausage:

Keywords: Brownian motion, Local times, Self-intersection

Nature: Original

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XX: 33, 515-531, LNM 1204 (1986)

A renormalized local time for multiple intersections of planar Brownian motion (Brownian motion)

Using Fourier techniques, the existence of a renormalized local time for $n$-fold self-intersections of planar Brownian motion is obtained, thus extending the case $n=2$, obtained in the pioneering work of Varadhan (Appendix to

Comment: Closely related to 2036. A general reference is Le Gall,

Keywords: Local times, Self-intersection

Nature: Original

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XX: 35, 543-552, LNM 1204 (1986)

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

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,

Keywords: Local times, Self-intersection, Previsible representation

Nature: Original proofs

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XXV: 33, 407-424, LNM 1485 (1991)

Second order limit laws for the local times of stable processes (Limit theorems)

Using the method of moments, a central limit theorem is established for the increments $L^x_t-L^0_t$ of the local times of a symmetric $\beta$-stable process ($\beta>1$). The limit law is that of a fractional Brownian sheet, with Hurst index $\beta-1$, time-changed via $L_t^0$ in its time variable

Comment: Another proof due to Eisenbaum 3120 uses Dynkin's isomorphism. Ray-Knight theorems for these local times can be found in Eisenbaum-Kaspi-Marcus-Rosen-Shi

Keywords: Local times, Stable processes, Method of moments, Fractional Brownian motion, Brownian sheet

Nature: Original

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XXIX: 26, 266-289, LNM 1613 (1995)

Une version sans conditionnement du théorème d'isomorphisme de Dynkin (Limit theorems)

After establishing an unconditional version of Dynkin's isomorphism theorem, the author applies this theorem to give a new proof of Ray-Knight theorems for Brownian local times, and also to give another proof to limit theorems due to Rosen 2533 concerning the increments of the local times of a symmetric $\beta$-stable process for $\beta>1$. Some results by Marcus-Rosen (

Comment: A general reference on the subject is Marcus-Rosen,

Keywords: Stable processes, Local times, Central limit theorem, Dynkin isomorphism, Fractional Brownian motion, Brownian sheet

Nature: Original

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XXXI: 20, 216-224, LNM 1655 (1997)

Théorèmes limites pour les temps locaux d'un processus stable symétrique (Limit theorems)

Using Dynkin's isomorphism, a central-limit type theorem is derived for the local times of a stable symmetric process of index $\beta$ at a finite number $n$ of levels. The limiting process is expressed in terms of a fractional, $n$-dimensional Brownian sheet with Hurst index $\beta-1$. The case when $n=1$ is due to Rosen 2533, and, for Brownian local times, to Yor 1709

Comment: This kind of result is now understood as a weak form of theorems à la Ray-Knight, describing the local times of a stable symmetric process: see Eisenbaum-Kaspi-Marcus-Rosen-Shi

Keywords: Stable processes, Local times, Central limit theorem, Dynkin isomorphism, Fractional Brownian motion, Brownian sheet

Nature: Original

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XXXIII: 20, 388-394, LNM 1709 (1999)

The distribution of local times of a Brownian bridge (Brownian motion)

Several useful identities for the one-dimensional marginals of local times of Brownian bridges are derived. This is a variation and extension on the well-known joint law of the maximum and the value of Brownian motion at a given time

Comment: Useful references are Borodin,

Keywords: Local times, Brownian bridge

Nature: Original

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XLII: 12, 331-363, LNM 1978 (2009)

Penalisation of the standard random walk by a function of the one-sided maximum of the local time or of the duration of the excursions (Theory of stochastic processes)

Keywords: Penalisation, Excursions, Local times

Nature: Original

XLIII: 03, 95-104, LNM 2006 (2011)

A stochastic calculus proof of the CLT for the $L^{2}$ modulus of continuity of local time (Theory of Brownian motion)

Keywords: Central Limit Theorem, Moduli of continuity, Local times, Brownian motion

Nature: Original

XLIII: 04, 105-126, LNM 2006 (2011)

On a zero-one law for the norm process of transient random walk (Theory of random walks)

Keywords: Zero-one law, Random walk, Local times, Jeulin's lemma

Nature: Original

XLIII: 09, 221-239, LNM 2006 (2011)

Simulation of a Local Time Fractional Stable Motion (Theory of processes)

Keywords: Stable processes, Self-similar processes, Shot noise series, Local times, Fractional Brownian motion, Simulation

Nature: Original

XLIV: 08, 167-190, LNM 2046 (2012)

Discrete approximation to solution flows of Tanaka's SDE related to Walsh Brownian motion (Stochastic calculus, Limit theorems)

Keywords: Walsh's Brownian motion, Tanaka's SDE, Local times

Nature: Original