Seminar za teoriju vjerojatnosti / Probability Seminar
Seminar za teoriju vjerojatnosti /
Probability Seminar
Department of Mathematics
University of Zagreb
Croatia |
2012./2013.:
Održani seminari:
Nasumicnost i Gross-Pitaevskiijeva hijerarhija
(Randomization and the Gross-Pitaevskii hierarchy)
SAZETAK: Gross-Pitaevskiijeva hijerarhija je beskonacan sustav
linearnih parcijalnih diferencijalnih jednadzbi koji se pojavljuje u
izvodu nelinearne Schrodingerove jednadzbe. U ovom predavanju cemo
promatrati GP hijerarhiju sa nasumicnim kolizijskim operatorom na
trodimenzionalnom torusu. Koristenjem nasumicnosti, dobit cemo veci
skup eksponenata regularnosti u kljucnoj ocjeni u prostoru i vremenu.
Navedenu ocjenu cemo dalje primjeniti u proucavanju Duhamelovog
razvoja u nasumicnoj GP hijerarhiji. Ovo je zajednicki rad sa
Gigliolom Staffilani./
ABSTRACT: The Gross-Pitaevskii hierarchy is an infinite system of
linear partial differential equations which occurs in the derivation
of the nonlinear Schrodinger equation. In this talk, we will consider
the GP hierarchy with randomized collisions on the three-dimensional
torus. Using randomness, we will extend the range of regularity
exponents in the key spacetime estimate. The latter bound will let us
control the Duhamel expansion associated to the randomized GP
hierarchy. This is a joint work with Gigliola Staffilani.
Cenzurirani procesi
(Censored processes)
Još jedna mjera sličnosti nizova znakova
(Yet Another String Similarity Measure)
Zadržavanje zajednickih alela u jednoj i dvije populacije
(Persistence of common alleles in one and two populations)
Demokratski sustavi II (zajedničko predavanje sa Seminarom za funkcionalnu analizu)
Demokratski sustavi I (zajedničko predavanje sa Seminarom za funkcionalnu analizu)
Granično ponašanje slučajnih varijabli opaženih u slučajn vremenu IV
(Limiting behaviour of random variables observed in random times IV)
| 7.5.2013. |
14:00-18:00 |
(002) | |
PROBABILITY AFTERNOON
14:00
Exit problems for spectrally negative Levy and
Markov Additive processes
Abstract:
In this lecture we present the solutions of exit problems for a
spectrally negative Markov additive process (MAP) and its reflections.
A particular example of MAP is a L\'{e}vy process.
At the beginning we give a review of some fluctuation theory for
spectrally negative
L\'{e}vy processes using for the most part martingale theory.
We unveil also the way so-called scale function appear in the solutions.
Later, in the more general context of MAPs, we present another
technique which is based on the occupation density formula. The scale
matrix, which is a generalization of the scale function of a
spectrally negative L\'{e}vy process, plays here also the central
role. We provide the probabilistic construction of the scale matrix,
and identify its transform. In addition, we generalize to the MAP
setting the relation between the scale function and the excursion
(height) measure. Our representation of the scale matrix
$W(x)=e^{-\Lambda x}\eL(x)$ in terms of
the matrix $L(x)$ of expected occupation times at $0$ up to the first
passage over $x$ and the transition rate matrix $\Lambda$ of the
governing Markov chain seen at this first passage times
opens up possibilities for further investigation of its properties.
The lecture is based on 3 papers:
[1] J. Ivanovs, Z. Palmowski
Occupation densities in solving exit problems for Markov additive
processes and their reflections,
Stochastic Processes and their Applications 122(9) (2012), 3342-3360.
[2] A. Kyprianou, Z. Palmowski
A martingale review of some fluctuation theory for spectrally
negative L\'{e}vy processes,
S\'{e}minaire de Probabilit\'{e}s Vol XXXVIII (2005), 16-29.
[3] A. Kyprianou, Z. Palmowski
Fluctuations of spectrally negative Markov Additive processes,
S\'{e}minaire de Probabilit\'{e}s Vol XLI (2008), 121-135.
16:00
Avoiding bubbles by Lévy processes
Abstract: Let X be a Lévy process in R^d starting at the origin satisfying
some mild conditions. We say that set A is avoidable for the process X if
there is a positive probability that X never hits the set A.
Let {B(x_n,r_n)} be a collection of disjoint balls in R^d and et A be the
union of these balls. In this talk we give necessary and sufficient
conditions for the set A to be avoidable.
17:00
Spectral theory and fluctuation theory for Lévy processes
Abstract: In 1957 Baxter and Donsker gave a formula for the Laplace
transform of the supremum of a Lévy process. I will describe how inverting
the Laplace transform leads to the spectral theory of the process killed
upon leaving half-line. My talk will be based on joint results with Jacek
Maecki and Micha Ryznar.
Granično ponašanje slučajnih varijabli opaženih u slučajnom vremenu III
(Limiting behaviour of random variables observed in random times III)
Granično ponašanje slučajnih varijabli opaženih u slučajnom vremenu II
(Limiting behaviour of random variables observed in random times II)
Moment (in)determinacy of probability distributions: recent progress
Abstract:
The discussion will be on heavy tailed distributions (of random variables or stochastic processes) whose all moments are finite. An important question in the classical moment problem is about the uniqueness. Either such a distribution is uniquely determined by its moments (M-determinate), or it is nonunique (M-indeterminate). Our goal is to describe the current state of art in this area. After a brief summary of known and widely used classical criteria (Cramer, Hausdorff, Carleman, Krein, …), we focus our attention on the following recent developments:
(a) Stieltjes classes for M-indeterminate distributions. Index of dissimilarity.
(b) New Hardy’s criterion for uniqueness. Multidimensional moment problem.
(c) Nonlinear transformations of random data and their moment (in)determinacy.
(d) Moment determinacy of distributions of stochastic processes defined by SDEs.
There will be results, hints for their proof, examples and counterexamples, and also open questions and conjectures.
Ekstremalna svojstva slabo zavisnih vremenskih i prostornih podataka;
Drago Špoljarić
Granično ponašanje slučajnih varijabli opaženih u slučajnom vremenu
(Limiting behaviour of random variables observed in random times)
O konvergenciji procesa u Skorohodovim prostorima
Laplaceova transformacija i eksponencijalno ponasanje reprezentirajucih mjera
(Laplace transforms and exponential behaviour of representing measures
Sažetak: U ovom predavanju promatramo slučajeve kada reprezentirajuća mjera Laplaceove transformacije ima eksponencijalno ponašanje. Dobivene ocjene se mogu primijeniti na ispitivanje asimptotskog ponašanja Lévyjevih mjera subordinatora i jezgri jedne klase nelokalnih (integralnih) operatora./
Abstract: We consider some classes of Laplace transforms withrepresenting measures that have exponential behavior. Estimates ofthese measures are used to obtain asymptotical properties of Lévymeasures of some subordinators and kernels of some non-local(integral) operators.
Uvjet prolaznosti za klasu jednodimenzionalnih simetričnih Levyjevih procesa
(A transience property for a class of one-dimensional symmetric Levyprocesses)
Granični Harnackov princip i Martinova granica u beskonačnosti
(Boundary Harnack principle and Martin boundary at infinity)
Greenova funkcija subordiniranog Brownovog gibanja
(On Green function of subordinate Brownian motions)
Sažetak:
U ovom seminaru će biti predstavljene ocjene Greenove funkcije u
omeđenim C^{1,1} skupovima za klasu subordiniranih Brownovih gibanja, koja,
osim stabilnih, sadrži i geometrijski stabilne Levyjeve procese. /
Abstract:
We present sharp estimates of Green function for a class of
subordinate Brownian motions, which includes stable and geometric
stable processes.
O jedinstvu rješenja trodimenzionalne periodične
Gross-Pitaevskiijeve hijerarhije
(On the uniqueness of solutions to the three-dimensional
periodic Gross-Pitaevskii hierarchy)
Hillov procjenitelj i primjena na cjelobrojnoj mrezi
(Hill estimator and an application to the integer grid)
Vjerojatnost propasti i distribucija supremuma za generalizirane procese rizika III
(Ruin probability and distribtion of supremum for generalized risk processes III)
Sustavi translacija i redundancija III
(Systems of translates and redundancy III)
G-Brownian Motion - Brownian Motion with Variance Uncertainty
Osnove Steinove metode za normalnu aproksimaciju
(Fundamentals of Stein's method for
normal approximation)
Vjerojatnost propasti i distribucija supremuma za generalizirane procese rizika II
(Ruin probability and distribtion of supremum for generalized risk processes II)
Vjerojatnost propasti i distribucija supremuma za generalizirane procese rizika
(Ruin probability and distribtion of supremum for generalized risk processes)
Sustavi translacija i redundancija II
(Systems of translates and redundancy II)
Sustavi translacija i redundancija
(Systems of translates and redundancy)
Sažetak:
Osnovni sustavi koji se pojavljuju u harmonijskoj analizi su sustavi
generirani translacijama, dilatacijama i modulacijama (npr. valični
sustavi, Gaborovi sustavi). Postojanje redundancije u takvim sustavima
pokazalo se, s praktične strane, kao vrlo poželjno svojstvo.
Karakterizacija raznih vrsta linearne nezavisnosti beskonačnog tipa u
terminima tzv. periodizacijske funkcije dio je potreban za razumijevanje
redundancije u kontekstu sustava translacija. Prezentirat ćemo rezultate
vezane uz problem l^p-linearne nezavisnosti te omega-linearne
nezavisnosti. /
Abstract:
Various systems generated from a single function by using three simple
operators - translation, dilation and modulation (e.g. wavelet systems,
Gabor systems) play an important role in harmonic analysis. It turned
out
to be useful if such systems allow redundancy. An important part for
understanding this problem in whole is expressing the linear
independence
of the system of translates in terms of the periodization function (we
consider various types of infinite linear independence, rather than
finite, which is trivial). We will present some results concerning
l^p-linear independence and omega-linear independence.
Markovljeva svojstva procesa na vremenolikim grafovima
i asimptotika Brownovog gibanja na kvadratnoj mreži
(Markov properties of processes on time-like graphs and
asymptotics of the Browninan motion on a square net)
Sažetak:
U prvom dijelu predavanja bavimo se konstrukcijom procesa čiji
je skup indeksa induciran posebnim vrstom grafova, tzv.
vremenolikim grafovima. Postavlja se pitanje uz koje uvjete
možemo konstruirati proces na takvim grafovima, koja svojstva
ima taj proces obzirom na strukturu samog grafa i uz koje
uvjete mu je distribucija jedinstvena.
U drugom dijelu analizirat ćemo takav tip procesa na kvadratnoj
mreži koju smo dobili diobom jediničnog kvadrata, kao i što
se zbiva s procesom kad broj kvadrata mreže postaje sve veći.
Prilikom analize, kovarijacijske strukutre ovog procesa rabit
ćemo brojne poznate
alate iz vjerojatnosti kao sto su slučajna šetnja i zakon velikih
brojeva, a za analizu graničnog procesa u slučaju
Brownovog gibanja morat ćemo analizirati i maksimum (nekoreliranih)
slučajnih varijabli koje imaju Gaussov rep.
Na kraju spomenut ćemo jos neke ideje vezane uz problematiku ovih
procesa.
Prošli seminari / Past seminars: