Title Ergodicity of diffusion processes
Title (croatian) Ergodičnost procesa difuzija
Author Petra Lazić
Mentor Nikola Sandrić (mentor)
Committee member Miljenko Huzak (predsjednik povjerenstva)
Committee member Nikola Sandrić (član povjerenstva)
Committee member Stjepan Šebek (član povjerenstva)
Committee member Snježana Lubura Strunjak (član povjerenstva)
Granter University of Zagreb Faculty of Science (Department of Mathematics) Zagreb
Defense date and country 2022-03-24, Croatia
Scientific / art field, discipline and subdiscipline NATURAL SCIENCES Mathematics
Universal decimal classification (UDC ) 51 - Mathematics
Abstract The topic of this work is ergodicity (stochastic stability) of various types of stochastic processes. The urge for analysis of random processes exists in every area of science and real life - medicine, biology, chemistry, physics, finance, etc., as many phenomena naturally exhibit some sort of random behaviour in their movement. Mathematical models used to describe those random movements are called stochastic differential equations (SDEs). Since the solutions of SDEs often have a very complicated structure or are impossible to obtain explicitly, it is usually hard to analyse them directly. Therefore, the emphasis has been placed on analysing their long-term stability. This includes detecting their equilibria (stacionary distributions) as well as the rate at which convergence occurs. The convergence is observed with respect to some appropriate distance function. In my work the emphasis has been placed on the quantitative aspect of this problem, namely, on finding explicit bounds on the rate of the convergence with respect to two distance functions: to total variation distance and the class of Wasserstein distances (which provides convergence in some weaker sense). As most of the existing results in this area correspond to the geometric ergodicity (that is, the case when the rate of the convergence is exponential), and known conditions ensuring sub-goemetric ergodicity are far from being optimal because there are many known examples of sub-geometrically ergodic systems that do not satisfy those conditions, the focus of my work is to find sharp and general conditions in terms of coefficients of the process that will ensure sub-geometric ergodicity of a wide range of processes. The results of my research can be divided in two parts. Firstly, I will consider classical diffusion processes - Markov processes with continuous trajectories (here, the class of processes with singular diffusion coefficients will be of special interest as they were not investigated so far). The second part of the research deals with somewhat more complicated random processes - diffusion processes with Markovian switching, which are processes that, beside the continuous, diffusive one, have a second, discrete component which changes the behaviour of the process at random times. This theory is relatively new so here we still have many interesting open questions and uninvestigated phenomena that are not characteristic for classical diffusion process. Also, in both cases, I will extend the results on a class of processes with jumps.
Abstract (croatian) Svrha mog rada je istražiti problem ergodičnosti (stohastičke stabilnosti) različitih tipova slučajnih procesa. Slučajni procesi iznimno su važni jer se koriste za modeliranje pojava u gotovo svim područjima znanosti i svakodnevnog života – primjenjuju se u medicini, biologiji, kemiji, fizici, financijama itd. Naime, u svim tim područjima često se dolazi do zaključka da se pojave ne mogu opisati determinističkim modelima jer su neki aspekti njihovog ponašanja slučajni. Matematički modeli koji se koriste za opisivanje ovakvih pojava su stohastičke diferencijalne jednadžbe (SDJ). Međutim, budući da rješenja SDJ-ova imaju jako kompliciranu strukturu i iznimno ih je teško analizirati direktnim putem, naglasak se stavlja na analizu njihove dugoročne stabilnosti. To uključuje određivanje njihovih ekvilibrija (stacionarnih distribucija), ali i brzine kojom konvergiraju prema tim ekvilibrijima. Konvergencija se promatra s obzirom na neku odredenu funkciju udaljenosti. U mojem istraživanju naglasak je stavljen na kvantitativni aspekt ovog problema, odnosno na eksplicitne ocjene brzine konvergencije s obzirom na dvije funkcije udaljenosti: udaljenost totalne varijacije i klasu Wassersteinovih udaljenosti (koja predstavlja konvergenciju u nešto slabijem smislu). Kako se većina dosadašnjih rezultata odnosi na geometrijsku ergodičnost (tj. slučaj kada je brzina konvergencije eksponencijalna), a poznati uvjeti za subgeometrijsku ergodičnost nisu blizu optimalnih jer su poznati mnogi sub-geometrijski ergodični sustavi koji te uvjete ne zadovoljavaju, fokus mog rada je pronaći oštre i opće uvjete u terminima koeficijenata samog procesa koji će osigurati subgeometrijsku ergodičnost široke klase procesa. Rezultate mog istraživanja mogu podijeliti u dvije cjeline. U prvoj ću proučavati klasične difuzije - Markovljeve procese neprekidnih trajektorija (gdje će od posebnog značaja biti klasa procesa sa singularnim difuzijskim koeficijentima koji do sada nisu razmatrani). Drugi dio rada proučava nešto složenije procese - difuzije sa slučajnim prebacivanjem, koji osim neprekidne, difuzijske komponente sadrže i drugu, diskretnu komponentu koja u slučajnim trenucima mijenja ponašanje procesa. Ova teorija je relativno nova pa tu ima još puno zanimljivih otvorenih pitanja i neistraženih pojava koje nisu karakteristične za klasične procese difuzija. Također, u oba slučaja, rezultate ću primijeniti na klasu procesa sa skokovima.
Keywords
stochastic differential equations
diffusion processes
diffusion processes with Markovian switching
ergodicity
sub-geometric ergodicity
\(\varphi\)-irreducibility
phi-irreducibility
aperiodicity
total variation distance
Wasserstein distance.
Keywords (croatian)
stohastičke diferencijalne jednadžbe
difuzije
difuzije sa slučajnim prebacivanjem
ergodičnost
subgeometrijska ergodičnost
\(\varphi\)-ireducibilnost
phi-ireducibilnost
aperiodičnost
udaljenost totalne varijacije
Wassersteinova udaljenost.
Language english
URN:NBN urn:nbn:hr:217:478121
Promotion 2022
Study programme Title: Mathematics Study programme type: university Study level: postgraduate Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
Type of resource Text
Extent viii, 155 str.
File origin Born digital
Access conditions Open access
Terms of use
Created on 2022-05-02 08:27:45