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Rough Paths, Probability and Related Topics

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Rough Paths, Probability and Related Topics Rough paths, probability and related topics vorgelegt von Master of Science Atul Shekhar geb. in Darbhanga Von der Fakult¨atII - Mathematik und Naturwissenschaften der Technischen Universit¨atBerlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr.rer.nat. genehmigte Dissertation Promotionsausschuss: Vorsitzende: Prof. Dr. Boris Springborn Berichter/Gutachter: Prof. Dr. Peter K. Friz Berichter/Gutachter: Prof. Dr. Terry Lyons Tag der wissenschaftlichen Aussprache: 09 December 2015 Berlin 2016 ii To my parents Anandi Das and Lakshman Lal Das v Acknowledgement Successful completion of my doctorate degree is indebted to many people to whom I express my sincere gratitude. First and foremost of all, a big thanks goes to my supervisior Prof. Dr. Peter K. Friz who has constantly supported me during this period. I admire him for the patience that he had with me. He constantly kept me motivated through many research discussion sessions, during which i learnt a lot from him. He has shaped me into a better mathematician and a researcher by giving me great ideas and directing me into right path. Due to his support, I attended many conferences and summer schools to deliver a talk, which made me a better teacher. His encouragement indeed has laid the foundational stones of current work. I would like to thank Prof. Terry Lyons for being the external examiners of my thesis which includes careful examining the results in this manuscript. I would also like to thanks Prof. Boris Springborn for accepting to act as a chairman in my PhD defense. In June 2014, I had the oppurtunity to visit university of Washington, Seattle, under the invitation of Prof. Steffen Rohde. He also visited Berlin for 3 months in the winter semester 2014-15 accompanied by a series of lectures on random geometry. Meeting and discussing mathematics with him was truly inspirational and I came to learn a lot about theory of Schramm-Loewner evolutions through him. In Nov 2014, I was invited as an speaker in the weekly seminar in Oxford-Man institute, Oxford by Prof. Terry Lyons. I also visited Oxford university during summer semester 2015 for two months as a research visitor, adding to my collaborating research experience. I thank Prof. Steffen Rohde and Prof. Terry Lyons for their invitation and taking their valuable time for fruitful discussions we had. Of course I wouldn't forget to mention the part of my friends in Berlin for making my stay so beautiful. Sharing mathematical ideas along with sharing jokes and fun moments during the coffee breaks leaves you wonder the meaning of stress. Ahappy brain is what all you need for mathematics afterall. Thank you Alberto Chiarini, Giovanni Conforti, Adrian Gonzalez, Giuseppe Cannizzaro, Matti Leimbach, Benedickt Rehle and Julie Meissner. I also thank the postdoctoral fellows in our research group for exposing me to different areas of research and to always be there for clearing my doubts and questions. Thank you Antoine Dalqvist, Romain Allez, Joscha Diehl and Paul Gassiat from Technical university of Berlin, Horatia Boedihardjo from Oxford university, Huy Tran from university of Washington and Laure Dumaz from Cambridge university. I thank Berlin Mathematical School and reseach training group 1845 for providing the financial support on the one hand and organising wonderful talks and conferences like BMS fridays, minicourses, RTG summer schools on the other. Last but not the least, all this couldn't have been possible without the support of my family, Anandi Das, Lakshman Lal Das, Leena Rajesh kumari, Uday Shekhar, Rashmi Kumari and Himanshu Shekhar. Thank you for having belief in me and supporting and suggesting all the important decisions of my career and life. vi Introduction The contents of this doctoral dissertation consists of results in two different areas of probability theory and its applications, namely Rough path theory and theory of Loewner evolutions. In order to describe best, we take a view on the two areas seperately and then present the results. Rough path theory Rough path theory, abbreviated RPT, was introduced by Terry Lyons in a seminal pa- per [54] in 1998 for a systematic study of continuous paths which are very irregular. Continuous paths, whether deterministic or random, appear frequently in real world as a mathematical model for something which evolves in time. It could be describing the motion of a particle in an external field, it could be some time series data from financial world, e.g. price of a financial commodity like call options or could it be the trajec- tory of a script alphabet being written on a touchscreen interactive device like computer tablets. In many applications, it is very essential to associate parameters to the path which can be used to derive informations it. One seeks for a set of parameters which is efficient and easy to implement and also should also contain relevant informations about the path which will distinguish it from other ones. For example, given a smooth path X : [0;T ] ! Rd, length of the path Z T _ L(X) = jXrjdr 0 is one such parameter measuring the length of the path. Other examples include winding number of a closed complex valued (or two dimensional) path, 1 Z 1 W (X; P ) = dz 2πi X z − P which measures number of times path X turns around point P . These examples are very useful for certain appliations but they do not cover all the aspects of the path. More generally, in 1957, K. Chen [[7], [8]] introduced signature of a path, which is collection of its all possible iterated integrals as a set of such parameters which would describe all the important aspects of the path in an efficient manner. d Definition .0.1. Given a path X : [0;T ] ! R , signature of X, denoted S(X)0;T is defined as ( Z Z Z ) S(X)0;T = 1; dXr1 ; dXr1 ⊗dXr2 ; :::; dXr1 ⊗dXr2 ⊗::⊗dXrn ; ::: 0<r1<T 0<r1<r2<T 0<r1<r2<::<rn<T given that all the integrals above are well defined, e.g. when X is a smooth path. vii viii Signature can be considered as the building block of Rough path theory. In 1936, L.C. Young proved the following remarkable result. Theorem .0.2 (Young, 1936). Let α; β 2 (0; 1] such that α+β > 1. If X 2 Cα([0;T ]; Rd) and Y 2 Cβ([0;T ]; Rn×d) are two H¨olderregular paths. Then Z T X YrdXr := lim Ys(Xt − Xs) jPj!0 0 [s;t]2P is well defined, where P denote a partition of [0;T ] and jPj its mesh size. Furthermore, there exist a constant C such that for all s < t, ⏐Z t ⏐ ⏐ ⏐ α+β ⏐ YrdXr − Ys(Xt − Xs)⏐ ≤ CjjXjjαjjY jjβjt − sj ⏐ s ⏐ Young's result guarantees that iterated integrals in definition of signature are well 1 defined wheneve X is α-H¨olderpath for α > 2 . But when X is even more irregular, it is not possible to make sense of signature of path X. The key idea of T. Lyons leading to emergence of RPT was abstractifying the concept of signature. When the path X is not regular enough, the integrals in the definition of signature are not well defined. But, if one could construct an objects which mimics signature of X upto some algebraic and analytical properties, then one could use that object in the place of signature for practically all relevant purposes. RPT also comes with a extension theorem which gurantees that it is only required to construct an objects which mimics signature only upto some finitely many levels and all the higher order iterated integrals can be naturally N d N d d constructed. Let T1 (R ) be the truncated tensor algebra defined by T1 (R ) := 1 ⊕ R ⊕ fRdg⊗2 ⊕ :: ⊕ fRdg⊗N with the multiplication g ⊗ h such that k X πk(g ⊗ h) = πi(g) ⊗ πk−i(h) i=0 N d ⊗k N d where πk is the projection map πk : T1 ! fR g . G (R ) denote the step N nilpotent group. α d 1 Definition .0.3. Let X 2 C ([0;T ]; R ) be a path and let N = [ α ]. A rough path N d associated to X is a path X : [0;T ] ! T1 (R ) such that π1(X) = X and for 1 ≤ k ≤ N, jπk(Xs;t)j jjπk(X)jjkα := sup kα < 1 s6=t jt − sj −1 where Xs;t := Xs ⊗ Xt. A rough path is called a geometric rough path if it takes value N d N d in G (R ) ⊂ T1 (R ). The rough path X indeed mimics the signature and it can be proved that when X is smooth, Z πk(X) = dXr1 ⊗ dXr2 ⊗ :: ⊗ dXrk 0<r1<r2<::<rk<T An another strategy for styding properties of a path is to construct calculus based on that path, i.e. make sense of integration Z YrdXr ix for appropriate paths Y . This strategy has proved to be very effective in many situations, e.g. many non-trivial properties of Brownian motion sample path can be derived using Ito stochastic calculus. The rough path X associated to X can also be used develop an integration theory called \rough integration". In particular, rough integration can be used to make sense of higher order iterated integrals Z dXr1 ⊗ dXr2 ⊗ :: ⊗ dXrn 0<r1<r2<::<rn<T for n > N, justying the fact that it is enough to construct the object X which mimics the signature upto N levels. The key idea behind rough integration is that since X is not regular enough to define integration, we enhance the path X via X, adding some extra information to it and X can be used to define the (rough) integration.
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