A large deviation principle for a RWRC in a box

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "A large deviation principle for a RWRC in a box"

Transcription

1 A large deviation principle for a RWRC in a box 7th Cornell Probability Summer School Michele Salvi TU Berlin July 12, 2011 Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

2 Outline 1 Large Deviations for dummies What are Large Deviations? A proper denition 2 Random Walk among Random Conductances The model The main theorem Related elds Some heuristics Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

3 Large Deviations for dummies What are Large Deviations? X 1, X 2,... i.i.d. R-valued random variables with E [X 1 ] = 0 and Var(X1) = σ 2 R. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

4 Large Deviations for dummies What are Large Deviations? X 1, X 2,... i.i.d. R-valued random variables with E [X 1 ] = 0 and Var(X1) = σ 2 R. Strong law of large numbers (SLLN): ( 1 P n n j=1 X j ) n 0 = 1; Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

5 Large Deviations for dummies What are Large Deviations? X 1, X 2,... i.i.d. R-valued random variables with E [X 1 ] = 0 and Var(X1) = σ 2 R. Strong law of large numbers (SLLN): ( 1 P n n j=1 Central limit theorem (CLT): ( 1 P σ n n j=1 X j ) n 0 = 1; ) n X j A 1 e y2 2 dy; 2π A Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

6 Large Deviations for dummies What are Large Deviations? X 1, X 2,... i.i.d. R-valued random variables with E [X 1 ] = 0 and Var(X1) = σ 2 R. Strong law of large numbers (SLLN): ( 1 P n n j=1 Central limit theorem (CLT): ( 1 P σ n n j=1 X j ) n 0 = 1; ) n X j A 1 e y2 2 dy; 2π A Large Deviation Principle (LDP) (+ nite exponential moments): ( 1 P n n j=1 ) X j x e ni(x), x 0. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

7 Large Deviations for dummies What are Large Deviations? Large Deviation Theory deals with asymptotic computation of small probabilities on an exponential scale. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

8 Large Deviations for dummies What are Large Deviations? Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

9 Large Deviations for dummies What are Large Deviations? The function I(x) is convex, has compact level sets (= is lower semi-continuous), I(x) 0 and equality holds i x = µ = E [X 1 ]. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

10 Large Deviations for dummies A proper denition Denition Let X be a Polish space. A function I : X [0, ] is called rate function if I I has compact level sets (= lower semicontinuous) Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

11 Large Deviations for dummies A proper denition Denition Let X be a Polish space. A function I : X [0, ] is called rate function if I I has compact level sets (= lower semicontinuous) Denition Let γ n be a sequence in R +. A sequence of probability measures {µ n } on (X, B(X )) satises a large deviation principle with rate function I and speed γ n if Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

12 Large Deviations for dummies A proper denition Denition Let X be a Polish space. A function I : X [0, ] is called rate function if I I has compact level sets (= lower semicontinuous) Denition Let γ n be a sequence in R +. A sequence of probability measures {µ n } on (X, B(X )) satises a large deviation principle with rate function I and speed γ n if 1 For every open set O, lim inf n 2 For every closed set C, lim sup n 1 log µ n (O) inf I(x); γ n x O 1 log µ n (C) inf I(x). γ n x C Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

13 Large Deviations for dummies A proper denition Remarks 1 I lower semi-continuous = attains a minimum on every compact set; Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

14 Large Deviations for dummies A proper denition Remarks 1 I lower semi-continuous = attains a minimum on every compact set; 2 for a "nice" set A µ n (A) e γn inf A I ; Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

15 Large Deviations for dummies A proper denition Remarks 1 I lower semi-continuous = attains a minimum on every compact set; 2 for a "nice" set A µ n (A) e γn inf A I ; 3 µ n (X ) = 1 = inf x X I(x) = min I(x) = 0; x X Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

16 Large Deviations for dummies A proper denition Remarks 1 I lower semi-continuous = attains a minimum on every compact set; 2 for a "nice" set A µ n (A) e γn inf A I ; 3 µ n (X ) = 1 = inf x X I(x) = min I(x) = 0; x X 4 if!x s.t. I(x) = 0, the LDP implies SLLN ; Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

17 Large Deviations for dummies A proper denition Remarks 1 I lower semi-continuous = attains a minimum on every compact set; 2 for a "nice" set A µ n (A) e γn inf A I ; 3 µ n (X ) = 1 = inf x X I(x) = min I(x) = 0; x X 4 if!x s.t. I(x) = 0, the LDP implies SLLN ; 5 in general no relation between LDP and CLT. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

18 The model The model Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

19 The model The model Consider the lattice Z d and assign to any bond (x, x + e) a random weight ω x,e such that ω x,e = ω x+e, e (symmetry), {ω x,e } x Zd,e E are i.i.d., ω x,e 0 (positivity). Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

20 The model The model Consider the lattice Z d and assign to any bond (x, x + e) a random weight ω x,e such that ω x,e = ω x+e, e (symmetry), {ω x,e } x Zd,e E are i.i.d., ω x,e 0 (positivity). Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

21 The model The model Consider the lattice Z d and assign to any bond (x, x + e) a random weight ω x,e such that ω x,e = ω x+e, e (symmetry), {ω x,e } x Zd,e E are i.i.d., ω x,e 0 (positivity). Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

22 The model The model Consider the lattice Z d and assign to any bond (x, x + e) a random weight ω x,e such that ω x,e = ω x+e, e (symmetry), {ω x,e } x Zd,e E are i.i.d., ω x,e 0 (positivity). Denition The Random Walk among Random Conductances (RWRC) is the continuous-time process generated by ω f (x) := ( ) ω x,e f (x + e) f (x). x Z d,e E Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

23 The model Let (X t ) t [0, ) be the RWRC. For x B Z d, B nite and connected set, dene the local time l t (x) := t 0 1l {Xs=x} ds. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

24 The model Let (X t ) t [0, ) be the RWRC. For x B Z d, B nite and connected set, dene the local time l t (x) := t 0 1l {Xs=x} ds. We want to study the annealed behaviour of l t : ( 1 ) P ω 0 t l t g 2 where g 2 M 1 (B) and is the expectation w.r.t. the conductances. The conductances can attain arbitrarily small values. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

25 The model Let (X t ) t [0, ) be the RWRC. For x B Z d, B nite and connected set, dene the local time l t (x) := t 0 1l {Xs=x} ds. We want to study the annealed behaviour of l t : ( 1 ) P ω 0 t l t g 2 where g 2 M 1 (B) and is the expectation w.r.t. the conductances. The conductances can attain arbitrarily small values. Three noises: the conductances; the waiting times; the embedded discrete-time RW. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

26 The main theorem Hypothesis: B Z d nite and connected; log Pr( ωx,e < ε ) ε η, for ε 0, η > 1. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

27 The main theorem Hypothesis: B Z d nite and connected; log Pr( ωx,e < ε ) ε η, for ε 0, η > 1. Theorem (joint work with Wolfgang König and Tilman Wol) The process of empirical measures ( 1l t t) t R + of the Random Walk among Random Conductances under the annealed law P ω( {supp(l 0 t) B}) satises a large deviation principle on M 1 (B) with speed γ t = t η η+1 and rate function J given by J(g 2 ) := C η g(z + e) g(z) 2η η+1 = C η g z,e for all g 2 M 1 (B), where C η := ( η ) η 1 1+η. 2η 1+η 2η 1+η Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

28 The main theorem This means P ω ( {1 0 t l t g 2} { supp(l t ) B }) 2η γt Cη g(z+e) g(z) η+1 e z,e. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

29 The main theorem This means P ω ( {1 0 t l t g 2} { supp(l t ) B }) 2η γt Cη g(z+e) g(z) η+1 e z,e. In particular: Corollary The annealed probability of non-exit from the box B for the Random Walk among Random Conductances for t 0 is log P ω 0 ( supp(l t ) B ) t η 1+η inf g 2 M 1 (B) C η z,e g(z + e) g(z) 2η η+1. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

30 Related elds Related elds: Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

31 Related elds Related elds: By a Fourier expansion: P ω 0 ( ) d #B supp(l t ) B = e tλω(b) k f k (0) f k, 1l e tλω 1 (B), k=1 where λ ω 1 (B) is the bottom of the spectrum of ω restricted to the box B. Relation with Random Schrödinger operators! Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

32 Related elds Related elds: By a Fourier expansion: P ω 0 ( ) d #B supp(l t ) B = e tλω(b) k f k (0) f k, 1l e tλω 1 (B), k=1 where λ ω 1 (B) is the bottom of the spectrum of ω restricted to the box B. Relation with Random Schrödinger operators! Parabolic Anderson model with random Laplace operator: { t u(x, t) = ω u(x, t) + ξ(x)u(x, t), t (0, ), x Z d u(x, 0) = δ 0 (x) x Z d. Feynman-Kac formula gives u(x, t) = E ω x is a RWRC. [ t ] e 0 ξ(xs)ds δ 0 (X t ), where X t Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

33 Some heuristics Sketch of the proof (heuristics): rescale the conductances (t r ω = ϕ); Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

34 Some heuristics Sketch of the proof (heuristics): rescale the conductances (t r ω = ϕ); combine "classical" LDP's for the empirical measure of a (weighted) random walk by Donsker and Varadhan (1979) (speed t 1 r ) and for the conductances (speed t rη ); Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

35 Some heuristics Sketch of the proof (heuristics): rescale the conductances (t r ω = ϕ); combine "classical" LDP's for the empirical measure of a (weighted) random walk by Donsker and Varadhan (1979) (speed t 1 r ) and for the conductances (speed t rη ); "physicists' trick" (t 1 r = t rη ); Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

36 Some heuristics Sketch of the proof (heuristics): rescale the conductances (t r ω = ϕ); combine "classical" LDP's for the empirical measure of a (weighted) random walk by Donsker and Varadhan (1979) (speed t 1 r ) and for the conductances (speed t rη ); "physicists' trick" (t 1 r = t rη ); optimization over the rescaled shape of the conductances. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

37 Some heuristics Technical obstacles: Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

38 Some heuristics Technical obstacles: Lower bound for open sets. Problem: need to understand the asymptotics of inf P ϕ( 1 ) ϕ A t l t. There seems to be no monotonicity, but there is some kind of continuity of the map ϕ P ϕ 0 ( ). Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

39 Some heuristics Technical obstacles: Lower bound for open sets. Problem: need to understand the asymptotics of inf P ϕ( 1 ) ϕ A t l t. There seems to be no monotonicity, but there is some kind of continuity of the map ϕ P ϕ 0 ( ). Upper bound for closed sets. Problem: t r ω is not bounded. We need a compactication argument for the space of rescaled conductances. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

40 Some heuristics Thank you! Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

The parabolic Anderson model on Z d with time-dependent potential: Frank s works

The parabolic Anderson model on Z d with time-dependent potential: Frank s works Weierstrass Institute for Applied Analysis and Stochastics The parabolic Anderson model on Z d with time-dependent potential: Frank s works Based on Frank s works 2006 2016 jointly with Dirk Erhard (Warwick),

More information

Annealed Brownian motion in a heavy tailed Poissonian potential

Annealed Brownian motion in a heavy tailed Poissonian potential Annealed Brownian motion in a heavy tailed Poissonian potential Ryoki Fukushima Research Institute of Mathematical Sciences Stochastic Analysis and Applications, Okayama University, September 26, 2012

More information

Annealed Brownian motion in a heavy tailed Poissonian potential

Annealed Brownian motion in a heavy tailed Poissonian potential Annealed Brownian motion in a heavy tailed Poissonian potential Ryoki Fukushima Tokyo Institute of Technology Workshop on Random Polymer Models and Related Problems, National University of Singapore, May

More information

Brownian survival and Lifshitz tail in perturbed lattice disorder

Brownian survival and Lifshitz tail in perturbed lattice disorder Brownian survival and Lifshitz tail in perturbed lattice disorder Ryoki Fukushima Kyoto niversity Random Processes and Systems February 16, 2009 6 B T 1. Model ) ({B t t 0, P x : standard Brownian motion

More information

General Theory of Large Deviations

General Theory of Large Deviations Chapter 30 General Theory of Large Deviations A family of random variables follows the large deviations principle if the probability of the variables falling into bad sets, representing large deviations

More information

A Note On Large Deviation Theory and Beyond

A Note On Large Deviation Theory and Beyond A Note On Large Deviation Theory and Beyond Jin Feng In this set of notes, we will develop and explain a whole mathematical theory which can be highly summarized through one simple observation ( ) lim

More information

A Representation of Excessive Functions as Expected Suprema

A Representation of Excessive Functions as Expected Suprema A Representation of Excessive Functions as Expected Suprema Hans Föllmer & Thomas Knispel Humboldt-Universität zu Berlin Institut für Mathematik Unter den Linden 6 10099 Berlin, Germany E-mail: foellmer@math.hu-berlin.de,

More information

C.7. Numerical series. Pag. 147 Proof of the converging criteria for series. Theorem 5.29 (Comparison test) Let a k and b k be positive-term series

C.7. Numerical series. Pag. 147 Proof of the converging criteria for series. Theorem 5.29 (Comparison test) Let a k and b k be positive-term series C.7 Numerical series Pag. 147 Proof of the converging criteria for series Theorem 5.29 (Comparison test) Let and be positive-term series such that 0, for any k 0. i) If the series converges, then also

More information

Uniformly Uniformly-ergodic Markov chains and BSDEs

Uniformly Uniformly-ergodic Markov chains and BSDEs Uniformly Uniformly-ergodic Markov chains and BSDEs Samuel N. Cohen Mathematical Institute, University of Oxford (Based on joint work with Ying Hu, Robert Elliott, Lukas Szpruch) Centre Henri Lebesgue,

More information

u( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3)

u( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3) M ath 5 2 7 Fall 2 0 0 9 L ecture 4 ( S ep. 6, 2 0 0 9 ) Properties and Estimates of Laplace s and Poisson s Equations In our last lecture we derived the formulas for the solutions of Poisson s equation

More information

Random Walks on Hyperbolic Groups III

Random Walks on Hyperbolic Groups III Random Walks on Hyperbolic Groups III Steve Lalley University of Chicago January 2014 Hyperbolic Groups Definition, Examples Geometric Boundary Ledrappier-Kaimanovich Formula Martin Boundary of FRRW on

More information

Weak convergence and large deviation theory

Weak convergence and large deviation theory First Prev Next Go To Go Back Full Screen Close Quit 1 Weak convergence and large deviation theory Large deviation principle Convergence in distribution The Bryc-Varadhan theorem Tightness and Prohorov

More information

MATH 6605: SUMMARY LECTURE NOTES

MATH 6605: SUMMARY LECTURE NOTES MATH 6605: SUMMARY LECTURE NOTES These notes summarize the lectures on weak convergence of stochastic processes. If you see any typos, please let me know. 1. Construction of Stochastic rocesses A stochastic

More information

Semi-strongly asymptotically non-expansive mappings and their applications on xed point theory

Semi-strongly asymptotically non-expansive mappings and their applications on xed point theory Hacettepe Journal of Mathematics and Statistics Volume 46 (4) (2017), 613 620 Semi-strongly asymptotically non-expansive mappings and their applications on xed point theory Chris Lennard and Veysel Nezir

More information

P (A G) dp G P (A G)

P (A G) dp G P (A G) First homework assignment. Due at 12:15 on 22 September 2016. Homework 1. We roll two dices. X is the result of one of them and Z the sum of the results. Find E [X Z. Homework 2. Let X be a r.v.. Assume

More information

Brownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539

Brownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539 Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory

More information

Generalised extreme value statistics and sum of correlated variables.

Generalised extreme value statistics and sum of correlated variables. and sum of correlated variables. Laboratoire Charles Coulomb CNRS & Université Montpellier 2 P.C.W. Holdsworth, É. Bertin (ENS Lyon), J.-Y. Fortin (LPS Nancy), S.T. Bramwell, S.T. Banks (UCL London). University

More information

A new proof of Gromov s theorem on groups of polynomial growth

A new proof of Gromov s theorem on groups of polynomial growth A new proof of Gromov s theorem on groups of polynomial growth Bruce Kleiner Courant Institute NYU Groups as geometric objects Let G a group with a finite generating set S G. Assume that S is symmetric:

More information

9.2 Branching random walk and branching Brownian motions

9.2 Branching random walk and branching Brownian motions 168 CHAPTER 9. SPATIALLY STRUCTURED MODELS 9.2 Branching random walk and branching Brownian motions Branching random walks and branching diffusions have a long history. A general theory of branching Markov

More information

Alignment processes on the sphere

Alignment processes on the sphere Alignment processes on the sphere Amic Frouvelle CEREMADE Université Paris Dauphine Joint works with : Pierre Degond (Imperial College London) and Gaël Raoul (École Polytechnique) Jian-Guo Liu (Duke University)

More information

Lecture No 1 Introduction to Diffusion equations The heat equat

Lecture No 1 Introduction to Diffusion equations The heat equat Lecture No 1 Introduction to Diffusion equations The heat equation Columbia University IAS summer program June, 2009 Outline of the lectures We will discuss some basic models of diffusion equations and

More information

f-divergence Estimation and Two-Sample Homogeneity Test under Semiparametric Density-Ratio Models

f-divergence Estimation and Two-Sample Homogeneity Test under Semiparametric Density-Ratio Models IEEE Transactions on Information Theory, vol.58, no.2, pp.708 720, 2012. 1 f-divergence Estimation and Two-Sample Homogeneity Test under Semiparametric Density-Ratio Models Takafumi Kanamori Nagoya University,

More information

Polymer Theory: Freely Jointed Chain

Polymer Theory: Freely Jointed Chain Polymer Theory: Freely Jointed Chain E.H.F. February 2, 23 We ll talk today about the most simple model for a single polymer in solution. It s called the freely jointed chain model. Each monomer occupies

More information

Percolation by cumulative merging and phase transition for the contact process on random graphs.

Percolation by cumulative merging and phase transition for the contact process on random graphs. Percolation by cumulative merging and phase transition for the contact process on random graphs. LAURENT MÉNARD and ARVIND SINGH arxiv:502.06982v [math.pr] 24 Feb 205 Abstract Given a weighted graph, we

More information

Stochastic Processes

Stochastic Processes Stochastic Processes A very simple introduction Péter Medvegyev 2009, January Medvegyev (CEU) Stochastic Processes 2009, January 1 / 54 Summary from measure theory De nition (X, A) is a measurable space

More information

1. Introduction Let fx(t) t 1g be a real-valued mean-zero Gaussian process, and let \kk" be a semi-norm in the space of real functions on [ 1]. In the

1. Introduction Let fx(t) t 1g be a real-valued mean-zero Gaussian process, and let \kk be a semi-norm in the space of real functions on [ 1]. In the Small ball estimates for Brownian motion under a weighted -norm by hilippe BERTHET (1) and Zhan SHI () Universite de Rennes I & Universite aris VI Summary. Let fw (t) t 1g be a real-valued Wiener process,

More information

Hypothesis testing for Stochastic PDEs. Igor Cialenco

Hypothesis testing for Stochastic PDEs. Igor Cialenco Hypothesis testing for Stochastic PDEs Igor Cialenco Department of Applied Mathematics Illinois Institute of Technology igor@math.iit.edu Joint work with Liaosha Xu Research partially funded by NSF grants

More information

LARGE DEVIATIONS FOR STOCHASTIC PROCESSES

LARGE DEVIATIONS FOR STOCHASTIC PROCESSES LARGE DEVIATIONS FOR STOCHASTIC PROCESSES By Stefan Adams Abstract: The notes are devoted to results on large deviations for sequences of Markov processes following closely the book by Feng and Kurtz ([FK06]).

More information

Characteristic Functions and the Central Limit Theorem

Characteristic Functions and the Central Limit Theorem Chapter 6 Characteristic Functions and the Central Limit Theorem 6.1 Characteristic Functions 6.1.1 Transforms and Characteristic Functions. There are several transforms or generating functions used in

More information

Lecture 2. We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales.

Lecture 2. We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales. Lecture 2 1 Martingales We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales. 1.1 Doob s inequality We have the following maximal

More information

18.175: Lecture 14 Infinite divisibility and so forth

18.175: Lecture 14 Infinite divisibility and so forth 18.175 Lecture 14 18.175: Lecture 14 Infinite divisibility and so forth Scott Sheffield MIT 18.175 Lecture 14 Outline Infinite divisibility Higher dimensional CFs and CLTs Random walks Stopping times Arcsin

More information

Diffusion for a Markov, Divergence-form Generator

Diffusion for a Markov, Divergence-form Generator Diffusion for a Markov, Divergence-form Generator Clark Department of Mathematics Michigan State University Arizona School of Analysis and Mathematical Physics March 16, 2012 Abstract We consider the long-time

More information

L 2 Geometry of the Symplectomorphism Group

L 2 Geometry of the Symplectomorphism Group University of Notre Dame Workshop on Innite Dimensional Geometry, Vienna 2015 Outline 1 The Exponential Map on D s ω(m) 2 Existence of Multiplicity of Outline 1 The Exponential Map on D s ω(m) 2 Existence

More information

semi-group and P x the law of starting at x 2 E. We consider a branching mechanism : R +! R + of the form (u) = au + bu 2 + n(dr)[e ru 1 + ru]; (;1) w

semi-group and P x the law of starting at x 2 E. We consider a branching mechanism : R +! R + of the form (u) = au + bu 2 + n(dr)[e ru 1 + ru]; (;1) w M.S.R.I. October 13-17 1997. Superprocesses and nonlinear partial dierential equations We would like to describe some links between superprocesses and some semi-linear partial dierential equations. Let

More information

Bielefeld Course on Nonlinear Waves - June 29, Department of Mathematics University of North Carolina, Chapel Hill. Solitons on Manifolds

Bielefeld Course on Nonlinear Waves - June 29, Department of Mathematics University of North Carolina, Chapel Hill. Solitons on Manifolds Joint work (on various projects) with Pierre Albin (UIUC), Hans Christianson (UNC), Jason Metcalfe (UNC), Michael Taylor (UNC), Laurent Thomann (Nantes) Department of Mathematics University of North Carolina,

More information

On Schrödinger equations with inverse-square singular potentials

On Schrödinger equations with inverse-square singular potentials On Schrödinger equations with inverse-square singular potentials Veronica Felli Dipartimento di Statistica University of Milano Bicocca veronica.felli@unimib.it joint work with Elsa M. Marchini and Susanna

More information

Lifshitz tail for Schödinger Operators with random δ magnetic fields

Lifshitz tail for Schödinger Operators with random δ magnetic fields Lifshitz tail for Schödinger Operators with random δ magnetic fields Takuya Mine (Kyoto Institute of Technology, Czech technical university in Prague) Yuji Nomura (Ehime University) March 16, 2010 at Arizona

More information

Notes on Mathematical Expectations and Classes of Distributions Introduction to Econometric Theory Econ. 770

Notes on Mathematical Expectations and Classes of Distributions Introduction to Econometric Theory Econ. 770 Notes on Mathematical Expectations and Classes of Distributions Introduction to Econometric Theory Econ. 77 Jonathan B. Hill Dept. of Economics University of North Carolina - Chapel Hill October 4, 2 MATHEMATICAL

More information

Harmonic Functions and Brownian motion

Harmonic Functions and Brownian motion Harmonic Functions and Brownian motion Steven P. Lalley April 25, 211 1 Dynkin s Formula Denote by W t = (W 1 t, W 2 t,..., W d t ) a standard d dimensional Wiener process on (Ω, F, P ), and let F = (F

More information

x log x, which is strictly convex, and use Jensen s Inequality:

x log x, which is strictly convex, and use Jensen s Inequality: 2. Information measures: mutual information 2.1 Divergence: main inequality Theorem 2.1 (Information Inequality). D(P Q) 0 ; D(P Q) = 0 iff P = Q Proof. Let ϕ(x) x log x, which is strictly convex, and

More information

Max-Planck-Institut fur Mathematik in den Naturwissenschaften Leipzig A nonlocal anisotropic model for phase transitions Part II: Asymptotic behaviour of rescaled energies by Giovanni Alberti and Giovanni

More information

Classical behavior of the integrated density of states for the uniform magnetic field and a randomly perturbed

Classical behavior of the integrated density of states for the uniform magnetic field and a randomly perturbed Classical behavior of the integrated density of states for the uniform magnetic field and a randomly perturbed lattice * Naomasa Ueki 1 1 Graduate School of Human and Environmental Studies, Kyoto University

More information

VISCOSITY SOLUTIONS OF ELLIPTIC EQUATIONS

VISCOSITY SOLUTIONS OF ELLIPTIC EQUATIONS VISCOSITY SOLUTIONS OF ELLIPTIC EQUATIONS LUIS SILVESTRE These are the notes from the summer course given in the Second Chicago Summer School In Analysis, in June 2015. We introduce the notion of viscosity

More information

FEM for Stokes Equations

FEM for Stokes Equations FEM for Stokes Equations Kanglin Chen 15. Dezember 2009 Outline 1 Saddle point problem 2 Mixed FEM for Stokes equations 3 Numerical Results 2 / 21 Stokes equations Given (f, g). Find (u, p) s.t. u + p

More information

Upper Bound for Intermediate Singular Values of Random Sub-Gaussian Matrices 1

Upper Bound for Intermediate Singular Values of Random Sub-Gaussian Matrices 1 Upper Bound for Intermediate Singular Values of Random Sub-Gaussian Matrices 1 Feng Wei 2 University of Michigan July 29, 2016 1 This presentation is based a project under the supervision of M. Rudelson.

More information

1 Introduction This work follows a paper by P. Shields [1] concerned with a problem of a relation between the entropy rate of a nite-valued stationary

1 Introduction This work follows a paper by P. Shields [1] concerned with a problem of a relation between the entropy rate of a nite-valued stationary Prexes and the Entropy Rate for Long-Range Sources Ioannis Kontoyiannis Information Systems Laboratory, Electrical Engineering, Stanford University. Yurii M. Suhov Statistical Laboratory, Pure Math. &

More information

Intertwinings for Markov processes

Intertwinings for Markov processes Intertwinings for Markov processes Aldéric Joulin - University of Toulouse Joint work with : Michel Bonnefont - Univ. Bordeaux Workshop 2 Piecewise Deterministic Markov Processes ennes - May 15-17, 2013

More information

Derivatives. Differentiability problems in Banach spaces. Existence of derivatives. Sharpness of Lebesgue s result

Derivatives. Differentiability problems in Banach spaces. Existence of derivatives. Sharpness of Lebesgue s result Differentiability problems in Banach spaces David Preiss 1 Expanded notes of a talk based on a nearly finished research monograph Fréchet differentiability of Lipschitz functions and porous sets in Banach

More information

LEBESGUE INTEGRATION. Introduction

LEBESGUE INTEGRATION. Introduction LEBESGUE INTEGATION EYE SJAMAA Supplementary notes Math 414, Spring 25 Introduction The following heuristic argument is at the basis of the denition of the Lebesgue integral. This argument will be imprecise,

More information

Some Semi-Markov Processes

Some Semi-Markov Processes Some Semi-Markov Processes and the Navier-Stokes equations NW Probability Seminar, October 22, 2006 Mina Ossiander Department of Mathematics Oregon State University 1 Abstract Semi-Markov processes were

More information

balls, Edalat and Heckmann [4] provided a simple explicit construction of a computational model for a Polish space. They also gave domain theoretic pr

balls, Edalat and Heckmann [4] provided a simple explicit construction of a computational model for a Polish space. They also gave domain theoretic pr Electronic Notes in Theoretical Computer Science 6 (1997) URL: http://www.elsevier.nl/locate/entcs/volume6.html 9 pages Computational Models for Ultrametric Spaces Bob Flagg Department of Mathematics and

More information

On the quantiles of the Brownian motion and their hitting times.

On the quantiles of the Brownian motion and their hitting times. On the quantiles of the Brownian motion and their hitting times. Angelos Dassios London School of Economics May 23 Abstract The distribution of the α-quantile of a Brownian motion on an interval [, t]

More information

Closest Moment Estimation under General Conditions

Closest Moment Estimation under General Conditions Closest Moment Estimation under General Conditions Chirok Han and Robert de Jong January 28, 2002 Abstract This paper considers Closest Moment (CM) estimation with a general distance function, and avoids

More information

Simultaneous vs. non simultaneous blow-up

Simultaneous vs. non simultaneous blow-up Simultaneous vs. non simultaneous blow-up Juan Pablo Pinasco and Julio D. Rossi Departamento de Matemática, F..E y N., UBA (428) Buenos Aires, Argentina. Abstract In this paper we study the possibility

More information

are harmonic functions so by superposition

are harmonic functions so by superposition J. Rauch Applied Complex Analysis The Dirichlet Problem Abstract. We solve, by simple formula, the Dirichlet Problem in a half space with step function boundary data. Uniqueness is proved by complex variable

More information

XXXV Konferencja Statystyka Matematyczna -7-11/ XII 2009

XXXV Konferencja Statystyka Matematyczna -7-11/ XII 2009 XXXV Konferencja Statystyka Matematyczna -7-11/ XII 2009 1 Statistical Inference for Image Symmetries Mirek Pawlak pawlak@ee.umanitoba.ca 2 OUTLINE I Problem Statement II Image Representation in the Radial

More information

HOMEWORK ASSIGNMENT 6

HOMEWORK ASSIGNMENT 6 HOMEWORK ASSIGNMENT 6 DUE 15 MARCH, 2016 1) Suppose f, g : A R are uniformly continuous on A. Show that f + g is uniformly continuous on A. Solution First we note: In order to show that f + g is uniformly

More information

Numerical Approximation of Phase Field Models

Numerical Approximation of Phase Field Models Numerical Approximation of Phase Field Models Lecture 2: Allen Cahn and Cahn Hilliard Equations with Smooth Potentials Robert Nürnberg Department of Mathematics Imperial College London TUM Summer School

More information

LECTURE 12 UNIT ROOT, WEAK CONVERGENCE, FUNCTIONAL CLT

LECTURE 12 UNIT ROOT, WEAK CONVERGENCE, FUNCTIONAL CLT MARCH 29, 26 LECTURE 2 UNIT ROOT, WEAK CONVERGENCE, FUNCTIONAL CLT (Davidson (2), Chapter 4; Phillips Lectures on Unit Roots, Cointegration and Nonstationarity; White (999), Chapter 7) Unit root processes

More information

GENERALIZED RAY KNIGHT THEORY AND LIMIT THEOREMS FOR SELF-INTERACTING RANDOM WALKS ON Z 1. Hungarian Academy of Sciences

GENERALIZED RAY KNIGHT THEORY AND LIMIT THEOREMS FOR SELF-INTERACTING RANDOM WALKS ON Z 1. Hungarian Academy of Sciences The Annals of Probability 1996, Vol. 24, No. 3, 1324 1367 GENERALIZED RAY KNIGHT THEORY AND LIMIT THEOREMS FOR SELF-INTERACTING RANDOM WALKS ON Z 1 By Bálint Tóth Hungarian Academy of Sciences We consider

More information

Statistical Learning Theory

Statistical Learning Theory Statistical Learning Theory Fundamentals Miguel A. Veganzones Grupo Inteligencia Computacional Universidad del País Vasco (Grupo Inteligencia Vapnik Computacional Universidad del País Vasco) UPV/EHU 1

More information

On It^o's formula for multidimensional Brownian motion

On It^o's formula for multidimensional Brownian motion On It^o's formula for multidimensional Brownian motion by Hans Follmer and Philip Protter Institut fur Mathemati Humboldt-Universitat D-99 Berlin Mathematics and Statistics Department Purdue University

More information

On Coarse Geometry and Coarse Embeddability

On Coarse Geometry and Coarse Embeddability On Coarse Geometry and Coarse Embeddability Ilmari Kangasniemi August 10, 2016 Master's Thesis University of Helsinki Faculty of Science Department of Mathematics and Statistics Supervised by Erik Elfving

More information

Large Sample Properties & Simulation

Large Sample Properties & Simulation Large Sample Properties & Simulation Quantitative Microeconomics R. Mora Department of Economics Universidad Carlos III de Madrid Outline Large Sample Properties (W App. C3) 1 Large Sample Properties (W

More information

CONSEQUENCES OF POWER SERIES REPRESENTATION

CONSEQUENCES OF POWER SERIES REPRESENTATION CONSEQUENCES OF POWER SERIES REPRESENTATION 1. The Uniqueness Theorem Theorem 1.1 (Uniqueness). Let Ω C be a region, and consider two analytic functions f, g : Ω C. Suppose that S is a subset of Ω that

More information

Delta Method. Example : Method of Moments for Exponential Distribution. f(x; λ) = λe λx I(x > 0)

Delta Method. Example : Method of Moments for Exponential Distribution. f(x; λ) = λe λx I(x > 0) Delta Method Often estimators are functions of other random variables, for example in the method of moments. These functions of random variables can sometimes inherit a normal approximation from the underlying

More information

Classical and quantum behavior of the integrated density of states for a randomly perturbed lattice

Classical and quantum behavior of the integrated density of states for a randomly perturbed lattice Classical and quantum behavior of the integrated density of states for a randomly perturbed lattice Ryoki Fukushima and Naomasa Ueki Abstract. The asymptotic behavior of the integrated density of states

More information

Equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains

Equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains Equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains J. Földes Department of Mathematics, Univerité Libre de Bruxelles 1050 Brussels, Belgium P. Poláčik School

More information

Exponential stability of families of linear delay systems

Exponential stability of families of linear delay systems Exponential stability of families of linear delay systems F. Wirth Zentrum für Technomathematik Universität Bremen 28334 Bremen, Germany fabian@math.uni-bremen.de Keywords: Abstract Stability, delay systems,

More information

Monte Carlo simulations of harmonic and anharmonic oscillators in discrete Euclidean time

Monte Carlo simulations of harmonic and anharmonic oscillators in discrete Euclidean time Monte Carlo simulations of harmonic and anharmonic oscillators in discrete Euclidean time DESY Summer Student Programme, 214 Ronnie Rodgers University of Oxford, United Kingdom Laura Raes University of

More information

Limit theorems for Hawkes processes

Limit theorems for Hawkes processes Limit theorems for nearly unstable and Mathieu Rosenbaum Séminaire des doctorants du CMAP Vendredi 18 Octobre 213 Limit theorems for 1 Denition and basic properties Cluster representation of Correlation

More information

Advanced Probability Theory (Math541)

Advanced Probability Theory (Math541) Advanced Probability Theory (Math541) Instructor: Kani Chen (Classic)/Modern Probability Theory (1900-1960) Instructor: Kani Chen (HKUST) Advanced Probability Theory (Math541) 1 / 17 Primitive/Classic

More information

Converse bounds for private communication over quantum channels

Converse bounds for private communication over quantum channels Converse bounds for private communication over quantum channels Mark M. Wilde (LSU) joint work with Mario Berta (Caltech) and Marco Tomamichel ( Univ. Sydney + Univ. of Technology, Sydney ) arxiv:1602.08898

More information

roots, e.g., Barnett (1966).) Quite often in practice, one, after some calculation, comes up with just one root. The question is whether this root is

roots, e.g., Barnett (1966).) Quite often in practice, one, after some calculation, comes up with just one root. The question is whether this root is A Test for Global Maximum Li Gan y and Jiming Jiang y University of Texas at Austin and Case Western Reserve University Abstract We give simple necessary and sucient conditions for consistency and asymptotic

More information

BRANCHING PROCESSES 1. GALTON-WATSON PROCESSES

BRANCHING PROCESSES 1. GALTON-WATSON PROCESSES BRANCHING PROCESSES 1. GALTON-WATSON PROCESSES Galton-Watson processes were introduced by Francis Galton in 1889 as a simple mathematical model for the propagation of family names. They were reinvented

More information

Lecture 5. 1 Chung-Fuchs Theorem. Tel Aviv University Spring 2011

Lecture 5. 1 Chung-Fuchs Theorem. Tel Aviv University Spring 2011 Random Walks and Brownian Motion Tel Aviv University Spring 20 Instructor: Ron Peled Lecture 5 Lecture date: Feb 28, 20 Scribe: Yishai Kohn In today's lecture we return to the Chung-Fuchs theorem regarding

More information

Solution Sheet 3. Solution Consider. with the metric. We also define a subset. and thus for any x, y X 0

Solution Sheet 3. Solution Consider. with the metric. We also define a subset. and thus for any x, y X 0 Solution Sheet Throughout this sheet denotes a domain of R n with sufficiently smooth boundary. 1. Let 1 p

More information

Deforming conformal metrics with negative Bakry-Émery Ricci Tensor on manifolds with boundary

Deforming conformal metrics with negative Bakry-Émery Ricci Tensor on manifolds with boundary Deforming conformal metrics with negative Bakry-Émery Ricci Tensor on manifolds with boundary Weimin Sheng (Joint with Li-Xia Yuan) Zhejiang University IMS, NUS, 8-12 Dec 2014 1 / 50 Outline 1 Prescribing

More information

Metastability for the Ising model on a random graph

Metastability for the Ising model on a random graph Metastability for the Ising model on a random graph Joint project with Sander Dommers, Frank den Hollander, Francesca Nardi Outline A little about metastability Some key questions A little about the, and

More information

Closest Moment Estimation under General Conditions

Closest Moment Estimation under General Conditions Closest Moment Estimation under General Conditions Chirok Han Victoria University of Wellington New Zealand Robert de Jong Ohio State University U.S.A October, 2003 Abstract This paper considers Closest

More information

Conditional Value-at-Risk (CVaR) Norm: Stochastic Case

Conditional Value-at-Risk (CVaR) Norm: Stochastic Case Conditional Value-at-Risk (CVaR) Norm: Stochastic Case Alexander Mafusalov, Stan Uryasev RESEARCH REPORT 03-5 Risk Management and Financial Engineering Lab Department of Industrial and Systems Engineering

More information

The Distributions of Stopping Times For Ordinary And Compound Poisson Processes With Non-Linear Boundaries: Applications to Sequential Estimation.

The Distributions of Stopping Times For Ordinary And Compound Poisson Processes With Non-Linear Boundaries: Applications to Sequential Estimation. The Distributions of Stopping Times For Ordinary And Compound Poisson Processes With Non-Linear Boundaries: Applications to Sequential Estimation. Binghamton University Department of Mathematical Sciences

More information

Classical and quantum behavior of the integrated density of states for a randomly perturbed lattice

Classical and quantum behavior of the integrated density of states for a randomly perturbed lattice Classical and quantum behavior of the integrated density of states for a randomly perturbed lattice Ryoki Fukushima Naomasa Ueki March 13, 2009 Abstract The asymptotic behavior of the integrated density

More information

ON LARGE DEVIATIONS OF MARKOV PROCESSES WITH DISCONTINUOUS STATISTICS 1

ON LARGE DEVIATIONS OF MARKOV PROCESSES WITH DISCONTINUOUS STATISTICS 1 The Annals of Applied Probability 1998, Vol. 8, No. 1, 45 66 ON LARGE DEVIATIONS OF MARKOV PROCESSES WITH DISCONTINUOUS STATISTICS 1 By Murat Alanyali 2 and Bruce Hajek Bell Laboratories and University

More information

Homogenization of a Hele-Shaw-type problem in periodic time-dependent med

Homogenization of a Hele-Shaw-type problem in periodic time-dependent med Homogenization of a Hele-Shaw-type problem in periodic time-dependent media University of Tokyo npozar@ms.u-tokyo.ac.jp KIAS, Seoul, November 30, 2012 Hele-Shaw problem Model of the pressure-driven }{{}

More information

5th-order differentiation

5th-order differentiation ARBITRARY-ORDER REAL-TIME EXACT ROBUST DIFFERENTIATION A. Levant Applied Mathematics Dept., Tel-Aviv University, Israel E-mail: levant@post.tau.ac.il Homepage: http://www.tau.ac.il/~levant/ 5th-order differentiation

More information

Mathematics 426 Robert Gross Homework 9 Answers

Mathematics 426 Robert Gross Homework 9 Answers Mathematics 4 Robert Gross Homework 9 Answers. Suppose that X is a normal random variable with mean µ and standard deviation σ. Suppose that PX > 9 PX

More information

Functional Properties of MMSE

Functional Properties of MMSE Functional Properties of MMSE Yihong Wu epartment of Electrical Engineering Princeton University Princeton, NJ 08544, USA Email: yihongwu@princeton.edu Sergio Verdú epartment of Electrical Engineering

More information

A PHYSICAL SPACE PROOF OF THE BILINEAR STRICHARTZ AND LOCAL SMOOTHING ESTIMATES FOR THE SCHRÖDINGER EQUATION

A PHYSICAL SPACE PROOF OF THE BILINEAR STRICHARTZ AND LOCAL SMOOTHING ESTIMATES FOR THE SCHRÖDINGER EQUATION A PHYSICAL SPACE PROOF OF THE BILINEAR STRICHARTZ AND LOCAL SMOOTHING ESTIMATES FOR THE SCHRÖDINGER EQUATION TERENCE TAO Abstract. Let d 1, and let u, v : R R d C be Schwartz space solutions to the Schrödinger

More information

Mixing time for a random walk on a ring

Mixing time for a random walk on a ring Mixing time for a random walk on a ring Stephen Connor Joint work with Michael Bate Paris, September 2013 Introduction Let X be a discrete time Markov chain on a finite state space S, with transition matrix

More information

Chapter 1. Optimality Conditions: Unconstrained Optimization. 1.1 Differentiable Problems

Chapter 1. Optimality Conditions: Unconstrained Optimization. 1.1 Differentiable Problems Chapter 1 Optimality Conditions: Unconstrained Optimization 1.1 Differentiable Problems Consider the problem of minimizing the function f : R n R where f is twice continuously differentiable on R n : P

More information

Nonconcave Penalized Likelihood with A Diverging Number of Parameters

Nonconcave Penalized Likelihood with A Diverging Number of Parameters Nonconcave Penalized Likelihood with A Diverging Number of Parameters Jianqing Fan and Heng Peng Presenter: Jiale Xu March 12, 2010 Jianqing Fan and Heng Peng Presenter: JialeNonconcave Xu () Penalized

More information

On duality theory of conic linear problems

On duality theory of conic linear problems On duality theory of conic linear problems Alexander Shapiro School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 3332-25, USA e-mail: ashapiro@isye.gatech.edu

More information

A SHORT INTRODUCTION TO BANACH LATTICES AND

A SHORT INTRODUCTION TO BANACH LATTICES AND CHAPTER A SHORT INTRODUCTION TO BANACH LATTICES AND POSITIVE OPERATORS In tis capter we give a brief introduction to Banac lattices and positive operators. Most results of tis capter can be found, e.g.,

More information

Introduction to Empirical Processes and Semiparametric Inference Lecture 08: Stochastic Convergence

Introduction to Empirical Processes and Semiparametric Inference Lecture 08: Stochastic Convergence Introduction to Empirical Processes and Semiparametric Inference Lecture 08: Stochastic Convergence Michael R. Kosorok, Ph.D. Professor and Chair of Biostatistics Professor of Statistics and Operations

More information

Linear stochastic approximation driven by slowly varying Markov chains

Linear stochastic approximation driven by slowly varying Markov chains Available online at www.sciencedirect.com Systems & Control Letters 50 2003 95 102 www.elsevier.com/locate/sysconle Linear stochastic approximation driven by slowly varying Marov chains Viay R. Konda,

More information

Unique equilibrium states for geodesic flows in nonpositive curvature

Unique equilibrium states for geodesic flows in nonpositive curvature Unique equilibrium states for geodesic flows in nonpositive curvature Todd Fisher Department of Mathematics Brigham Young University Fractal Geometry, Hyperbolic Dynamics and Thermodynamical Formalism

More information

Implementation of Sparse Wavelet-Galerkin FEM for Stochastic PDEs

Implementation of Sparse Wavelet-Galerkin FEM for Stochastic PDEs Implementation of Sparse Wavelet-Galerkin FEM for Stochastic PDEs Roman Andreev ETH ZÜRICH / 29 JAN 29 TOC of the Talk Motivation & Set-Up Model Problem Stochastic Galerkin FEM Conclusions & Outlook Motivation

More information

Correlation Detection and an Operational Interpretation of the Rényi Mutual Information

Correlation Detection and an Operational Interpretation of the Rényi Mutual Information Correlation Detection and an Operational Interpretation of the Rényi Mutual Information Masahito Hayashi 1, Marco Tomamichel 2 1 Graduate School of Mathematics, Nagoya University, and Centre for Quantum

More information

Tail inequalities for additive functionals and empirical processes of. Markov chains

Tail inequalities for additive functionals and empirical processes of. Markov chains Tail inequalities for additive functionals and empirical processes of geometrically ergodic Markov chains University of Warsaw Banff, June 2009 Geometric ergodicity Definition A Markov chain X = (X n )

More information