Stochastic Loewner evolution with branching and the Dyson superprocess
|
|
- Gloria Harvey
- 5 years ago
- Views:
Transcription
1 Stochastic Loewner evolution with branching and the Dyson superprocess Govind Menon (Brown University) Vivian Olsiewski Healey (Brown/University of Chicago) This talk is based on Vivian's Ph.D thesis (May 2017) as well as more recent joint work. The help of Steffen Rohde (U. of Washington) is gratefully acknowledged. Random Matrix Theory, IAS/PCMI summer school, NSF , NSF
2 Preview Galton-Watson trees The Loewner equation
3 Preview Galton-Watson trees Describe the genealogy of birthdeath processes. The Loewner equation
4 Preview Galton-Watson trees Describe the genealogy of birthdeath processes. The Loewner equation
5 Preview Galton-Watson trees Describe the genealogy of birthdeath processes. The Loewner equation
6 Preview Galton-Watson trees Describe the genealogy of birthdeath processes. The Loewner equation
7 Preview Galton-Watson trees Describe the genealogy of birthdeath processes. The Loewner equation Continuum Random Tree The CRT is a random metric space that is a universal scaling limit of these trees, conditioned to be large.
8 Preview Galton-Watson trees Describe the genealogy of birthdeath processes. The Loewner equation Gives a bijection between (certain) growth processes in the upper half-plane and (certain) evolving measures on the real line. Continuum Random Tree The CRT is a random metric space that is a universal scaling limit of these trees, conditioned to be large.
9 Preview Galton-Watson trees Describe the genealogy of birthdeath processes. The Loewner equation Gives a bijection between (certain) growth processes in the upper half-plane and (certain) evolving measures on the real line. Continuum Random Tree The CRT is a random metric space that is a universal scaling limit of these trees, conditioned to be large.
10 Preview Galton-Watson trees Describe the genealogy of birthdeath processes. The Loewner equation Gives a bijection between (certain) growth processes in the upper half-plane and (certain) evolving measures on the real line. Continuum Random Tree The CRT is a random metric space that is a universal scaling limit of these trees, conditioned to be large.
11 Preview Galton-Watson trees Describe the genealogy of birthdeath processes. The Loewner equation Gives a bijection between (certain) growth processes in the upper half-plane and (certain) evolving measures on the real line. Continuum Random Tree The CRT is a random metric space that is a universal scaling limit of these trees, conditioned to be large.
12 The main questions Q1. Can we use the Loewner equation to construct natural graph embeddings of Galton-Watson trees in the upper half plane? Q2. Can we construct a graph embedding of the CRT as a scaling limit of these embeddings of finite Galton-Watson trees? Q3. What does this construction say about "true trees" (conformally balanced embeddings) and the Brownian map? At first sight, there is no random matrix theory here. But the above problems are closely related to map enumeration.
13 Outline (a) A brief introduction to the CRT. (b) Loewner evolution with branching (tree embedding). (c) Scaling limits: the SPDE in the case of a Feller diffusion. (d) Some remark on true trees and the Brownian map. (a) is (necessary) background. Basic reference: Le Gall (1999).
14 The continuum random tree Definition: A plane tree is a rooted combinatorial tree for which the edges are assigned a cyclic order about each vertex.
15 The continuum random tree Definition: A plane tree is a rooted combinatorial tree for which the edges are assigned a cyclic order about each vertex. Each plane tree has a corresponding contour function.
16 The continuum random tree Definition: A plane tree is a rooted combinatorial tree for which the edges are assigned a cyclic order about each vertex. Each plane tree has a corresponding contour function.
17 The continuum random tree Def: A real tree is a pointed compact metric space with the tree property.
18 The continuum random tree Def: A real tree is a pointed compact metric space with the tree property. Construction: Let f :[0, 1] R 0 be continuous, and f (0) = f (1) = 0. Consider and d(u, v) =f(u)+f(v) 2 min f(s) s [u,v] u f v d(u, v) =0.
19 The continuum random tree Def: A real tree is a pointed compact metric space with the tree property. Construction: Let f :[0, 1] R 0 be continuous, and f (0) = f (1) = 0. Consider and d(u, v) =f(u)+f(v) 2 min f(s) s [u,v] u f v d(u, v) =0. Def: T f =[0, 1]/ f is the real tree coded by f.
20 The continuum random tree Definition: The continuum random tree (CRT) is the random real tree coded by the normalized Brownian excursion.
21 The continuum random tree Definition: The continuum random tree (CRT) is the random real tree coded by the normalized Brownian excursion. As the uniform distribution on rescaled Dyck paths of length 2n converges to, the uniform distribution on plane trees conditioned to have n edges converges to the CRT. 1 2n C n(2nt) 0 t 1 n (d) ( t ) 0 t 1 The CRT was introduced by Aldous ( ).
22 Chordal Loewner evolution Let γ : (0, T] H be a simple curve with γ(0) R. (t) g t ) U(t)
23 Chordal Loewner evolution Let γ : (0, T] H be a simple curve with γ(0) R. (t) g t ) The Riemann mapping theorem implies that for each t there is a unique conformal mapping g t such that 1) g t : H \ γ((0, t]) H U(t)
24 Chordal Loewner evolution Let γ : (0, T] H be a simple curve with γ(0) R. (t) g t ) The Riemann mapping theorem implies that for each t there is a unique conformal mapping g t such that 1) g t : H \ γ((0, t]) H 2) g t (z) =z + b(t) z + O 1 z 2, z. U(t)
25 Chordal Loewner evolution Let γ : (0, T] H be a simple curve with γ(0) R. (t) g t ) The Riemann mapping theorem implies that for each t there is a unique conformal mapping g t such that 1) g t : H \ γ((0, t]) H 2) g t (z) =z + b(t) z Loewner (1920s): + O 1 z 2, z. U(t) g t satisfies the initial value problem ġ t (z) = ḃ(t) g t (z) U(t), g 0(z) =z.
26 Chordal Loewner evolution General version: Let g t (z) denote the solution to the initial value problem ġ t (z) = R μ t (du) g t (z) u, g 0(z) =z. Let H t = {z H} for which g t (z) H is well defined. Then g t is the unique conformal map from H t onto H with the hydrodynamic normalization. The hull is the set H\H t = K t. Idea: The measure is supported on points that are escaping H. Conditions: {µ t } t 0 is a family of nonnegative Borel measures on R that is right continuous with left limits in the weak topology. For each t, µ s (R) and supp (µ s ) are each uniformly bounded for 0 s t.
27 Examples 1) $ t = δ U(t) N 2) μ t = produces the multislit equation (Schleissinger 13): i=1 δ Ui (t) ġ t (z) = N i=1 1 g t (z) U i (t). 3) μ t = δ κbt generates SLE κ. Question: Which measures generate embeddings of trees?
28 What's new The general form of Loewner evolution is rarely used. However, it is central to our approach. The one line summary of our work is the following conjecture: Graph embeddings of continuum trees are generated by Loewner evolution when the driving measure is a suitable superprocess. The simplest example in this class is what we call the Dyson superprocess. It is the free probability analogue of the Dawson-Watanabe superprocess.
29 Loewner evolution driven by Dyson BM with branching. (Simulation courtesy of Vivian Healey and Brent Werness.)
30 SPDE for the Dyson superprocess µ t (dx) = (x, t) dx, x (, ),t>0, t + x ( H )= Ẇ, where Ẇ is space-time white noise and is the Hilbert transform H (Hµ t )(x) = p.v. 1 x s µ t(ds). The SPDE is formal, but convenient. The measure valued process is actually defined through a martingale problem.
31 Comparison with the Dawson-Watanabe superprocess The Dawson-Watanabe superprocess is the scaling limit of branching Brownian motion when the discrete branching processes converges to the Feller diffusion. The spatial motion of each particle is independent. t = Ẇ, x R d,t>0. The Dyson superprocess is the free probability version of this SPDE: t + x ( H )= Ẇ, x R,t>0. Unlike Dawson-Watanabe, this is a superprocess of interacting particles.
32 The SPDE and stochastic Loewner evolution Consider the Cauchy-Stieltjes transform f(z,t) = 1 z s µ t(ds), z H. Define the Gaussian analytic function h(z,t),z H with covariance kernel E h(z,t) h(w, t ) = (t t ) z 1 s w 1 s µ t(ds).
33 The Dyson superprocess and Loewner evolution Let Ḃ denote white-noise (in time alone). Then (formally) tf + f z f = hḃ, z H,t>0. This SPDE may be solved by the method of characteristics dz dt = f(z,t), df = h(z,t) db, z H. The stochastic Loewner evolution is given by the subordination formula ġ t (z) =f(g t (z),t), g 0 (z) =z, z H.
34 Absolute continuity w.r.t. Lebesgue measure µ t (dx)? = (x, t) dx Dawson-Watanabe superprocess: µ t d =1 is absolutely continuous (Konno-Shiga, Reimers, 1988). µ t d 2 is singular (Perkins, 1988). Dyson superprocess: we don't know yet. The basic regularity estimates for free convolution with a semicircular law were obtained by Biane (1997). Unlike Dawson-Watanabe we would like µ t to be singular with respect to Lebesgue measure. If a density exists, then the hull cannot be a tree, so this is a crucial property.
35 Tree Embedding Question: In the deterministic setting, what conditions guarantee that the hull is a tree? Fundamental step: What conditions on the driving measure guarantee that the generated hull is a union of two simple curves in H that meet at a single point on R at nontrivial angles (not 0 or #)?
36 Tree Embedding: (α, β)-approach Setup: Let U 1,..., U n be n continuous functions U i : [0,T] R that are mutually nonintersecting U i (t) < U i+1 (t) for all i = 1,..., n and all t [0,T], except for U j (0) = U j+1 (0). Let $ t be the discrete measure μ t = c n i=1 δ Ui (t)
37 Tree Embedding: (α, β)-approach Setup: Let U 1,..., U n be n continuous functions U i : [0,T] R that are mutually nonintersecting U i (t) < U i+1 (t) for all i = 1,..., n and all t [0,T], except for U j (0) = U j+1 (0). Let $ t be the discrete measure μ t = c Definition: Let α, β (0, #) such that α + β < '. n i=1 δ Ui (t)
38 Tree Embedding: (α, β)-approach Setup: Let U 1,..., U n be n continuous functions U i : [0,T] R that are mutually nonintersecting U i (t) < U i+1 (t) for all i = 1,..., n and all t [0,T], except for U j (0) = U j+1 (0). Let $ t be the discrete measure μ t = c n i=1 δ Ui (t) Definition: Let α, β (0, #) such that α + β < '. We say that K t approaches R at U j (0) in (α, β)-direction if for each ε > 0 there is s = s ε > 0 such that there are exactly two connected components of K s that have U j (0) as a boundary point, and K j s ε {z H : π β ε < arg(z U j (0)) < π β + ε}, K j+1 s ε {z H : α ε < arg(z U j (0)) < α + ε}. (Motivated by Schleissinger 12.)
39 Tree Embedding: (α, β)-approach β α
40 Tree Embedding: (α, β)-approach Theorem (Healey): In the setting above, the hulls K t approach R in (α, β)-direction at U j (0) if U j (t) U j (0) lim t 0 t U j+1 (t) U j+1 (0) lim t 0 t = φ 1 (α, β) φ 2 (α, β) = φ 1 (α, β)+φ 2 (α, β), where φ 1 (α, β) and φ 2 (α, β) are explicitly computable functions.
41 Tree Embedding: (α, β)-approach Balanced case: If 0 < α = β < '/2, then φ 1 and φ 2 simplify to φ 1 (α, α) =0 and φ 2 (α, α) = 2c π 2α. α Intuitively: Loewner scaling If $ t generates hulls K t, then ρ$ t/ρ 2 generates the hulls ρk t. So we expect to see t whenever a hull is preserved under dilation.
42 A driving measure for any tree Let T = {ν, h(ν)} be a marked plane tree. (Think of h(ν) as the time of death of ν.) Let $ t be indexed by the elements of T alive at t: μ t = c ν Δ t T δ Uν (t).
43 Tree Embedding Let T = {ν, h(ν)} be a marked plane tree. (Think of h(ν) as the time of death of ν.) Let $ t be indexed by the elements of T alive at t: μ t = c δ Uν (t). ν Δ t T On time intervals without branching, chose the U ν to evolve according to U ν (t) = ν=η Δ t T c 1 U ν (t) U η (t).
44 The driving measure Let T = {ν, h(ν)} be a marked plane tree. (Think of h(ν) as the time of death of ν.) Let $ t be indexed by the elements of T alive at t: μ t = c δ Uν (t). ν Δ t T On time intervals without branching, chose the U ν to evolve according to U ν (t) = ν=η Δ t T c 1 U ν (t) U η (t). Theorem (Healey) : If T is a binary tree such that h ν h η, then for each 0 s max{h(ν)}, the hull K s generated at time s by the Loewner equation driven by $ t is a graph embedding of the subtree T s = {ν T : h(p(ν)) < s} in H, with the image of the root on the real line, and K s K s if s < s.
45 Tree Embedding Proof (idea): The proof relies on ODE results about the particle system U ν (t) = ν=η Δ t T c 1 U ν (t) U η (t). Extend the solution backward to the initial condition U j (0) = U j+1 (0). Show that the solution generates curves away from t = 0. Show that the generated hull approaches R in (α, α)-direction for α = π 2 + c 1 2c.
46 Example: Galton-Watson Trees with deterministic repulsion If θ is distributed as a binary Galton-Watson tree with exponential lifetimes, then the theorem guarantees that the Loewner equation driven by $ t generates a graph embedding of θ with probability one. A sample of a binary Galton- Watson tree with exponential lifetimes. (Simulation courtesy of Brent Werness.)
47 Embedding the CRT? Question 2: Let {θ k } be a sequence of random trees that (when appropriately rescaled) converges in distribution to the CRT when θ k is conditioned on having k edges. Does the law of the generated hulls converge to a scaling limit?
48 Embedding the CRT? Question 2: Let {θ k } be a sequence of random trees that (when appropriately rescaled) converges in distribution to the CRT when θ k is conditioned on having k edges. Does the law of the generated hulls converge to a scaling limit? First step: Find the scaling limit of the corresponding sequence of random driving measures.
49 Choosing a Sequence of Measures Let {T k } be a sequence of random trees, and let {c k } and {c k 1} be two sequences with elements in R +. For each k, define μ k t = ck ν Δ t T k δ Uν (t),
50 Choosing a Sequence of Measures Let {T k } be a sequence of random trees, and let {c k } and {c k 1} be two sequences with elements in R +. For each k, define μ k t = ck ν Δ t T k δ Uν (t), where the U ν (t) evolve according to U ν (t) = ν=η Δ t T k ck 1 U ν (t) U η (t).
51 Choosing a Sequence of Measures Let {T k } be a sequence of random trees, and let {c k } and {c k 1} be two sequences with elements in R +. For each k, define μ k t = ck ν Δ t T k δ Uν (t), where the U ν (t) evolve according to U ν (t) = ν=η Δ t T k ck 1 U ν (t) U η (t). Same setting as tree embedding theorem. How do we choose the trees {T k }?
52 Galton-Watson trees to the CRT Theorem (Aldous): If θ k is distributed as a critical binary Galton-Watson 1 tree with exponential lifetimes of mean 2 k, conditioned to have k edges, then θ k converges in distribution to the CRT as k.
53 Galton-Watson trees to the CRT Theorem (Aldous): If θ k is distributed as a critical binary Galton-Watson 1 tree with exponential lifetimes of mean 2 k, conditioned to have k edges, then θ k converges in distribution to the CRT as k. Question 2a: Can we find a scaling limit of {μ k t } defined by μ k t = ck ν Δ t θ k δ Uν (t), where the U ν (t) evolve according to U ν (t) = ν=η Δ t θ k ck 1 U ν (t) U η (t), if for each k, θ k is distributed as a critical binary Galton-Watson tree 1 with exponential lifetimes of mean, conditioned to have k edges? 2 k
54 The Scaling Limit For each k, the driving measure μ k t is really a measure-valued process defined for t [0, ). By convergence of driving measures we mean convergence in the Skorokhod space D Mf (R)[0, ) of functions from [0, ) to M f (R) with càdlàg paths (right continuous with left limits). Called superprocesses. How to prove convergence of superprocesses? Tightness Prokhorov: tight relatively compact subset of D Mf (R)[0, ). In particular, there is at least one limit point. Uniqueness of the limit point. (Conv. of finite dimensional marginals.)
55 Scaling limits: tightness Theorem (Healey, M.): (a) For each k, let θ k be distributed as a critical 1 binary Galton-Watson tree with exponential lifetimes of mean 2 k, conditioned to have k edges, and let {$ k } be the corresponding sequence of measures. If the scaling constants are c k = c k 1 = 1 k then the sequence {$ k } is tight in D Mf (ˆR) [0, ).
56 Scaling limits: tightness Theorem (Healey, M.): (a) For each k, let θ k be distributed as a critical 1 binary Galton-Watson tree with exponential lifetimes of mean 2 k, conditioned to have k edges, and let {$ k } be the corresponding sequence of measures. If the scaling constants are c k = c k 1 = 1 k then the sequence {$ k } is tight in D Mf (ˆR) [0, ).
57 Scaling limits: tightness Theorem (Healey, M. ): For each k, let θ k be distributed as a critical 1 binary Galton-Watson tree with exponential lifetimes of mean 2 k, conditioned to have k edges, and let {$ k } be the corresponding sequence of measures. If the scaling constants are c k = c k 1 = 1 k then the sequence {$ k } is tight in D Mf (ˆR) [0, ). Why these constants? Choose c k = c k 1, since the ratio c k 1/c k determines the branching angle. c k = 1/ k is the rescaling for which the total population process of θ k converges to L t, the local time at level t of the normalized Brownian excursion.
58 The CRT scaling limit Contour function for critical binary GW trees time
59 The CRT scaling limit Contour function for critical binary GW trees time
60 The CRT scaling limit Contour function for critical binary GW trees time Galton-Watson process
61 The Scaling limit Contour function for critical binary GW trees time Galton-Watson process Normalized Brownian excursion ( ) 0 t 1 Local time at level t of normalized Brownian excursion: L t
62 The Scaling limit Theorem (Pitman): If for each k, N k is the total population process of θ k, then N k t L t, k as k, in the sense of convergence in distribution of random variables in D Mf (R)[0, ).
63 Conditioned v. unconditioned limits Theorem (Pitman): If for each k, N k is the total population process of θ k, then N k t L t, k as k, in the sense of convergence in distribution of random variables in D Mf (R)[0, ). Standard unconditioned result: For each k, let N k t be a discrete critical Galton-Watson process (all lifetimes of length one) whose offspring distribution has finite variance. Then N k kt X t, k as k, in the sense of convergence in distribution of random variables in D Mf (R)[0, ), where X t is the Feller diffusion. (Need N k 0/k x 0 > 0, since the Feller diffusion is absorbing at 0.)
64 The scaling limit: characterization Theorem (Healey, M. ): In the unconditioned case, each subsequential limit solves the martingale problem for the Dyson superprocess: µ t, = µ 0, + t 0 R R (x) x y (y) µ s (dx)µ s (dy)+m t ( ), where the local martingale M has quadratic variation [M( )] t = 0 t µ s, 2 ds. Remark: (a) Don't know yet if the solution to the martingale problem is unique. (b) Similar result for the conditioned case (Pitman, Serlet).
65 Conformally balanced trees in C Joel Barnes (Ph.D, 2014, U. Washington) A planar tree is conformally balanced if (a) each edge has equal harmonic measure from infinity (b) edge subsets have the same measure from either side. Balanced trees are in 1-1 correspondence with Shabat polynomials (Bishop, Biane). However, these polynomials are poorly understood.
66 Conformally balanced trees in C ( ) D C B E A F H G Fig. 1. A noncrossing partition B D C E A B H F G C A Fig. 3. The domain B C A Fig. 2. The planar tree Biane (2009): builds a conformal mapping of the exterior domain by welding edges in pairs. End result is a conformal mapping, continuous onto the boundary, that gives exactly the non-crossing partition.
67 The CVS bijection and the Brownian map t t 111 t t 112 t t 12 2 t t t 2 3 t 3 t 2 t 1 t 2 t t 1? 1 t Figure 2. The Cori-Vauquelin-Schae er bijection. On the left side, a well-labeled tree (the framed numbers are the labels assigned to the vertices). On the right side, the edges of the associated quadrangulation Q appear in thick curves. Image from Le Gall (ICM, 2014).
68 Conformal CVS
69 Conformal CVS
70 Given the conformal map, each labeling gives a nested family of geodesics in the upper half plane with endpoints on its preimage. The image of these geodesics under the conformal map is a quadrangulation of the upper half plane.
The Brownian map A continuous limit for large random planar maps
The Brownian map A continuous limit for large random planar maps Jean-François Le Gall Université Paris-Sud Orsay and Institut universitaire de France Seminar on Stochastic Processes 0 Jean-François Le
More informationBrownian surfaces. Grégory Miermont based on ongoing joint work with Jérémie Bettinelli. UMPA, École Normale Supérieure de Lyon
Brownian surfaces Grégory Miermont based on ongoing joint work with Jérémie Bettinelli UMPA, École Normale Supérieure de Lyon Clay Mathematics Institute conference Advances in Probability Oxford, Sept
More informationLocality property and a related continuity problem for SLE and SKLE I
Locality property and a related continuity problem for SLE and SKLE I Masatoshi Fukushima (Osaka) joint work with Zhen Qing Chen (Seattle) October 20, 2015 Osaka University, Σ-hall 1 Locality property
More informationScaling limit of random planar maps Lecture 2.
Scaling limit of random planar maps Lecture 2. Olivier Bernardi, CNRS, Université Paris-Sud Workshop on randomness and enumeration Temuco, Olivier Bernardi p.1/25 Goal We consider quadrangulations as metric
More informationPlan 1. This talk: (a) Percolation and criticality (b) Conformal invariance Brownian motion (BM) and percolation (c) Loop-erased random walk (LERW) (d
Percolation, Brownian Motion and SLE Oded Schramm The Weizmann Institute of Science and Microsoft Research Plan 1. This talk: (a) Percolation and criticality (b) Conformal invariance Brownian motion (BM)
More information4.5 The critical BGW tree
4.5. THE CRITICAL BGW TREE 61 4.5 The critical BGW tree 4.5.1 The rooted BGW tree as a metric space We begin by recalling that a BGW tree T T with root is a graph in which the vertices are a subset of
More informationItô s excursion theory and random trees
Itô s excursion theory and random trees Jean-François Le Gall January 3, 200 Abstract We explain how Itô s excursion theory can be used to understand the asymptotic behavior of large random trees. We provide
More informationExcursion Reflected Brownian Motion and a Loewner Equation
and a Loewner Equation Department of Mathematics University of Chicago Cornell Probability Summer School, 2011 and a Loewner Equation The Chordal Loewner Equation Let γ : [0, ) C be a simple curve with
More information9.2 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 informationThe Contour Process of Crump-Mode-Jagers Branching Processes
The Contour Process of Crump-Mode-Jagers Branching Processes Emmanuel Schertzer (LPMA Paris 6), with Florian Simatos (ISAE Toulouse) June 24, 2015 Crump-Mode-Jagers trees Crump Mode Jagers (CMJ) branching
More informationScaling limits for random trees and graphs
YEP VII Probability, random trees and algorithms 8th-12th March 2010 Scaling limits for random trees and graphs Christina Goldschmidt INTRODUCTION A taste of what s to come We start with perhaps the simplest
More informationQLE. Jason Miller and Scott Sheffield. August 1, 2013 MIT. Jason Miller and Scott Sheffield (MIT) QLE August 1, / 37
QLE Jason Miller and Scott Sheffield MIT August 1, 2013 Jason Miller and Scott Sheffield (MIT) QLE August 1, 2013 1 / 37 Surfaces, curves, metric balls: how are they related? FPP: first passage percolation.
More informationThe range of tree-indexed random walk
The range of tree-indexed random walk Jean-François Le Gall, Shen Lin Institut universitaire de France et Université Paris-Sud Orsay Erdös Centennial Conference July 2013 Jean-François Le Gall (Université
More informationFast-slow systems with chaotic noise
Fast-slow systems with chaotic noise David Kelly Ian Melbourne Courant Institute New York University New York NY www.dtbkelly.com May 1, 216 Statistical properties of dynamical systems, ESI Vienna. David
More informationPlan 1. Brownian motion 2. Loop-erased random walk 3. SLE 4. Percolation 5. Uniform spanning trees (UST) 6. UST Peano curve 7. Self-avoiding walk 1
Conformally invariant scaling limits: Brownian motion, percolation, and loop-erased random walk Oded Schramm Microsoft Research Weizmann Institute of Science (on leave) Plan 1. Brownian motion 2. Loop-erased
More informationGEOMETRIC AND FRACTAL PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (SLE)
GEOMETRIC AND FRACTAL PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (SLE) Triennial Ahlfors-Bers Colloquium Gregory F. Lawler Department of Mathematics Department of Statistics University of Chicago 5734 S.
More informationRandom trees and applications
Probability Surveys Vol. 2 25) 245 311 ISSN: 1549-5787 DOI: 1.1214/15495785114 Random trees and applications Jean-François Le Gall DMA-ENS, 45 rue d Ulm, 755 PARIS, FRANCE e-mail: legall@dma.ens.fr Abstract:
More informationScaling limits of anisotropic random growth models
Scaling limits of anisotropic random growth models Amanda Turner Department of Mathematics and Statistics Lancaster University (Joint work with Fredrik Johansson Viklund and Alan Sola) Overview 1 Generalised
More informationMean-field dual of cooperative reproduction
The mean-field dual of systems with cooperative reproduction joint with Tibor Mach (Prague) A. Sturm (Göttingen) Friday, July 6th, 2018 Poisson construction of Markov processes Let (X t ) t 0 be a continuous-time
More informationWalsh Diffusions. Andrey Sarantsev. March 27, University of California, Santa Barbara. Andrey Sarantsev University of Washington, Seattle 1 / 1
Walsh Diffusions Andrey Sarantsev University of California, Santa Barbara March 27, 2017 Andrey Sarantsev University of Washington, Seattle 1 / 1 Walsh Brownian Motion on R d Spinning measure µ: probability
More informationIn terms of measures: Exercise 1. Existence of a Gaussian process: Theorem 2. Remark 3.
1. GAUSSIAN PROCESSES A Gaussian process on a set T is a collection of random variables X =(X t ) t T on a common probability space such that for any n 1 and any t 1,...,t n T, the vector (X(t 1 ),...,X(t
More informationBessel-like SPDEs. Lorenzo Zambotti, Sorbonne Université (joint work with Henri Elad-Altman) 15th May 2018, Luminy
Bessel-like SPDEs, Sorbonne Université (joint work with Henri Elad-Altman) Squared Bessel processes Let δ, y, and (B t ) t a BM. By Yamada-Watanabe s Theorem, there exists a unique (strong) solution (Y
More informationFunctional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals
Functional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals Noèlia Viles Cuadros BCAM- Basque Center of Applied Mathematics with Prof. Enrico
More informationConvergence of loop erased random walks on a planar graph to a chordal SLE(2) curve
Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve Hiroyuki Suzuki Chuo University International Workshop on Conformal Dynamics and Loewner Theory 2014/11/23 1 / 27 Introduction(1)
More informationBrownian Motion with Darning applied to KL and BF equations for planar slit domains
Brownian Motion with Darning applied to KL and BF equations for planar slit domains Masatoshi Fukushima with Z.-Q. Chen and S. Rohde September 26, 2012 at Okayama University Stochastic Analysis and Applications
More informationFirst Passage Percolation
First Passage Percolation (and other local modifications of the metric) on Random Planar Maps (well... actually on triangulations only!) N. Curien and J.F. Le Gall (Université Paris-Sud Orsay, IUF) Journées
More informationGEODESICS IN LARGE PLANAR MAPS AND IN THE BROWNIAN MAP
GEODESICS IN LARGE PLANAR MAPS AND IN THE BROWNIAN MAP Jean-François Le Gall Université Paris-Sud and Institut universitaire de France Revised version, June 2009 Abstract We study geodesics in the random
More informationExtremal process associated with 2D discrete Gaussian Free Field
Extremal process associated with 2D discrete Gaussian Free Field Marek Biskup (UCLA) Based on joint work with O. Louidor Plan Prelude about random fields blame Eviatar! DGFF: definitions, level sets, maximum
More informationInvariance Principle for Variable Speed Random Walks on Trees
Invariance Principle for Variable Speed Random Walks on Trees Wolfgang Löhr, University of Duisburg-Essen joint work with Siva Athreya and Anita Winter Stochastic Analysis and Applications Thoku University,
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationRAYLEIGH PROCESSES, REAL TREES, AND ROOT GROWTH WITH RE-GRAFTING
RAYLEIGH PROCESSES, REAL TREES, AND ROOT GROWTH WITH RE-GRAFTING STEVEN N. EVANS, JIM PITMAN, AND ANITA WINTER Abstract. The real trees form a class of metric spaces that extends the class of trees with
More informationA. Bovier () Branching Brownian motion: extremal process and ergodic theorems
Branching Brownian motion: extremal process and ergodic theorems Anton Bovier with Louis-Pierre Arguin and Nicola Kistler RCS&SM, Venezia, 06.05.2013 Plan 1 BBM 2 Maximum of BBM 3 The Lalley-Sellke conjecture
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationThe dimer model: universality and conformal invariance. Nathanaël Berestycki University of Cambridge. Colloque des sciences mathématiques du Québec
The dimer model: universality and conformal invariance Nathanaël Berestycki University of Cambridge Colloque des sciences mathématiques du Québec The dimer model Definition G = bipartite finite graph,
More informationarxiv: v1 [math.dg] 28 Jun 2008
Limit Surfaces of Riemann Examples David Hoffman, Wayne Rossman arxiv:0806.467v [math.dg] 28 Jun 2008 Introduction The only connected minimal surfaces foliated by circles and lines are domains on one of
More informationReversibility of Some Chordal SLE(κ; ρ) Traces
Reversibility of Some Chordal SLE(κ; ρ Traces Dapeng Zhan May 14, 010 Abstract We prove that, for κ (0, 4 and ρ (κ 4/, the chordal SLE(κ; ρ trace started from (0; 0 + or (0; 0 satisfies the reversibility
More informationOn semilinear elliptic equations with measure data
On semilinear elliptic equations with measure data Andrzej Rozkosz (joint work with T. Klimsiak) Nicolaus Copernicus University (Toruń, Poland) Controlled Deterministic and Stochastic Systems Iasi, July
More informationFractal Properties of the Schramm-Loewner Evolution (SLE)
Fractal Properties of the Schramm-Loewner Evolution (SLE) Gregory F. Lawler Department of Mathematics University of Chicago 5734 S. University Ave. Chicago, IL 60637 lawler@math.uchicago.edu December 12,
More informationI forgot to mention last time: in the Ito formula for two standard processes, putting
I forgot to mention last time: in the Ito formula for two standard processes, putting dx t = a t dt + b t db t dy t = α t dt + β t db t, and taking f(x, y = xy, one has f x = y, f y = x, and f xx = f yy
More informationStochastic Volatility and Correction to the Heat Equation
Stochastic Volatility and Correction to the Heat Equation Jean-Pierre Fouque, George Papanicolaou and Ronnie Sircar Abstract. From a probabilist s point of view the Twentieth Century has been a century
More informationDynamics of the evolving Bolthausen-Sznitman coalescent. by Jason Schweinsberg University of California at San Diego.
Dynamics of the evolving Bolthausen-Sznitman coalescent by Jason Schweinsberg University of California at San Diego Outline of Talk 1. The Moran model and Kingman s coalescent 2. The evolving Kingman s
More informationHard-Core Model on Random Graphs
Hard-Core Model on Random Graphs Antar Bandyopadhyay Theoretical Statistics and Mathematics Unit Seminar Theoretical Statistics and Mathematics Unit Indian Statistical Institute, New Delhi Centre New Delhi,
More informationLecture 21 Representations of Martingales
Lecture 21: Representations of Martingales 1 of 11 Course: Theory of Probability II Term: Spring 215 Instructor: Gordan Zitkovic Lecture 21 Representations of Martingales Right-continuous inverses Let
More informationErdős-Renyi random graphs basics
Erdős-Renyi random graphs basics Nathanaël Berestycki U.B.C. - class on percolation We take n vertices and a number p = p(n) with < p < 1. Let G(n, p(n)) be the graph such that there is an edge between
More informationApplications of Ito s Formula
CHAPTER 4 Applications of Ito s Formula In this chapter, we discuss several basic theorems in stochastic analysis. Their proofs are good examples of applications of Itô s formula. 1. Lévy s martingale
More informationTHE DENSITY OF THE ISE AND LOCAL LIMIT LAWS FOR EMBEDDED TREES. CNRS, Université Bordeaux 1 and Uppsala University
Submitted to the Annals of Applied Probability THE DENSITY OF THE ISE AND LOCAL LIMIT LAWS FOR EMBEDDED TREES By Mireille Bousquet-Mélou and Svante Janson CNRS, Université Bordeaux 1 and Uppsala University
More informationJean-François Le Gall. Laboratoire de Mathématiques. Ecole Normale Supérieure. Based on lectures given in the Nachdiplomsvorlesung, ETH Zürich
SPATIAL BRANCHING PROCESSES RANDOM SNAKES AND PARTIAL DIFFERENTIAL EQUATIONS Jean-François Le Gall Laboratoire de Mathématiques Ecole Normale Supérieure Based on lectures given in the Nachdiplomsvorlesung,
More informationTHE CENTER OF MASS OF THE ISE AND THE WIENER INDEX OF TREES
Elect. Comm. in Probab. 9 (24), 78 87 ELECTRONIC COMMUNICATIONS in PROBABILITY THE CENTER OF MASS OF THE ISE AND THE WIENER INDEX OF TREES SVANTE JANSON Departement of Mathematics, Uppsala University,
More information(B(t i+1 ) B(t i )) 2
ltcc5.tex Week 5 29 October 213 Ch. V. ITÔ (STOCHASTIC) CALCULUS. WEAK CONVERGENCE. 1. Quadratic Variation. A partition π n of [, t] is a finite set of points t ni such that = t n < t n1
More informationExercises. T 2T. e ita φ(t)dt.
Exercises. Set #. Construct an example of a sequence of probability measures P n on R which converge weakly to a probability measure P but so that the first moments m,n = xdp n do not converge to m = xdp.
More informationRegularization by noise in infinite dimensions
Regularization by noise in infinite dimensions Franco Flandoli, University of Pisa King s College 2017 Franco Flandoli, University of Pisa () Regularization by noise King s College 2017 1 / 33 Plan of
More informationCutting down trees with a Markov chainsaw
Cutting down trees with a Markov chainsaw L. Addario-Berry N. Broutin C. Holmgren October 9, 2013 Abstract We provide simplified proofs for the asymptotic distribution of the number of cuts required to
More informationBrownian 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 informationImaginary Geometry and the Gaussian Free Field
Imaginary Geometry and the Gaussian Free Field Jason Miller and Scott Sheffield Massachusetts Institute of Technology May 23, 2013 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian
More informationLECTURE Itineraries We start with a simple example of a dynamical system obtained by iterating the quadratic polynomial
LECTURE. Itineraries We start with a simple example of a dynamical system obtained by iterating the quadratic polynomial f λ : R R x λx( x), where λ [, 4). Starting with the critical point x 0 := /2, we
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationSPATIAL STRUCTURE IN LOW DIMENSIONS FOR DIFFUSION LIMITED TWO-PARTICLE REACTIONS
The Annals of Applied Probability 2001, Vol. 11, No. 1, 121 181 SPATIAL STRUCTURE IN LOW DIMENSIONS FOR DIFFUSION LIMITED TWO-PARTICLE REACTIONS By Maury Bramson 1 and Joel L. Lebowitz 2 University of
More informationSpectral asymptotics for stable trees and the critical random graph
Spectral asymptotics for stable trees and the critical random graph EPSRC SYMPOSIUM WORKSHOP DISORDERED MEDIA UNIVERSITY OF WARWICK, 5-9 SEPTEMBER 2011 David Croydon (University of Warwick) Based on joint
More informationCones of measures. Tatiana Toro. University of Washington. Quantitative and Computational Aspects of Metric Geometry
Cones of measures Tatiana Toro University of Washington Quantitative and Computational Aspects of Metric Geometry Based on joint work with C. Kenig and D. Preiss Tatiana Toro (University of Washington)
More informationlim n C1/n n := ρ. [f(y) f(x)], y x =1 [f(x) f(y)] [g(x) g(y)]. (x,y) E A E(f, f),
1 Part I Exercise 1.1. Let C n denote the number of self-avoiding random walks starting at the origin in Z of length n. 1. Show that (Hint: Use C n+m C n C m.) lim n C1/n n = inf n C1/n n := ρ.. Show that
More informationZdzislaw Brzeźniak. Department of Mathematics University of York. joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York)
Navier-Stokes equations with constrained L 2 energy of the solution Zdzislaw Brzeźniak Department of Mathematics University of York joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York) LMS
More informationLogFeller et Ray Knight
LogFeller et Ray Knight Etienne Pardoux joint work with V. Le and A. Wakolbinger Etienne Pardoux (Marseille) MANEGE, 18/1/1 1 / 16 Feller s branching diffusion with logistic growth We consider the diffusion
More informationTowards conformal invariance of 2-dim lattice models
Towards conformal invariance of 2-dim lattice models Stanislav Smirnov Université de Genève September 4, 2006 2-dim lattice models of natural phenomena: Ising, percolation, self-avoiding polymers,... Realistic
More informationStochastic Calculus. Kevin Sinclair. August 2, 2016
Stochastic Calculus Kevin Sinclair August, 16 1 Background Suppose we have a Brownian motion W. This is a process, and the value of W at a particular time T (which we write W T ) is a normally distributed
More informationOn trees with a given degree sequence
On trees with a given degree sequence Osvaldo ANGTUNCIO-HERNÁNDEZ and Gerónimo URIBE BRAVO Instituto de Matemáticas Universidad Nacional Autónoma de México 3rd Bath-UNAM-CIMAT workshop 16-05-2016 Our discrete
More informationInfinitely iterated Brownian motion
Mathematics department Uppsala University (Joint work with Nicolas Curien) This talk was given in June 2013, at the Mittag-Leffler Institute in Stockholm, as part of the Symposium in honour of Olav Kallenberg
More informationNumerical simulation of random curves - lecture 1
Numerical simulation of random curves - lecture 1 Department of Mathematics, University of Arizona Supported by NSF grant DMS-0501168 http://www.math.arizona.edu/ e tgk 2008 Enrage Topical School ON GROWTH
More informationA NOTE ON STOCHASTIC INTEGRALS AS L 2 -CURVES
A NOTE ON STOCHASTIC INTEGRALS AS L 2 -CURVES STEFAN TAPPE Abstract. In a work of van Gaans (25a) stochastic integrals are regarded as L 2 -curves. In Filipović and Tappe (28) we have shown the connection
More informationPathwise construction of tree-valued Fleming-Viot processes
Pathwise construction of tree-valued Fleming-Viot processes Stephan Gufler November 9, 2018 arxiv:1404.3682v4 [math.pr] 27 Dec 2017 Abstract In a random complete and separable metric space that we call
More information1 Informal definition of a C-M-J process
(Very rough) 1 notes on C-M-J processes Andreas E. Kyprianou, Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY. C-M-J processes are short for Crump-Mode-Jagers processes
More informationLecture 19 L 2 -Stochastic integration
Lecture 19: L 2 -Stochastic integration 1 of 12 Course: Theory of Probability II Term: Spring 215 Instructor: Gordan Zitkovic Lecture 19 L 2 -Stochastic integration The stochastic integral for processes
More informationLecture 12. F o s, (1.1) F t := s>t
Lecture 12 1 Brownian motion: the Markov property Let C := C(0, ), R) be the space of continuous functions mapping from 0, ) to R, in which a Brownian motion (B t ) t 0 almost surely takes its value. Let
More informationStrong Markov property of determinatal processes
Strong Markov property of determinatal processes Hideki Tanemura Chiba university (Chiba, Japan) (August 2, 2013) Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 1 / 27 Introduction The
More informationFast-slow systems with chaotic noise
Fast-slow systems with chaotic noise David Kelly Ian Melbourne Courant Institute New York University New York NY www.dtbkelly.com May 12, 215 Averaging and homogenization workshop, Luminy. Fast-slow systems
More informationWiener Measure and Brownian Motion
Chapter 16 Wiener Measure and Brownian Motion Diffusion of particles is a product of their apparently random motion. The density u(t, x) of diffusing particles satisfies the diffusion equation (16.1) u
More informationCritical percolation on networks with given degrees. Souvik Dhara
Critical percolation on networks with given degrees Souvik Dhara Microsoft Research and MIT Mathematics Montréal Summer Workshop: Challenges in Probability and Mathematical Physics June 9, 2018 1 Remco
More informationPart II Probability and Measure
Part II Probability and Measure Theorems Based on lectures by J. Miller Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
More informationBurgers equation in the complex plane. Govind Menon Division of Applied Mathematics Brown University
Burgers equation in the complex plane Govind Menon Division of Applied Mathematics Brown University What this talk contains Interesting instances of the appearance of Burgers equation in the complex plane
More informationAn Introduction to Percolation
An Introduction to Percolation Michael J. Kozdron University of Regina http://stat.math.uregina.ca/ kozdron/ Colloquium Department of Mathematics & Statistics September 28, 2007 Abstract Percolation was
More informationRandom trees and branching processes
Random trees and branching processes Svante Janson IMS Medallion Lecture 12 th Vilnius Conference and 2018 IMS Annual Meeting Vilnius, 5 July, 2018 Part I. Galton Watson trees Let ξ be a random variable
More informationSome Tools From Stochastic Analysis
W H I T E Some Tools From Stochastic Analysis J. Potthoff Lehrstuhl für Mathematik V Universität Mannheim email: potthoff@math.uni-mannheim.de url: http://ls5.math.uni-mannheim.de To close the file, click
More informationThe Chaotic Character of the Stochastic Heat Equation
The Chaotic Character of the Stochastic Heat Equation March 11, 2011 Intermittency The Stochastic Heat Equation Blowup of the solution Intermittency-Example ξ j, j = 1, 2,, 10 i.i.d. random variables Taking
More informationBrownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
More information{σ x >t}p x. (σ x >t)=e at.
3.11. EXERCISES 121 3.11 Exercises Exercise 3.1 Consider the Ornstein Uhlenbeck process in example 3.1.7(B). Show that the defined process is a Markov process which converges in distribution to an N(0,σ
More informationOn pathwise stochastic integration
On pathwise stochastic integration Rafa l Marcin Lochowski Afican Institute for Mathematical Sciences, Warsaw School of Economics UWC seminar Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic
More information1. If 1, ω, ω 2, -----, ω 9 are the 10 th roots of unity, then (1 + ω) (1 + ω 2 ) (1 + ω 9 ) is A) 1 B) 1 C) 10 D) 0
4 INUTES. If, ω, ω, -----, ω 9 are the th roots of unity, then ( + ω) ( + ω ) ----- ( + ω 9 ) is B) D) 5. i If - i = a + ib, then a =, b = B) a =, b = a =, b = D) a =, b= 3. Find the integral values for
More informationOn conformally invariant CLE explorations
On conformally invariant CLE explorations Wendelin Werner Hao Wu arxiv:1112.1211v2 [math.pr] 17 Dec 2012 Abstract We study some conformally invariant dynamic ways to construct the Conformal Loop Ensembles
More informationZdzislaw Brzeźniak. Department of Mathematics University of York. joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York)
Navier-Stokes equations with constrained L 2 energy of the solution Zdzislaw Brzeźniak Department of Mathematics University of York joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York) Stochastic
More informationTHE STANDARD ADDITIVE COALESCENT 1. By David Aldous and Jim Pitman University of California, Berkeley
The Annals of Probability 1998, Vol. 6, No. 4, 1703 176 THE STANDARD ADDITIVE COALESCENT 1 By David Aldous and Jim Pitman University of California, Berkeley Regard an element of the set { = x 1 x x 1 x
More informationPercolation on random triangulations
Percolation on random triangulations Olivier Bernardi (MIT) Joint work with Grégory Miermont (Université Paris-Sud) Nicolas Curien (École Normale Supérieure) MSRI, January 2012 Model and motivations Planar
More informationLiouville quantum gravity as a mating of trees
Liouville quantum gravity as a mating of trees Bertrand Duplantier, Jason Miller and Scott Sheffield arxiv:1409.7055v [math.pr] 9 Feb 016 Abstract There is a simple way to glue together a coupled pair
More informationA view from infinity of the uniform infinite planar quadrangulation
ALEA, Lat. Am. J. Probab. Math. Stat. 1 1), 45 88 13) A view from infinity of the uniform infinite planar quadrangulation N. Curien, L. Ménard and G. Miermont LPMA Université Paris 6, 4, place Jussieu,
More informationLecture 22 Girsanov s Theorem
Lecture 22: Girsanov s Theorem of 8 Course: Theory of Probability II Term: Spring 25 Instructor: Gordan Zitkovic Lecture 22 Girsanov s Theorem An example Consider a finite Gaussian random walk X n = n
More informationExtrema of discrete 2D Gaussian Free Field and Liouville quantum gravity
Extrema of discrete 2D Gaussian Free Field and Liouville quantum gravity Marek Biskup (UCLA) Joint work with Oren Louidor (Technion, Haifa) Discrete Gaussian Free Field (DGFF) D R d (or C in d = 2) bounded,
More informationRandom maps Lecture notes for the course given at the INI Random Geometry programme
Random maps Lecture notes for the course given at the INI Random Geometry programme preliminary draft Grégory Miermont January 23, 2015 Contents 1 Maps as embedded graphs 2 1.1 Duality and Euler s formula...................
More informationn [ F (b j ) F (a j ) ], n j=1(a j, b j ] E (4.1)
1.4. CONSTRUCTION OF LEBESGUE-STIELTJES MEASURES In this section we shall put to use the Carathéodory-Hahn theory, in order to construct measures with certain desirable properties first on the real line
More informationParacontrolled KPZ equation
Paracontrolled KPZ equation Nicolas Perkowski Humboldt Universität zu Berlin November 6th, 2015 Eighth Workshop on RDS Bielefeld Joint work with Massimiliano Gubinelli Nicolas Perkowski Paracontrolled
More informationdynamical zeta functions: what, why and what are the good for?
dynamical zeta functions: what, why and what are the good for? Predrag Cvitanović Georgia Institute of Technology November 2 2011 life is intractable in physics, no problem is tractable I accept chaos
More information11 COMPLEX ANALYSIS IN C. 1.1 Holomorphic Functions
11 COMPLEX ANALYSIS IN C 1.1 Holomorphic Functions A domain Ω in the complex plane C is a connected, open subset of C. Let z o Ω and f a map f : Ω C. We say that f is real differentiable at z o if there
More informationPotential Theory on Wiener space revisited
Potential Theory on Wiener space revisited Michael Röckner (University of Bielefeld) Joint work with Aurel Cornea 1 and Lucian Beznea (Rumanian Academy, Bukarest) CRC 701 and BiBoS-Preprint 1 Aurel tragically
More information