a quenched CLT for certain ballistics RWRE-s in dimension geq 4.
Partly based on joint work with Ofer Zeitouni.
No knowledge of RWRE will be assumed.
Time: Friday 2/23/2007 at 2:30 pm.
Location: PDL C-401
Speaker: Yan-Xia Ren (Beijing University, visiting University
of Oregon)
Title: Limit theorems for super-diffusions corresponding to the
operator $Lu+\beta u-ku^2$
Abstract:
Consider a superdiffusion $X$ corresponding to the operator
$Lu+\beta u-ku^2$, where $\beta(x)$ is bounded from above and is
in the Kato class, and $k(x)\ge 0$ is bounded on compact subset of
${\bf R}^d$. Let $-\Lambda $ be the $L^{\infty}$-spectral radius of the
semigroup $Q_t$ corresponding to the Schrodinger operator $Lu+\beta u$.
We prove that if $\Lambda >0$, the exponential growth rate of the total
mass of $X$ is $\Lambda $; if $\Lambda <0$, the exponential decay rate of
the total mass of $X$ is $\Lambda <0$. We also describe the limiting
behavior of $\exp(-\Lambda t)X_t({\bf R}^d)$, where $X_t({\bf R}^d)$ is the
total mass of $X$ at time $t$. In particular, in the case $\Lambda =0$,
under some restrict conditions on $\beta$, we give a sufficient and
necessary condition for the superdiffusion $X$ exhibiting weak extinction.
It turns out that the branching rate function $k$ affects the weak
extinction, this should be compared with the known result that $k$
does not affects the weak local extinction of $X$.
Time: Monday 2/12/2007 at 2:30 pm.
Location: SAV 151
Speaker: Alexander Holroyd (University of British Columbia)
Title: Random Sorting Networks
Abstract:
See http://www.math.ubc.ca/~holroyd/sort for pictures. Joint work with
Omer Angel, Dan Romik and Balint Virag.
Sorting a list of items is perhaps the most celebrated problem in computer
science. If one must do this by swapping neighboring pairs, the worst
initial condition is when the n items are in reverse order, in which case n
choose 2 swaps are needed. A sorting network is any sequence of n choose 2
swaps which achieves this.
We investigate the behavior of a uniformly random n-item sorting network as
n->infinity. We prove a law of large numbers for the space-time process of
swaps. Exact simulations and heuristic arguments have led us to astonishing
conjectures. For example, the half-time permutation matrix appears to be
circularly symmetric, while the trajectories of individual items appear to
converge to a famous family of smooth curves. We prove the more modest
results that, asymptotically, the support of the matrix lies within a
certain octagon, while the trajectories are Holder-1/2. A key tool is a
connection with Young tableaux.
Time: Monday 2/5/2007 at 2:30 pm.
Location: SAV 151
Speaker: Abraham Flaxman (Microsoft Research)
Title: Expansion and lack thereof in randomly perturbed graphs
Abstract:
This talk will investigate the expansion properties of randomly
perturbed graphs. These graphs are formed by, for example, adding a
random 1-out or very sparse Erdos-Renyi graph to an arbitrary
connected graph.
When any connected n-vertex base graph is perturbed by adding a random
1-out then, with high probability, the resulting graph has expansion
properties. When the perturbation is by a sparse Erdos-Renyi graph,
the expansion of the perturbed graph depends on the structure of the
base graph.
The same techniques also apply to bound the expansion in the small
worlds graphs described by Watts and Strogatz in [Nature 292 (1998),
440--442] and by Kleinberg in [Proc. of 32nd ACM Symposium on Theory
of Computing (2000), 163--170]. Analysis of Kleinberg's model reveals
a phase transition: the graph stops being an expander exactly at the
point where a decentralized algorithm is effective in constructing a
short path.
The proofs of expansion rely on a way of summing over subsets of
vertices which allows an argument based on the First Moment Method to
succeed.
Time: Monday 1/29/2007 at 2:30 pm.
Location: SAV 151
Speaker: Gabor Pete (Microsoft Research)
Title: Corner percolation on \Z^2, and other linear
entropy models
Abstract:
Corner percolation is a strongly dependent percolation model introduced by
Bálint Tóth, exhibiting some unusual critical behaviour.
We prove that all connected components are finite cycles almost surely, but
the expected diameter of the cycle containing the origin is infinite.
Moreover, we derive the following critical exponents: the tail probability
\Pr(diameter of the cycle of the origin > n) \approx n^{-\gamma}, and the
expectation \E(length of a cycle conditioned on having diameter n) \approx
n^\delta, with \gamma=(5-\sqrt{17})/4=0.219... and
\delta=(\sqrt{17}+1)/4=1.28... The value of \delta comes from the solution
of a singular sixth order ODE, while the relation \gamma+\delta=3/2
corresponds to the fact that the scaling limit of the natural height
function in the model is the Additive Brownian Motion, whose level sets have
Hausdorff dimension 3/2.
I will discuss many exciting open problems, e.g. on the conformal invariance
of some similar linear entropy models.
Time: Monday 1/22/2007 at 2:30 pm.
Location: SAV 151
Speaker: Ioana Dumitriu (University of Washington)
Title: Central Limit Theorems for \beta-Hermite and \beta-Laguerre
ensembles (C^1 functions)
Abstract:
The \beta-Hermite and -Laguerre ensembles are generalizations of the
"classical" Gaussian and central Wishart ones, and have applications in
statistical physics (from the theory of log-Coulomb gases to traffic
patterns, parked car spacings, etc.)
The (scaled) empirical distributions of these ensembles converge to the
famous Wigner semicircle law (in the case of Hermite), respectively,
Mar\v{c}enko-Pastur law (for Laguerre). I will show that the fluctuations
from these laws describe Gaussian processes on C^1 functions. Then I
will comment on a "natural barrier" found at C^{1/2}, and express
consternation and a certain degree of frustration with it.
This is joint work with Ofer Zeitouni.
Time: Monday 1/8/2007 at 2:30 pm.
Location: SAV 151
Speaker: Krzysztof Burdzy (University of Washington)
Title: Pathwise uniqueness for reflected Brownian motion in
$C^{1+\gamma}$ domains
Abstract:
A domain in $R^d$ is called $C^{1+\gamma}$ if its boundary can be
locally represented as the graph of a function whose gradient is
Holder with exponent $\gamma$. For $d\geq 3$, pathwise uniqueness
holds for the SDE representing reflected Brownian motion in a
$C^{1+\gamma}$ domain provided $\gamma > 1/2$. For every $\gamma <
1/2$, there exists $d$ and a $C^{1+\gamma}$ domain $D$ in $R^d$
such that pathwise uniqueness fails in $D$.
Work in progress. Joint with R. Bass.
Time: Monday, December 4, 2006 at 2:30 pm.
Location: MGH 288
Speaker: Yuval Peres (Microsoft Research, UC Berkeley and UW)
Title: GRAVITATIONAL ALLOCATION TO POISSON POINTS
Abstract:
We consider the Poisson fair allocation problem: Given a
realization of a Poisson point process, allocate to each point of
the process a unit of volume, in a deterministic
translation-invariant way, so that the diameter of the region
allocated to each point is stochastically as small as possible.
One approach to this problem, studied in previous work with C.
Hoffman and A. Holroyd, uses the stable marriage algorithm of Gale
and Shapley. Here we show that in dimensions 3 and higher, gravity
without inertia yields a very satisfying solution. The argument
starts with the classical calculation by Chandrasekar of the total
force acting on a point, which has a symmetric stable law. The
fairness of the allocation is a consequence of the divergence
theorem; The diameters of the allocated regions are analyzed using
methods from percolation theory. (Joint work with S. Chatterjee,
R. Peled, D. Romik).
Time: Monday, November 27, 2006 at 2:30 pm.
Location: MGH 288
Speaker: Gunnar Gunnarsson (University of Washington)
Title: SPDE MODELS FOR HIGHWAY TRAFFIC
Abstract:
We introduce a new stochastic partial differential equation model
for multi-lane highway traffic. We prove well-posedness of the
model, derive a numerical method for calculating its solutions and
estimate some of its parameters.
Time: Monday, November 20, 2006 at 2:30 pm.
Location: MGH 288
Speaker: Yuan-Chung Sheu (National Chiao Tung University, Hsinchu,
Taiwan)
Title: AN ODE APPROACH FOR THE EXPECTED DISCOUNTED PENALTY
AT RUIN IN JUMP DIFFUSION MODEL
Abstract:
For a general penalty function, the expected discounted penalty at
ruin was considered by, for example, Gerber and Shiu(1998) and
Gerber and Landry (1998) in insurance literature. On the other
hand, many pricing functionals in mathematical finance can be
formulated in terms of expected discounted penalties. Under the
assumption that the asset value follows a jump diffusion, we show
the expected discounted penalty satisfies an ODE and obtain a
general form for the expected discounted penalty. In particular,
if only downward phase-type jumps are allowed, we get an explicit
formula in terms of the penalty function and jump distribution. On
the other hand, if downward jump distribution is a mixture of
exponential distributions (and upward jumps are determined by a
general L\'{e}vy measure), we obtain closed form solutions for the
expected discounted penalty. As an application, we work out an
example in Leland's structural model with jumps. For earlier and
related results, see Gerber and Landry(1998), Hilberink and
Rogers(2002), Mordecki(2002), Kou and Wang(2004), Asmussen et
al.(2004) and others.
Time: Monday, November 13, 2006 at 2:30 pm.
Location: MGH 288
Speaker: Tomoyuki Shirai (Kyushu University)
Title: ON THE NUMBER OF EIGENVALUES OF GINIBRE MATRIX ENSEMBLE
Abstract:
Ginibre matrix ensemble is the random matrix whose entries are
i.i.d. complex Gaussian random variables. Its random eigenvalues
form a determinantal point process (associated with exponential
kernel) on the whole complex plane. We discuss the large deviation
principle for the number of its eigenvalues inside a ball and also
its conditional expectation given that there are no eigenvalues in
a big ball.
Time: Monday, November 6, 2006 at 2:30 pm.
Location: MGH 288
Speaker: Tusheng Zhang (University of Manchester)
Title: GLOBAL FLOWS FOR STOCHASTIC DIFFERENTIAL EQUATIONS
WITHOUT GLOBAL LIPSCHITZ CONDITIONS
Abstract:
We consider stochastic differential equations driven by Brownian
motion. The coefficients are supposed to satisfy only local
Lipschitz conditions. The Lipschitz constants of the drift, valid
on balls of radius $R$, are supposed to grow not faster than $\log
R$, those of diffusion coefficient not faster than $\sqrt{\log
R}$. Under these conditions, we establish the existence of a
global flow for the stochastic differential equation.
Time: Monday, October 30, 2006 at 2:30 pm.
Location: MGH 288
Speaker: Kavita Ramanan (Carnegie Mellon University)
Title: EXISTENCE AND UNIQUENESS OF SOLUTIONS
TO A CLASS OF STOCHASTIC DIFFERENTIAL INCLUSIONS
Abstract:
We establish sufficient conditions for pathwise uniqueness and
existence of strong solutions to a class of stochastic
differential equations in $R^n$ with discontinuous drift
(interpreted as stochastic differential inclusions), that are
possibly reflected in a polyhedral domain with piecewise constant
reflection field. In contrast to previous results, we do not
impose any uniform ellipticity assumptions, and our results also
yield new conditions for uniqueness of solutions to ordinary
differential inclusions with drifts that have multiple
intersecting surfaces of discontinuity. We will also briefly
discuss the application of our results to the study of large
deviations of a class of jump Markov processes that arise
naturally in applications. This is joint work with Rami Atar and
Amarjit Budhiraja.
MATHEMATICS COLLOQUIUM (PIMS 10th Anniversary Distinguished Lecturer)
Time: Tuesday, October 24, 2006 at 4 pm.
Location: Smith 304
Speaker: Gregory Lawler (University of Chicago)
Title: CONFORMAL INVARIANCE AND TWO-DIMENSIONAL
STATISTICAL PHYSICS
Abstract:
A number of lattice models in two-dimensional statistical physics are
conjectured to exhibit conformal invariance in the scaling limit at criticality.
In this talk, I will try to explain what the previous sentence means,
focusing on three elementary examples: simple random walk, self-avoiding walk,
loop-erased random walk. I will describe the limit objects,
Schramm-Loewner Evolution (SLE), the Brownian loop soup,
and the normalized partition functions, and show how conformal invariance
can be used to calculate quantities ("critical exponents") for the model.
I will also describe why (in some sense) there is only a one-parameter family
of conformally invariant limits.
In conformal field theory, this family is parametrized by central charge.
This talk is for a general mathematical audience.
No knowledge of statistical physics will be assumed.
Time: Friday, October 20, 2006 at 2:30 pm.
Location: THO 235
Speaker: Eulalia Nualart (Universit\'e Paris 13)
Title: HITTING PROBABILITIES FOR SYSTEMS
OF NONLINEAR STOCHASTIC HEAT EQUATIONS
Abstract:
In this talk we develop potential theory for a system of non-linear
stochastic heat equations in spatial dimension one and driven by
a space-time white noise. In particular, we prove upper and lower bounds
on hitting probabilities of the process which is solution of this system
of equations, in terms of respectively Hausdorff measure and
Newtonian capacity. These estimates make it possible to discuss polarity
for points and to compute the Hausdorff dimension of the range and the
level sets of this process. In order to prove the hitting probabilities
estimates, we need to establish Gaussian type bounds for the bivariate
density of the process in order to quantify its degeneracy.
For this, we use techniques of Malliavin calculus.
Time: Monday, October 9, 2006 at 2:30 pm.
Location: MGH 288
Speaker: Lionel Levine (University of California, Berkeley)
Title: THE SCALING LIMIT OF DIACONIS-FULTON ADDITION
Abstract:
Given two sets $A$ and $B$ in the lattice, the Diaconis-Fulton sum
is a random set obtained by putting one particle in every point of
the symmetric difference, and two particles in every point of the
intersection, of $A$ and $B$. Each "extra" particle performs
random walk until it reaches an unoccupied site, where it settles.
The law of the resulting random occupied set $A+B$ does not depend
on the order of the walks. We find the (deterministic) scaling
limit of the sums $A+B$ when $A$ and $B$ consist of the lattice
points in some overlapping domains in Euclidean space. The limit
is described by focusing on the "odometer" of the process, which
solves a free boundary obstacle problem for the Laplacian. Joint
work with Yuval Peres.
Time: Monday, October 2, 2006 at 2:30 pm.
Location: MGH 288 (Mary Gates Hall)
Speaker: Peter Moerters (University of Bath)
Title: LOCALIZATION OF MASS IN THE PARABOLIC ANDERSON MODEL
Abstract:
The parabolic Anderson model is the Cauchy problem for the heat
equation with random potential. After a gentle introduction of
some basic features of the model, I will discuss a recent result
showing that, for a particular class of potentials, at large times
the mass becomes localised in a single point in space, which moves
in time.
The talk is based on joint work with W Koenig and N Sidorova.
Time: Thursday 8/3/2006 at 2:30 pm.
Location: LOW 111
Speaker: Panki Kim (University of Illinois at Urbana-Champaign)
Title: Intrinsic Ultracontractivity for Non-symmetric Levy Processes
Abstract:
Recently the concept of intrinsic ultracontractivity to non-symmetric
semigroups has been introduced by Kim and Song. In this talk, we study
the intrinsic ultracontractivity for
non-symmetric discontinuous Levy processes.
Time: Wednesday May 24, 2006 at 2:30 pm.
Location: THO 211
Speaker: Takashi Kumagai (Kyoto University)
Title: On the existence of cut points for Brownian motion on fractals
Abstract:
Let $B(t)$ be a Brownian motion on some metric space $K$.
A time $t\in [0,1]$ is called a cut time for $B[0,1]$ if
$B[0,t)\cap B(t,1]$ is empty, and $B(t)$ is called a cut point
if $t$ is a cut time. Let $L$ be the set of cut points for $B[0,1]$.
When $K=R^d (d=2,3)$, Burdzy showed that non-trivial cut points exist,
and later Lawler obtained the Hausdorff dimension of $L$ in terms of
the intersection exponent. On the other hand,
it is known that when $d=1$, there is no non-trivial cut points.
In this talk, we will discuss the existence of cut points for Brownian
motion on fractals. We will prove that there is a family of fractals whose
Brownian motions are point recurrent, but have non-trivial cut points.
This is a on-going joint work with Peter Morters (Bath).
Time: Wednesday May 17, 2006 at 2:30 pm.
Location: THO 211
Speaker: Kavita Ramanan (Carnegie-Mellon University)
Title: A Concentration Inequality for Weakly Contracting Markov Chains
Abstract:
Given a Markov chain X on a countable state space S and a
Lipschitz function f on S^n, we derive a concentration inequality for the
function f around its mean. We use the method of bounded martingale
differences to derive this concentration inequality. To set this result in
context, we will also provide a brief survey of concentration of measure
results in the i.i.d. setting.
This is joint work with Leonid Kontorovich.
Time: Monday May 8, 2006 at 2:30 pm.
Location: MEB 238
Speaker: Kittipat Wong (Chulalongkorn University, Thailand)
Title: Large time behavior of Dirichlet heat kernels
Abstract:
We will discuss the asymptotic behavior of the transition density
$p^D(t,x,y)$ of killed Brownian motions in $D$ where $D \subset R^d,
d \ge 2$ is an unbounded domain above the graph of a bounded
Lipschitz function.
Time: Wednesday May 3, 2006 at 2:30 pm.
Location: THO 211
Speaker: Rami Atar (Technion and University of Washington)
Title: Large Deviations and Related PDE
Abstract:
It is well known that dynamic control problems for Markov processes,
in which large deviations are heavily penalized, lead in an
appropriate scaling limit to differential games and, in turn, to
Hamilton-Jacobi type PDE. I will review results in the field and present new
queueing networks applications.
Time: Monday, April 24, 2006 at 2:30 pm.
Location: MEB 238
Speaker: Hanna Jankowski (University of Washington)
Title: Central Limit Theorem for a Zero Range Tagged Particle
Abstract:
To further understand the scaling limit of an interacting particle system,
one would like to establish a limit law for the evolution of a tagged
particle and the associated central limit theorem. Tagging the particles
makes this a difficult problem, and one method to deal with it is to
consider first an intermediate step: limit laws and fluctuations for the
colour empirical densities. I will discuss the nonequilibrium
fluctuations of the coloured version of the symmetric zero range particle
system, and what this implies about the CLT for the tagged particle.
Time: Monday, April 17, 2006 at 2:30 pm.
Location: MEB 238
Speaker: Krzysztof Burdzy (University of Washington)
Title: Shy couplings
Abstract:
Given Markovian transition
probabilities, can one construct two processes
$X$ and $X'$ with these transition probabilities,
such that the distance between $X$ and $X'$
stays always above some $\eps>0$?
A pair of proceses with the above property
is called a shy coupling. I will give examples of Markovian transition
probabilities
that admit shy couplings and examples
when shy couplings do not exist.
Joint work with I. Benjamini and Z. Chen.
Time: Monday, April 3, 2006 at 2:30 pm.
Location: MEB 238
Speaker: David Wilson (Microsoft Research)
Title: Boundary Connections in Trees and Dimers
Abstract:
We study groves on planar graphs, which are forests in which every
tree contains one or more of a special set of vertices on the outer
face. Each grove partitions the set of special vertices. When a
random grove is selected, we show how to compute the various
partition probabilities as functions of the electrical properties of
the graph when viewed as a resistor network. We prove that for any
partition sigma, Pr[grove has type sigma] / Pr[grove is a tree] is a
dyadic-coefficient polynomial in the pairwise resistances between
the special vertices, and Pr[grove has type sigma] / Pr[grove has
maximal number of trees] is an integer-coefficient polynomial in the
entries of the Dirichlet-to-Neumann matrix. We give analogous
polynomial formulas for the double-dimer model. These partition
probabilities are relevant to multichordal SLE_8, SLE_2, and SLE_4.
(Joint work with Richard Kenyon.)
Note the unusual day of the week.
Time: Friday, March 10, 2006 at 2:30 pm.
Location: PDL C-401
Speaker: Jin Ma (Purdue University)
Title: STOCHASTIC CONTROL PROBLEMS
DRIVEN BY NORMAL MARTINGALES
Abstract:
We study a class of stochastic control problems in which the
control of the jump size is essential. Such a model is a
generalized version for various applied problems, such as the
optimal reinsurance selections for general insurance models. The
main novel nature of such a control problem is that by changing
the jump size of the system, one essentially changes the type of
the driving martingale. Such a feature does not seem to have been
investigated in any existing stochastic control literature.
Assuming that the driving normal martingale is one-dimensional, we
prove the Bellman Principle for such a control problem, and derive
the corresponding Hamilton-Jacobi-Bellman (HJB) equation, which in
this case is a mixed second-order partial differential/difference
equation. We prove that the value function is the {\it unique}
viscosity solution of such an HJB equation and discuss the
uniqueness of such PDDE.
This is a join work with Rainer Buckdahn and Catherine Rainer.
Time: Monday, February 27, 2006 at 2:30 pm.
Location: PDL C-401
Speaker: Kathryn Temple (Central Washington University)
Title: PARTICLE REPRESENTATIONS OF EXIT MEASURES
Abstract:
The connection between diffusions and certain linear elliptic
boundary value problems is well-known. More recently, a
connection between superdiffusions and a class of nonlinear
elliptic PDEs has been developed and exploited. Making this
connection requires a construction of the superdiffusion which
allows additional structure, such as Dynkin's Branching Exit
Markov Systems, the Dawson-Perkins Historical Process, or LeGall's
Brownian Snake. We use an adaptation of a particle construction
of Kurtz and Rodrigues to construct an exit measure and therefore
solutions of the PDEs associated with a certain class of
superdiffusions. As time permits, we will give an application of
this construction to a simple homogenization problem.
Note the unusual day of the week and unusual room.
Time: Wednesday, February 22, 2006 at 2:30 pm.
Location: SIG 226
Speaker: Yuichi Shiozawa (Tohoku University)
Title: EXTINCTION OF BRANCHING SYMMETRIC $\alpha$-STABLE PROCESSES
Abstract:
We give a criterion for extinction of branching symmetric
$\alpha$-stable processes in terms of the principal eigenvalue
for Schr\"odinger type operators. Here the branching rate and the
branching mechanism are spatially dependent. In particular, the
branching rate is allowed to be singular with respect to the
Lebesgue measure. We also study the exponential growth of the
number of particles for these processes.
Time: Monday, February 13, 2006 at 2:30 pm.
Location: PDL C-401
Speaker: Oded Schramm (Microsoft)
Title: THE DISCRETE GAUSSIAN FREE FIELD AND ITS LEVEL LINES
Abstract:
An instance of the Gaussian free field is a random function
defined in a domain in $R^d$. It is a generalization of Brownian
motion to the case where time is multi-dimensional, and it is
useful for modelling many different kinds of random surfaces. In
two dimension, it exhibits conformal invariance.
During the talk we will define the Gaussian free field and also
the discrete Gaussian free field and describe joint work with
Scott Sheffield identifying the scaling limit of the level lines
of the discrete Gaussian free field in two dimensions (as the
lattice mesh tends to zero). These limits are random curves having
Hausdorff dimension 3/2 a.s.
Time: Monday, February 6, 2006 at 2:30 pm.
Location: PDL C-401
Speaker: Nicole Immorlica (Microsoft)
Title: MARRIAGE, HONESTY AND STABILITY
Abstract:
Many centralized two-sided markets, such as the medical residency
market or online dating services, form a matching between
participants by running a stable marriage algorithm. It is a
well-known fact that no matching mechanism based on a stable
marriage algorithm can guarantee truthfulness as a dominant
strategy for participants. However, as we will show in this talk,
in a certain probabilistic setting, truthfulness is (in some
sense) the best strategy for the participants.
We show this by proving that in our setting the set of stable
marriages is small. We derive several corollaries of this result.
First, we show that, with high probability, in a stable marriage
mechanism, the truthful strategy is the best response for a given
player when the other players are truthful. Then we analyze
equilibria of the deferred acceptance stable marriage game. We
show that the game with complete information has an equilibrium in
which a $(1-o(1))$ fraction of the strategies are truthful in
expectation. In the more realistic setting of a game of
incomplete information, we will show that the set of truthful
strategies form a $(1+o(1))$-approximate Bayesian-Nash
equilibrium. Our results have implications in many practical
settings and were inspired by experimental observations in a paper
of Roth and Peranson (1999) concerning the National Residency
Matching Program.
This is joint work with Mohammad Mahdian.
Time: Monday, January 30, 2006 at 2:30 pm.
Location: PDL C-401
Speaker: Omer Angel (University of British Columbia)
Title: ONE DIMENSIONAL DLA
Abstract:
We consider a variation on DLA (diffusion limited aggregation) in
1 dimension generated by a random walk with large jumps. The
growth rate of the diameter of the $n$ particle aggregate depends
on the tail of the step distribution, and exhibits three phase
transitions when the steps have 1, 2 or 3 finite moments.
Time: Monday, January 23,, 2006 at 2:30 pm.
Location: PDL C-401
Speaker: Rami Atar (Technion and University of Washington)
Title: ANALYSIS OF MIRROR COUPLINGS IN SMOOTH DOMAINS
Abstract:
We analyze a coupling of two reflecting Brownian motions in (the
closure of) a smooth domain, with the property that whenever both
processes are in the domain, the motions form mirror images of
each other. Under appropriate geometric conditions, we show that
the motion of the mirror is limited in the sense that there are
parts of the domain that it will never intersect (for suitable
initial conditions). This will be described along with
consequences regarding monotonicity of the Neumann-Laplacian
eigenfunctions a la the `hot spots' problem.
Joint work with Chris Burdzy.
Note the unusual day of the week and time.
This will be also a colloquium talk.
Time: 4:00 p.m., Thursday, January 19, 2006
Location: Smith 105
Speaker: Jin Feng (University of Massachusetts at Amherst)
Title: LARGE DEVIATIONS FOR MARKOV PROCESSES AND RELATED VARIATIONAL
PROBLEMS
Abstract:
Large deviation estimates are probabilistic limit theorems which
are used to describe atypical behavior of random systems. Markov
processes are a rich class of probabilistic models. Their
generators constitute a link between probability, classical
analysis and *linear* partial differential equations. I will
describe a method for deriving large deviation estimates for a
sequence of Markov processes through convergence of some
*nonlinear* transforms of their generators. This is a previously
unknown connection between probability and certain topics in
nonlinear analysis such as Hamilton-Jacobi equations, viscosity
solutions, and optimal control theory. I will offer examples where
probability can help with analytical problems and vice versa. I
will use examples ranging from small random perturbations of ODEs
to the more physically motivated ones such as macroscopic
description of multi-scale microscopic interacting particle
systems.
Time: Monday, January 9, 2006 at 2:30 pm.
Location: PDL C-401
Speaker: Jason Swanson (University of Wisconsin, Madison)
Title: ASYMPTOTIC BEHAVIOR OF A GENERALIZED TCP
CONGESTION AVOIDANCE ALGORITHM
Abstract:
The Transmission Control Protocol (TCP) is designed to guarantee
reliable and sequential delivery of packets of data from a sender
to a receiver on a computer network. To avoid network congestion,
TCP uses various variations on an
additive-increase-multiplicative-decrease algorithm, which means
that the rate of data flow is increased linearly until a packet
loss is detected, at which point the rate is reduced by some
constant multiplicative factor. I will present a generalization of
this TCP Congestion Avoidance algorithm, which is due to Teunis J.
Ott of the New Jersey Institute of Technology. This model
includes, as a special case, the so-called "Scalable TCP," which
is a multiplicative-increase-multiplicative-decrease scheme.
The general model is given by a Markov chain $\{W_n\}$, which
represents the size of the congestion window (that is, the rate of
data flow) at the time of the delivery of the $n$-th packet.
Implicit in this model is a parameter $p$, which denotes the
probability of a packet loss. Under suitable scaling of time and
space, this process converges weakly as $p\to 0$ to the solution
of a stochastic differential equation. The form of the equation
depends on the values of certain parameters in the model. For some
values, it is the Ornstein-Uhlenbeck equation; for the others, it
is an equation driven by a Poisson process. I will sketch the
proof of this result and discuss further unanswered question. This
is joint work with Teun Ott.
Time: Wednesday, December 7, 2005 at 2:30 pm.
Location: PDL C-401
Speaker: Jeremy Quastel (University of Toronto)
Title: Asymptotic speed of traveling fronts in the KPP equation with
noise
Abstract:
The most basic reaction-diffusion equation, KPP, was introduced by Fisher to
model the spread of an advantageous gene. In this
context it is very natural to consider what happens to traveling fronts
when the equation is perturbed by an appropriate noise. Brunet and Derrida
observed, by simulations and physical arguments, that the noise leads to an
unexpectedly large slowdown of the traveling fronts. We
describe the problem and proofs of some of their conjectures. This is joint
work with Carl Mueller (Rochester) and Leonid Mytnik (Technion).
Time: Monday, November 21, 2005 at 2:30 pm.
Location: PDL C-401
Speaker: Jim Burke (University of Washington)
Title: Estimation of Constrained Mixture Densities
Abstract:
We consider the constrained mixture density estimation
problem over a parametric family of densities. It is shown how this
problem can be posed as a convex optimization problem on the
space of regular Borel measures. By combining this formulation with
Caratheodory's Theorem for convex sets in finite dimensions it
is possible to pose an equivalent finite dimensional optimization
problem. The problem is then amenable to a variety of numerical
optimization routines. One such approach based on interior point
technology is discussed and numerical results are presented.
This research is the result of a joint work between the speaker
and Yeongcheon Baek.
Time: Monday, November 14, 2005 at 2:30 pm.
Location: PDL C-401
Speaker: Peter D. Hoff (University of Washington)
Title: Random Graph Models for Relational Network Data
Abstract:
Relational data measure the presence or absence of links
among a set of objects. This data structure is very general
and has many applications in the social and biological sciences.
In this talk we consider two families of probability models
which are used to make inference from network data.
The first family includes exponentially parameterized models,
which are motivated by simplicity and maximum entropy considerations.
Models in the second family are motivated by a certain type of row-and-column
exchangeability for arrays, and typically have
a very large parameter space. Finally, we compare the appropriateness
of these two model classes for a variety of inferential goals.
Time: Monday, November 7, 2005 at 2:30 pm.
Location: PDL C-401
Speaker: Chris Hoffman (University of Washington)
Title: Dynamic random walk in two dimesnions
Abstract:
Benjamini, Haggstrom, Peres and Steif introduced the concept of a
dynamical random walk. This is a continuous family of random walks in d
dimensions, S_n(t), indexed by a real number t. They asked which
properties of simple random walk that hold for almost every t hold for all
t. They proved that if d=3,4 then there exist (random) times t such that
the random walk returns to the origin infinitely often, but if d>4 then
the random walk returns to the origin only finitely often for all t. We
prove that if d=2 then there exists (random) times t such that the random
walk returns to the origin only finitely often. This result is related to
a theorem of Adelman, Burdzy and Pemantle in which they prove that
although Browninan motion in three dimensions spends an infinite amount of
time in a given infinite cylinder of radius one almost surely, but there
exist (random) cylinders where Brownian motion spends only a finite amount
of time almost surely.
Time: Monday, Oct 31, 2005 at 2:30 pm.
Location: PDL C-401
Speaker: Vlada Limic (University of British Columbia)
Title: The spatial Lambda-coalescent
Abstract:
This talk is based on a joint paper with Anja Sturm, and it
will describe the extension of the Lambda-coalescent of Pitman (1999)
to the spatial setting. The partition elements of the spatial
Lambda-coalescent migrate in a (finite) geographical space and may only
coalesce if located at the same site of the space. We characterize the
Lambda-coalescents that come down from infinity, in an analogous way to
Schweinsberg (2000). Surprisingly, all spatial coalescents that come down
from infinity, also come down from infinity in a uniform way. This enables
us to study space-time asymptotics of spatial Lambda-coalescents on large
tori in transient dimensions.
Our results generalize and strengthen those of Greven et al. (2005), who
studied the spatial Kingman coalescent in this context.
Time: Monday, Oct 24, 2005 at 2:30 pm.
Location: PDL C-401
Speaker: David Levin (University of Oregon)
Title: A Coupling, and a Conjecture of Darling and Erdos
Abstract:
I will describe a new coupling of the running maximum
of an Ornstein-Uhlenbeck process and the running maximum
of an explicit iid sequence. This coupling can be used
to resolve a conjecture of Darling and Erdos (1956).
Joint work with D. Khoshnevisan.
Time: Monday, Oct 17, 2005 at 2:30 pm.
Location: PDL C-401
Speaker: Ken Alexander (University of Southern California)
Title: The pinning transition for a polymer in the presence of a
random potential
Abstract:
We consider a polymer, with monomer locations modeled by the trajectory
of a Markov chain, in the presence of a potential that interacts with the
polymer when it visits a particular site 0. We assume the probability of an
excursion of length $n$ from 0, in the absence of the potential, decays like
$n^{-c}$ for some $c>1$. Disorder is introduced by, having the interaction
vary from one monomer to another, as a constant $u$ plus i.i.d. mean-0
randomness. There is a critical value of $u$ above which the polymer is
pinned, placing a positive fraction, called the contact fraction, of its
monomers at 0 with high probability. We obtain bounds for the contact
fraction near the critical point and examine the effect of the disorder on
the specific heat exponent, which describes the approach to 0 of the contact
fraction at the critical point. Our results are consistent with predictions
in the physics literature that the effect of disorder is quite different in
the cases $c<3/2$ and $c>3/2$.
Time: Wednesday, Oct 12, 2005 at 2:30 pm.
Location: SMI 311
Speaker: Bartek Dyda (Technical Univ of Wroclaw)
Title: On fractional order Hardy inequalities
Abstract:
We will state the fractional order Hardy inequality for e.g.
bounded Lipschitz domains, and prove it in the simplest
case when the domain is R^d \setminus \{0\}.
This inequality will give us transcience of the censored stable process
(with the stability index > 1) in bounded Lipschitz domains.
This result was obtained earlier (by another methods) in the paper
of Bogdan, Burdzy and Chen.
We will also mention some further results concerning
fractional order Hardy inequalities.
Time: Wednesday, August 24, 2005 at 3:30 pm.
Location: Padelford C-36
Speaker: Wojbor A. Woyczy\'nski (Case Western Reserve University)
Title: STOCHASTIC PARTICLE METHODS FOR NONLINEAR
EVOLUTION EQUATIONS INVOLVING L'EVY GENERATORS
Abstract:
I will consider a class of nonlinear integro-differential
equations driven by L\'evy infinitesimal generators and involving
nonlocal nonlinearities represented by singular integral
operators. We associate a nonlinear diffusion in the McKean sense
with the original singular equation and are able to prove the
convergence of cut-off interacting particle systems to the law of
this nonlinear singular diffusion.
Time: Monday, May 23, 2005 at 2:30 pm.
Location: SMI 102
Speaker: Elton Hsu (Northwestern University)
Title: MASS TRANSPORTATION PROBLEM
AND COUPLING OF DIFFUSION PROCESSES
Abstract:
We will discuss the problem of existence and uniqueness of maximal
Markov coupling of diffusion processes, in particular, Brownian
motion on Riemannian manifolds. We will show how this problem is
related to the mass transportation problem with a cost function
determined by the heat kernel of the diffusion process. The
problem is completely solved only in the Euclidean case.
A part of this talk is the joint work with Theo Sturm (Bonn).
Time: Monday, May 16, 2005 at 2:30 pm.
Location: SMI 102
Speaker: Matthew Stephens (University of Washington)
Title: EXCHANGEABLE AND NON-EXCHANGEABLE DISTRIBUTIONS
ARISING FROM APPLICATIONS IN POPULATION GENETICS
Abstract:
Population genetics is the study of genetic variation in random
samples of individuals from a population. I will give an informal
introduction to some of the mathematical theory underlying recent
progress in this discipline. Much of the theory is derived under
simplistic and unrealistic assumptions. I will describe work to
relax these assumptions to capture important aspects of real data.
Although this work has had important impact in several practical
applications, it raises several interesting and unresolved
theoretical problems.
Time: Monday, May 9, 2005 at 2:30 pm.
Location: SMI 102
Speaker: Karoly Simon (Technical University of Budapest)
Title: THE SIZE OF THE ALGEBRAIC DIFFERENCE
OF TWO RANDOM CANTOR SETS
Abstract:
In this talk we consider some families of random Cantor sets on
the line and investigate the question whether the condition that
the sum of Hausdorff dimension is larger than one, implies the
existence of interior points in the difference set. We prove that
this is the case for the so called Mandelbrot percolation. On the
other hand the same is not always true if we apply a construction
of random Cantor sets similar to the Mandelbrot percolation but
with different probabilities.
Time: Monday, May 2, 2005 at 4:30 pm.
Location: THO 119
Speaker: Michael Taksar (University of Missouri)
Title: SINGULAR STOCHASTIC CONTROL AND RELATED PDE WITH
GRADIENT CONSTRAINTS IN PORTFOLIO OPTIMIZATION MODELS
IN MATHEMATICAL FINANCE
Abstract:
In the modern mathematical finance the stock prices are modeled by
stochastic differential equations, whose solutions produce
logarithmic Brownian motions. This is the backbone of what is
nowadays became the classical Black-Scholes option pricing theory
and Merton's investment/consumption theory. We consider a
dynamical portfolio optimization model in the spirit of the
latter. The portfolio consists of several risky assets (Stocks)
and one risk-free asset (Bond). The rate of return on Bond is
constant while the rate of return of Stocks is governed by SDE of
the logarithmic Brownian motion type. Funds can be transferred
from one asset to another, however such transaction involves
penalty (brokerage fees) proportional to the size of the
transaction. The objective is to find the policy which maximizes
the expected rate of growth of funds.
The main mathematical tool in the solution of this problem is
singular stochastic control theory. In this theory the control
functionals are represented by processes of bounded variation, and
the optimal control consists of functionals which reflect the
process from an a priori unknown boundary. They are continuous
but singular (not absolutely continuous) with respect to time.
The analytical part of the solution to singular control is related
to a free boundary problem for an elliptic PDE with gradient
constraints, similar to the ones encountered in elastic-plastic
torsion problems. The existence of the classical $C^2$ solution
cannot be proved in general but one can show an existence of
viscosity solution to this equation.
The optimal policy is to keep the vector of fractions of funds
invested in different assets in an optimal (a priori unknown)
boundary. We show how to find these boundaries explicitly in the
case of one risky and one risk-free asset when the problem becomes
one dimensional. In this case the free boundary problem can be
reduced to a Stephan problem for an ODE.
Time: Thursday, April 21, 2005 at 2:30 pm.
Location: SAV 131
Speaker: Wlodzimierz Bryc (University of Cincinnati)
Title: QUADRATIC HARNESSES, Q-COMMUTATIONS, AND ORTHOGONAL
MARTINGALE POLYNOMIALS
Abstract:
Harnesses were introduced by Hammersley (1967) as random fields that
model `long-range misorientation'. They were studied for example by
Williams (1973), and recently by Mansuy-Yor (2005) who considered
harnesses indexed by t>0 and defined them by a reverse-martingale
condition for the normalized increments of the process.
Quadratic harnesses are related to Hammersley-Manuy-Yor harnesses
by mimicking the relation between the martingale and the quadratic
martingale conditions.
Examples of quadratic harnesses are the Poisson, Gamma, Pascal
(negative binomial), and the Wiener process. Lesser known examples are
Markov processes associated with certain free Levy processes, and with
non-commutative q-Gaussian processes introduced in the physics
literature by Frisch-Bourret (1970) and constructed rigorously by
Bozejko-Kummerer-Speicher {1997}.
It is plausible that generic quadratic harnesses are in fact
five-parameter Markov processes.
The five parameters arise as the coefficients in the q-commutation
equation for the Jacobi matrix of the associated orthogonal
martingale polynomials, which exist under minimal assumptions. At
present, constructions of the corresponding Markov transition
probabilities are known only for special choices of these
parameters, and explicit answers are even less frequent. One such
explicit example of a two-parameter quadratic harness is obtained by
appropriately re-scaling a certain non-homogeneous pure-birth Markov
process that `explodes' at a deterministic moment of time, and then
`returns back from infinity' as a pure-death process.
This talk is based on joint research with W. Matysiak and J. Wesolowski.
Time: Monday, April 18, 2005 at 2:30 pm.
Location: SMI 102
Speaker: Fausto Di Biase (Universit\`a `G.d'Annunzio', Pescara, Italy)
Title: THE STOLZ APPROACH IS SHARP, ISN'T IT?
Abstract:
We consider the following question: how sharp is the Stolz
approach region for the almost everywhere convergence of bounded
harmonic functions in the unit disc in the plane or in NTA domains
in $R^n$? The issue was first settled in the rotation invariant
case in the unit disc by Littlewood in 1927 and later examined,
under less stringent conditions, by Aikawa in 1991. We will
present results that are, in a certain sense, sharp.
A joint work in collaboration with A. Stokolos, O. Svensson
and T. Weiss.
Time: Monday, April 11, 2005 at 2:30 pm.
Location: SMI 102
Speaker: Joan Lind (University of Washington)
Title: THE GEOMETRY OF THE LOEWNER EVOLUTION
Abstract:
The Loewner differential equation provides a connection between
continuous, real-valued functions (called driving terms) and
families of domains in the complex plane (said to be generated by
the driving term.) The recent interest in this 80-year-old
equation is due to Oded Schramm's introduction of the SLE
processes, which have allowed mathematicians to prove many results
predicted by physicists as well as Mandelbrot's conjecture that
the outer boundary of a planar Brownian path has Hausdorff
dimension 4/3. I will discuss how the geometry of the generated
domains is related to the driving function, both in the
deterministic setting and for SLE.
Time: Monday, April 4, 2005 at 2:30 pm.
Location: SMI 102
Speaker: Julien Dub\'edat (Courant Institute, New York University)
Title: COMMUTATION OF SLE's
Abstract:
Schramm-Loewner Evolutions (SLEs) have proved to be a powerful
tool to describe the scaling limit of a conformally invariant
simple curve. In several instances (percolation, uniform spanning
tree ...), one can define in a discrete setting several simple
curves. We will discuss questions pertaining to the joint law of
these curves in the scaling limit.
Time: Tuesday, March 22, 2005 at 2:30 pm.
Location: PDL C-36
Speaker: Jason Swanson (University of Wisconsin)
Title: WEAK CONVERGENCE OF THE MEDIAN OF INDEPENDENT BROWNIAN MOTIONS
Abstract:
In a model of Spitzer (1968) model, we begin with countably many
particles distributed on the real line according to a Poisson
distribution. Each particle then begins moving with a random
velocity; these velocities are independent and have mean zero. The
particles interact through elastic collisions: when particles
meet, they exchange trajectories. We then choose a particular
particle (the tagged particle) and let X(t) denotes its
trajectory. Spitzer showed that, under a suitable rescaling of
time and space, X(t) converges weakly to Brownian motion.
Harris (1965) demonstrated a similar result: if the individual
particles move according to Brownian motion, then X(t) converges
to fractional Brownian motion. These results were generalized even
further by D\"{u}rr, Goldstein, and Lebowitz in 1985.
All of these results rely heavily on the initial distribution of
the particles. The initial Poisson distribution provides tractable
computations in the models of Spitzer, Harris, and D\"{u}rr et al;
In this talk, I will outline the following result: the (scaled)
median of independent Brownian motions converges to a centered
Gaussian process whose covariance function can be written down
explicitly. As with the other results, proving tightness is the
chief difficulty. Unlike the other results, the initial
distribution of the particles does not play a key role. Tightness
is proved in this model by direct estimates on the median process
itself. It is my hope that these techniques can be generalized and
used to extend the current family of results to models with
arbitrary initial particle distributions.
Time: Monday, March 7, 2005 at 2:30 pm.
Location: PDL C-401
Speaker: Roman Kotecky (Charles University, Prague, and Microsoft Research)
Title: BIRTH OF EQUILIBRIUM DROPLET
Abstract:
We consider large deviations for Ising model in the phase
coexistence region. The typical behavior, inside the coexistence
region and far from its edge, is governed by droplet
configurations. However, a new type of transition is experienced
close to the edge of the coexistence. It stems from the
competition between droplet contributions and supersaturated phase
and leads to an abrupt occurrence of a droplet. Its proof needs a
careful evaluation of typical configurations in different regimes.
Main results will be explained and some idea of the proofs will be
given without assuming any preliminary knowledge about the Ising
model.
The talk is based on joint papers with Marek Biskup and Lincoln
Chayes as well as a recent work with Ostap Hryniv and Dima Ioffe.
Time: Monday, February 28, 2005 at 2:30 pm.
Location: THO 135
Speaker: Ryan O'Donnell (Microsoft Research)
Title: NOISE STABILITY OF BOOLEAN FUNCTIONS WITH LOW INFLUENCES
Abstract:
Suppose we model a close election (such as Bush/Gore '00 or
Gregoire/Rossi '04) by assuming n voters vote independently and
50-50 between two candidates, 0 and 1. An ``election scheme'' is
a boolean function $f : \{0,1\}^n \to \{0,1\}$ mapping the votes
to a winner; e.g., $f$ = Majority, or $f$ = Electoral College.
Analyzing the properties of such functions is hampered by the fact
that the trivial ``dictator'' schemes, $f(x_1, ..., x_n) = x_i$,
are often extremal. It is thus both natural and desirable to
restrict attention to $f$'s in which each voter has small
``influence''.
We resolve two conjectures regarding functions with low
influences: 1. The ``Majority Is Stablest'' conjecture from
theoretical computer science, which states that if each vote is
mis-recorded independently with probability epsilon (butterfly
ballot? Diebold?) then Majority is the low-influence election
scheme most likely to preserve the outcome. 2. The ``It Ain't
Over Till It's Over'' conjecture from social choice in economics,
which states that if a random 99\% of the votes are in (from
Florida, say) then for any low-influence election scheme, the
outcome is still overwhelmingly likely to be ``too close to
call''.
This is joint work with Elchanan Mossel (Berkeley) and Krzysztof
Oleszkiewicz (Warsaw).
Time: Tuesday, February 22, 2005 at 2:30 pm.
Location: SAV 311
Speaker: Dayue Chen (Peking University)
Title: THE ANCHORED EXPANSION CONSTANT OF RANDOM GRAPHS
Abstract:
The anchored expansion constant is a variant of the Cheeger
constant; its positivity implies positive lower speed for the
simple random walk, as shown by Virag (2000). We prove that if $G$
has a positive anchored expansion constant then so does every
infinite cluster of independent percolation with parameter $p$
sufficiently close to 1. We also show that positivity of the
anchored expansion constant is preserved under a random stretch
if, and only if, the stretching law has an exponential tail.
This talk is based on my joint work with Yuval Peres and Gabor
Pete of the University of California, Berkeley. The paper entitled
``Anchored Expansion, Percolation and Speed'' just appeared in the
Annals of Probability.
Time: Monday, February 14, 2005 at 2:30 pm.
Location: THO 135
Speaker: Assaf Naor (Microsoft Research)
Title: MARKOV CHAINS IN METRIC SPACES
AND THEIR APPLICATIONS TO METRIC GEOMETRY
Abstract:
In this talk we will discuss the notion of Markov type: an
important bi-Lipschitz invariant of metric spaces which was
introduced by K. Ball. This invariant measures the affect of the
geometry of a metric space $X$ on the speed of certain Markov
chains taking values in $X$. We will describe some of the
applications of Markov type, and proceed to present a proof of the
recent solution of Ball's Markov type 2 problem, showing that
$L_p, p>2$, has Markov type 2. This result completes the work of
Ball and settles a conjecture of Johnson and Lindestrauss from
1992 by showing that every Lipschitz function defined on a subset
of $L_p, p>2$, with values in $L_q, 1 < q <2$, extends to a Lipschitz
function defined on all of $L_p$.
Based on joint work with Yuval Peres, Oded Schramm and Scott
Sheffield.
Time: Monday, February 7, 2005 at 2:30 pm.
Location: THO 135
Speaker: Uriel Feige (Weizmann Institute and Microsoft Research)
Title: ON SUMS OF INDEPENDENT RANDOM VARIABLES
Abstract:
We prove the following inequality: for every positive integer $n$
and every collection $X_1, \ldots, X_n$ of nonnegative independent
random variables that each has expectation~1, the probability that
their sum remains below $n+1$ is at least $\alpha > 0$. Our proof
produces a value of $\alpha = 1/13 \simeq 0.077$, but we
conjecture that the inequality also holds with $\alpha = 1/e
\simeq 0.368$.
Time: Monday, January 31, 2005 at 2:30 pm.
Location: THO 135
Speaker: Richard Bass (University of Connecticut)
Title: RENORMALIZED SELF-INTERSECTION LOCAL TIME
AND THE RANGE OF RANDOM WALKS
Abstract:
Self-intersection local time $\beta_t$ is a measure of how often a
Brownian motion (or other process) crosses itself. Since Brownian
motion in the plane intersects itself so often, a renormalization
is needed in order to get something finite. LeGall proved that $E
e^{\gamma \beta_1}$ is finite for small $\gamma$ and infinite for
large $\gamma$. It turns out that the critical value is related to
the best constant in a Gagliardo-Nirenberg inequality. I will
discuss this result (joint work with Xia Chen) as well as large
deviations for $\beta_1$ and $-\beta_1$ and LILs for $\beta_t$ and
$-\beta_t$. The range of random walks is closely related to
self-intersection local times, and I will also discuss joint work
with Jay Rosen making this idea precise.
Time: Tuesday, January 25, 2005 at 12:30 pm.
Location: MOR 219
Speaker: Bruce Erickson (University of Washington)
Title: FINITENESS OF INTEGRALS OF FUNCTIONS OF LEVY PROCESSES
Abstract:
We determine necessary and sufficient conditions under which
various integrals of functions of the supremum absolute value of a
Levy process are finite. The conditions are expressed
analytically in terms of the canonical Levy measure of the
process. An application of some of these results leads to an
interesting local (small time) comparison of the above mentioned
supremum process with the supremum absolute jump process.
This is a brief report on some work in progress with R. Maller.
Time: Wednesday, January 12, 2005 at 3:30 pm.
Location: Padelford C-36
Speaker: Krzysztof Burdzy (University of Washington)
Title: THE ROBIN PROBLEM AND RAY-KNIGHT THEOREM
Abstract:
The ``Robin problem'' or the ``third boundary problem'' is a
mathematical model for the flow of a substance (or heat) out of a
domain through a semi-permeable membrane. I will address the
question of when the concentration of the substance (or the
temperature) is bounded below by a constant over the whole domain.
The argument is based in part on a very non-trivial (although old)
result known as "Ray-Knight theorem". It describes the
distribution of the local time of Brownian motion as a function of
the space variable.
Time: Monday, Dec 6, 2004 at 2:30 pm.
Location: PDL C-401
Speaker: David Wilson (Microsoft Research)
Title: BALANCED BOOLEAN FUNCTIONS THAT CAN BE EVALUATED
SO THAT EVERY INPUT BIT IS UNLIKELY TO BE READ
Abstract:
A Boolean function of n bits is balanced if it takes the value 1
with probability 1/2. We exhibit a balanced Boolean function with
a randomized evaluation procedure (with probability 0 of making a
mistake) so that on uniformly random inputs, no input bit is read
with probability more than $\Theta(n^{-1/2} \sqrt{\log n})$. We
give a balanced monotone Boolean function for which the
corresponding probability is $\Theta(n^{-1/3} \log n)$. We then
show that for any randomized algorithm for evaluating a balanced
Boolean function, when the input bits are uniformly random, there
is some input bit that is read with probability at least
$\Theta(n^{-1/2})$. For balanced monotone Boolean functions, there
is some input bit that is read with probability at least
$\Theta(n^{-1/3})$.
Joint work with Itai Benjamini and Oded Schramm.
Time: Monday, Nov. 29, 2004 at 2:30 pm.
Location: PDL C-401
Speaker: David White (University of Washington)
Title: INTERACTIONS OF A BROWNIAN PARTICLE AND AN INERT PARTICLE
Abstract:
Frank Knight recently introduced a model for the motion of an
inert particle that is impinged on one side by a Brownian
particle. Interesting behavior can result when the two types of
particles are combined in different configurations. We describe
the stochastic process resulting from these interactions for a few
possible configurations.
Time: Monday, Nov. 15 and Monday, Nov. 22, 2004 at 2:30 pm.
Location: PDL C-401
Speaker: Charles J. Geyer (University of Minnesota)
Title: PROBABILITY AND STATISTICS USING NELSON-STYLE
NONSTANDARD ANALYSIS (parts I and II)
Abstract:
Edward Nelson's book Radically Elementary Probability Theory
shows how to `radically' simplify both nonstandard analysis
(NSA) and probability theory. In this approach all sample spaces
are finite and all nonempty events have nonzero probability. So
all nontrivial collections of random variables are finite. We have
only with finite sequences of random variables, stochastic
processes with finite carriers, etc. This eliminates any need for
measure theory. Nevertheless, infinitesimal and unlimited numbers
allow analogs of phenomena of conventional probability theory,
such as continuous random variables, laws of large numbers,
central limit theorems, invariance principles, and Brownian
motion.
In part I (Nov 15) we will introduce Nelson's `radically elementary'
version of NSA, which is simple enough to teach to undergraduates,
and using de Finetti's theorem as an example, show how this
approach can greatly simplify probabilistic reasoning. We also
start a survey of the definitions and main results of Nelson's
book.
In part II (Nov 22) we will discuss some of our work: weak convergence in
metric spaces, the portmanteau theorem, the continuous mapping
theorem, the Glivenko-Cantelli theorem, Prohorov metric strong
consistency of the empirical process, spatial point processes
(redone Nelson-style).
Time: Monday, Nov. 8, 2004 at 2:30 pm.
Location: PDL C-401
Speaker: Benjamin Morris (Indiana University and Microsoft Research)
Title: THE MIXING TIME FOR THE THORP SHUFFLE
Abstract:
In 1973, Thorp introduced the following card shuffling procedure.
Cut the deck into two equal piles. Drop the first card from the
left pile or the right pile according to the outcome of a fair
coin flip; then drop from the other pile. Continue this way,
flipping an independent coin each time, until both piles are
empty.
Despite its simple description, the Thorp shuffle has been hard to
analyze. It has long been believed that the mixing time is
polynomial in log of the number of cards. We prove the first such
bound.
Time: Monday, Nov. 1, 2004 at 2:30 pm.
Location: PDL C-401
Speaker: Chris Hoffman (University of Washington)
Title: HOW WELL CAN WE PREDICT A NEAREST NEIGHBOR PROCESS?
Abstract:
We consider the class of integer valued nearest neighbor
processes. These are the processes $S(t)$ such that for all $t$
$|S(t)-S(t+1)|=1$ a.s. In this problem we are given the past
$S(0),S(-1),...$ and we try to predict $S(k)$. We want to bound
the maximum probability that we predict correctly. Define
$$v(k)=\sup_{n,S(0),S(-1),S(-2),\dots}
P(S(0)-S(k)=n|S(0),S(-1),S(-2),\dots).$$
We will show that for any decreasing sequence $f(k)$ such that
$\sum_{k=1}^{\infty}f(k)=\infty$ there does not exist a nearest
neighbor process such that $$v(k)<{1\over kf(k)}$$ for all $k$ and
that this condition is tight. We will also talk about how this
problem relates to random walks on percolation clusters.
Time: Monday, Oct 18, 2004 at 2:30 pm.
Location: PDL C-401
Speaker: Zhenqing Chen (University of Washington)
Title: EIGENVALUES FOR SUBORDINATE PROCESSES IN DOMAINS
Abstract:
Abstract: When studying spectral properties for nonlocal operator in
domains, one often faces many new challenges. For example, in
contrast to the Laplace operator case, even the first eigenvalue
for 1-dimensional fractional Laplace operator in unit interval is
not explicitly known. In this talk, we will show that it is
possible to obtain two-sided eigenvalue estimates for fractional
Laplacian in terms of the eigenvalues of the Dirichlet Laplacian
in the domain. In fact, the above result holds more generally,
with Laplacian being replaced by a self-adjoint operator $L$ that
is the generator of a strong Markov process, and the fractional
Laplacian being replaced by $-\phi (-L)$, where $\phi$ is the
Laplace exponent of a subordinator. We further show that the
eigenvalues of $\phi (-L)$ in a domain with Dirichlet exterior
condition depends continuously on $\phi$.
This talk is based on joint work with R. Song.
Time: Monday, Oct 11, 2004 at 2:30 pm.
Location: PDL C-401
Speaker: Nati Linial (Hebrew University of Jerusalem and Microsoft)
Title: RANDOM LIFTS OF GRAPHS AND WHAT THEY ARE GOOD FOR
Abstract:
Time: Monday, Oct 11, 2004 at 2:30 pm.
Location: PDL C-401
Speaker: Nati Linial (Hebrew University of Jerusalem and Microsoft)
Title: RANDOM LIFTS OF GRAPHS AND WHAT THEY ARE GOOD FOR
Abstract:
The most thoroughly studied random graph model is the classical
Erd\"os-Renyi G(n,p) model where edges are placed independently
and with probability p between any pair from among n vertices.
There is a strong feeling in many parts of discrete mathematics
that this is not enough, and new models for random graphs are
needed. The quest for new models often comes from a desire to
depict some physical phenomena, or just to understand the typical
behavior of a certain family of graphs. Random lifts are a new
class of random graphs, and the chief reason for introducing them
was the hope to construct graphs with special desired properties,
which existing methods seem unable to achieve. Our research into
these graphs has revealed lots of beautiful phenomena and raised
new intriguing problems. However, only recently did it become
possible to use Random Lifts to construct graphs "by design" i.e.,
graphs with some desired extremal properties. Specifically, they
were utilized to explicitly construct graphs with very good
spectral properties ("Quasi-Ramanujan" graphs). In this talk I
will explain what random lifts are. I will also survey the main
known results and some of the open problems.
Time: Monday, October 4, 2004 at 2:30 pm.
Location: PDL C-36
Speaker: Krzysztof Burdzy (University of Washington)
Title: LENSES IN SKEW BROWNIAN MOTION`
Abstract:
Skew Brownian motion is a version of Brownian motion which makes
more positive excursions from 0 than negative excursions (or vice
versa). I will discuss a stochastic flow of skew Brownian motions,
i.e., a family of skew Brownian motions driven by the same
``noise.'' Individual processes in the flow have a tendency to
coalesce so this leads to a number of questions about the
topological and stochastic nature of the flow. Lenses are pairs of
distinct processes in the flow that start from the same point,
diverge, and then converge.
Time: Thursday, August 26, 2004 at 3:00 pm.
Location: PDL C-36
Speaker: Wojbor A. Woyczynski (Case Western Reserve University)
Title: Critical nonlinearity exponent for Levy conservation laws
Abstract:
Behavior of solutions of nonlinear evolutions of the form u_t=Lu-\nabla
Nu, where L is a linear diffusive operator and N is a nonlinear
operator depends on the interaqction and relative strength of L and N.
In the case when L is an infinitesimal generator of a Levy process
there is a critical nonlinearity which causes the asymptotic behavior
of all solutions to be like the asymptotic behavior of special
selfsimilar solutions. Statistical implications of this facts are
indicated.
Time: Thursday, May 27, 2004 at 2:30 pm.
Location: THO 331
Speaker: Richard Sowers (University of Illinois at Urbana-Champaign)
Title: Random Perturbations of Pseudoperiodic Flows
Abstract:
Arnol'd in 1991 characterized ``pseudoperiodic'' flows on the 2-dimensional torus.
We consider small random perturbations of such flows. Under appropriate
scaling of time, we
search for an averaged picture which describes the evolution of local ``energies''.
Under certain circumstances, we identify a certain
limiting Markov process with glueing conditions
(as suggested by Freidlin in 1996) which characterizes energy evolution.
Time: Monday, May 24, 2004 at 2:30 pm.
Location: THO 125
Speaker: Zoran Vondracek (University of Zagreb, Croatia)
Title: Ruin Probabilities for General Perturbed Risk Processes
Abstract:
We study a general perturbed risk process with
cumulative claims modeled by a subordinator with finite expectation,
and the perturbation being a spectrally negative Levy process with
zero expectation. We derive a Pollaczek-Hinchin type formula for
the survival probability of that risk process, and give an
interpretation of the formula based on the decomposition of the
dual risk process at modified ladder epochs. We also give a formula
that the ruin will occur by a jump of the subordinator.
Notice the different time and location
Time: Tuesday, May 18, 2004 at 2:30 pm.
Location: THO 331
Speaker: Renming Song (University of Illinois at Urbana-Champaign)
Title: Potential Theory of Geometric Stable Processes
Abstract:
Geometric stable processes form a special class of
Levy processes. Geometric stable processes are very
useful in modeling heavy-tailed phenomena and have
been used by various researchers in option pricing.
In this talk we will present some recent results
on the potential theory of geometric stable processes.
In particular, we will talk about asymptotic behaviors
of the Green function and jumping functions of geometric
stable processes, and estimates on the capacities
of small balls with respect to geometric stable
processes. We also show that the Harnack inequality is
valid for positive harmonic functions of geometric
stable processes.
Time: Monday, May 10, 2004 at 2:30 pm.
Location: THO 125
Speaker: Chris Hoffman (University of Washington)
Title: Coexistence for Richardson Type Competing Spatial Growth Models
Abstract:
In the Richardson growth model the vertices in Z^d can take on three
possible states 0,1, and 2. Vertices in states 1 and 2 remain in their
states forever, while vertices in state 0 which are adjacent to a vertex
in state 1 (or state 2) can switch to state 1 (or state 2). We think of
the vertices in states 1 and 2 as infected with one of two infections
while the vertices in state 0 are considered uninfected. We start the
models with a single vertex in state 1 and a single vertex is in state 2.
We show that with positive probability state 1 reaches an infinite number
of vertices and state 2 also reaches an infinite number of vertices. The
key tool is applying the ergodic theorem to stationary first passage
percolation.
Time: Monday, May 3, 2004 at 2:30 pm.
Location: THO 125
Speaker: Tadeusz Kulczycki (Wroclaw Technical University,
visiting Purdue University)
Title: The Cauchy Process and the Steklov Problem
Abstract:
Let $X_t$ be the Cauchy process in $R^d$. We investigate some of the fine
properties of the semigroup of this process killed upon leaving a domain
$D$. We establish a connection between the semigroup of this process and a
mixed boundary value problem for the Laplacian in one dimension higher,
known as the "Mixed Steklov Problem". Using this we derive a variational
characterization for the eigenvalues of the Cauchy process in $D$. This
characterization leads to many detailed properties of the eigenvalues and
eigenfunctions for the Cauchy process inspired by those for the Brownian
motion. The results are new even in the simplest geometric setting of the
interval $D=(-1,1)$ where we obtain more precise information on the size
of the second eigenvalue and on the geometry of the corresponding
eigenfunction.
Time: Monday, April 26, 2004 at 2:30 pm.
Location: THO 125
Speaker: Gordon Slade (University of British Columbia, Visiting Microsoft)
Title: The phase transition for random subgraphs of the n-cube
Abstract:
We describe recent results, obtained in collaboration with
C. Borg, J. T. Chayes, R. van der Hofstad and J. Spencer, which
provide a detailed description of the phase transition for
random subgraphs of the n-cube.
Time: Monday, April 19, 2004 at 2:30 pm.
Location: THO 125
Speaker: Jason Swanson (UW)
Title: The Signed Variations of the Solution to
a Stochastic Heat Equation
Abstract:
We will consider the solution to a certain stochastic heat equation. This
solution is a random function of time and space. For a fixed point in
space, the resulting random function of time (call it $F(t)$) has the same
local behavior as a fractional Brownian motion with Hurst parameter
$H=1/4$. (Roughly speaking, this means that, almost surely, the function
$F(t)$ is H\"{o}lder continuous with index $\alpha$ for all $\alpha<1/4$,
but not for $\alpha\ge1/4$.) The function $F(t)$, therefore, has an
infinite quadratic variation and, hence, is not a semimartingale. It
follows then that the classical Ito calculus does not apply to $F(t)$.
Heuristic ideas about a possible new calculus for this process will be
presented. These heuristics lead us in a natural way to define and study
what will be called the ``signed" quadratic variation of $F(t)$. The main
result to be presented is that, although the traditional quadratic
variation of $F(t)$ is infinite, the signed quadratic variation converges
to a Brownian motion.
Time: Monday, April 5, 2004 at 2:30 pm.
Location: THO 125
Speaker: Paul Shields
Title: Two New Methods of Markov Order Estimation
Abstract:
Given an ergodic finite state Markov chain of unknown order K,
the goal is to make an estimate k(n) of K from observation of a sample path
of length n, such that k(n) is eventually almost sure equal to K. An
important and widely used estimation method is Schwarz's BIC, which is based
on Bayesian ideas. Recently, in joint work with Csiszar, BIC consistency
was established without the usual prior bound assumption. The proof,
however, is quite complex. I will discuss two new methods that are simple to
describe, require no prior order bound, involve less computation than the
BIC, and whose consistency proofs use only classical probability and
information theory concepts. The first method focuses on maximum
fluctuations of empirical conditional probabilities of order k over slowly
growing intervals of values of k. The second method compares empirical
conditional entropy of order k with an auxiliary, upwardly biased, entropy
estimator. In both cases, the focused quantities have, eventually almost
surely, a qualitative change in behaviour for the first time at the place
where $k=K$. Extensions to random fields and to hidden Markov chains will
also be discussed. (This is joint work with Yuval Peres.)
Time: Wednesday, March 24, 2004 at 2:30 pm.
Location: PDL C-36
Speaker: Marek Biskup (UCLA)
Title: Graph Distance in Long-range Percolation
Abstract:
In 1967, using an ingenious sociological experiment,
S. Milgram studied the length of acquaintance chains between
"geometrically distant" individuals.
The results led him to the famous conclusion that average two Americans
are about six acquaintances (or "six handshakes") away from each other.
We will model the situation in terms of long-range percolation on
Zd, where the nearest neighbor bonds represent
the acquaintances due to geometric proximity -- people living in the house
next door -- while long bonds are acquaintances established by other
means -- e.g., friends from college.
The question is: What is the minimal number of bonds one needs to traverse
to get from site x to site y.
Thus, in addition to the usual connections between nearest neighbors on
Zd, any two sites x,y in Zd
at Euclidean distance |x-y| will be connected by an occupied
bond independently with probability proportional to
|x-y|-s, where s>0 is a parameter.
Using D(x,y) to denote the length of the shortest occupied path between
x and y, the main question boils down to the asymptotic
scaling of D(x,y) as |x-y| tends to infinity.
I will discuss a variety of possible behaviors and mention known results
and open problems. Then I will sketch the proof of the fact that,
when s in the interval (d,2d),
the distance D(x,y) scales like (log|x-y|)Delta,
where Delta-1 is the binary logarithm of 2d/s.
SPECIAL COLLOQUIUM/PROBABILITY SEMINAR
Time: Tuesday, March 16, 2004 at 2:30 pm.
Location: SMI 205
Speaker: Wendelin Werner (Universite Paris-Sud)
Title: CONFORMAL FIELD THEORY VIA BROWNIAN LOOP SOUP
Abstract:
We will (briefly) show how to relate Schramm-Loewner Evolutions
and to define Conformal Field Theories using the Brownian
loop-soup, a random conformally invariant countable family of
overlapping Brownian loops in a domain that we introduced in a
joint paper with Greg Lawler.
Time: Monday, March 8, 2004 at 2:30 pm.
Location: LOW 217
Speaker: Fadoua Balabdaoui (University of Washington)
Title: NONPARAMETRIC ESTIMATION OF A $K$-MONOTONE DENSITY
ASYMPTOTIC DISTRIBUTION THEORY
Abstract:
Let $k \geq 1$ be an integer. We consider nonparametric estimation
of a $k$-monotone density on $(0,\infty)$ via the methods of
Maximum Likelihood and Least Squares. Under the assumption that at
a fixed point $x_0 > 0$, the true density $g_0$ satisfies
$g^{(k)}_0(x_0) \ne 0$, we establish asymptotic minimax lower
bounds for estimation of the $j$-th derivative of $g_0$ for $j=0,
\cdots, k-1$. These bounds show that the rates of convergence of
any estimator of $g^{(j)}_0(x_0)$ can be {\it at most}
$n^{-(k-j)/(2k+1)}$. Furthermore, we are close to proving that the
ML and LS estimators, $\hat{g}_n$ and $\tilde{g}_n$, achieve these
rates, and that the limiting distributions depend on smooth
stochastic processes $H_k$ that stay above (or below) $Y_k$, the
$(k-1)$-fold integral of two-sided Brownian motion +
$\left(k!/(2k)!\right) t^{2k}, t \in {\bf R}$ when $k$ is even (or
odd). The key remaining difficulty for $k > 2$ consists in showing
that the distance between two successive jump points $\tau^{-}_n$
and $\tau^{+}_n$ of $\hat{g}^{(k-1)}_n$ or $\tilde{g}^{(k-1)}_n$
(in the neighborhood of $x_0$) is $O_p(n^{-1/(2k+1)})$ as $n \to
\infty$. Similarly, if $H_{k,c}$ denotes the envelope (or the
\lq\lq invelope\rq\rq) of $Y_k$ when $k$ is odd (or even) on
$\lbrack -c,c \rbrack$, $c > 0 $, it remains to prove that the
distance between two successive points of touch $\tau^{-}_c$ and
$\tau^{+}_c$, before and after a point $-c < t < c$, between the
processes $H_{k,c}$ and $Y_k$ is $O_p(1)$ as $c \to \infty$.
Several numerical examples will be shown to illustrate the
theoretical results and conjectures. For that, we will introduce a
$(2k-1)$-th iterative spline algorithm developed to compute the ML
and LS estimators, and to obtain an approximation of the process
$H_k$ on $\lbrack -c,c \rbrack $.
Joint work with Jon Wellner.
Time: Monday, March 1, 2004 at 2:30 pm.
Location: LOW 217
Speaker: Martin Barlow (University of British Columbia)
Title: RANDOM WALKS ON CRITICAL PERCOLATION CLUSTERS
Abstract:
It is now known that random walks on supercritical
($p>p_c$) percolation clusters in $Z^2$ behave in many ways like
the simple random walk on $Z^d$. The critical case ($p=p_c$) is
much harder. One needs to consider the ``incipient infinite
cluster''; in the cases where this has been defined it has a
fractal structure. The easiest case is that of trees; this was
studied by Kesten in 1986, but we can now revisit this problem
with new techniques.
This is a joint Math Department Colloquium and
Probability Seminar Talk
Time: Monday, February 23, 2004 at 2:30 pm.
Location: LOW 217
Speaker: Masatoshi Fukushima (Kansai University)
Title: POISSON POINT PROCESSES ATTACHED TO SYMMETRIC DIFFUSIONS
Abstract:
Let $a$ be a non-isolated point of a topological space $S$ and
$X^0$ be a symmetric diffusion on the complementary set $S_0$
which approaches to $a$ in finite time before killing with
positive probability. By making use of Poisson point processes
taking values in the spaces of excursions around $a$ whose
characteristic measures are uniquely determined by $X^0$, we
construct a symmetric diffusion on $S$ with no killing inside $S$
which extends $X^0$ on $S_0.$ We also prove that such an extension
is unique in law and its resolvent and Dirichlet form admit
explicit expressions in terms of $X^0.$
Time: Monday, February 9, 2004 at 2:30 pm.
Location: LOW 217
Speaker: Tilmann Gneiting (University of Washington)
Title: CONVOLUTION ROOTS OF COMPACTLY SUPPORTED
RADIAL POSITIVE DEFINITE FUNCTIONS
Abstract:
A classical theorem of Boas, Kac, and Krein states that a
characteristic function $\varphi$ with $\varphi(x) = 0$ for $|x|
\geq \tau$ admits a representation of the form
$$
\varphi(x) = \int u(y) \overline{u(y+x)} dy, x \in \real
$$
where $u \in L_2(\real)$ is complex-valued with $u(x) = 0$ for
$|x| \geq \tau/2$. The result can be expressed equivalently as a
factorization theorem for entire functions of finite exponential
type. This talk examines the Boas-Kac representation under
additional constraints: If $\varphi$ is real-valued and even, can
the convolution root $u$ be chosen as a real-valued and/or even
function?
A complete answer in terms of the zeros of the Fourier transform
of $\varphi$ is obtained. Furthermore, the analogous problem for
radially symmetric functions is solved. Perhaps surprisingly,
there are compactly supported, radial positive definite functions
that do not admit a convolution root with half support. Related
results include pointwise and integral bounds on compactly
supported positive definite functions and the solution to an
associated minimization problem. Applications to spatial moving
average processes, geostatistical simulation, crystallography,
optics, and phase retrieval are noted.
This is joint work with Werner Ehm and Donald Richards.
Time: Monday, February 2, 2004 at 2:30 pm.
Location: LOW 217
Speaker: David Galvin (Microsoft Research)
Title: THE ``ENTROPY METHOD'' IN COMBINATORICS
Abstract:
The binary entropy of a finite-range uniform random variable is
exactly the log of the size of the range space. For this reason,
entropy methods have become quite popular recently for tackling
certain enumerative problems in combinatorics---any bounds that
can be put on the entropy of a uniformly chosen member of a set
correspond immediately to bounds on the size of the set.
In this talk I will introduce an entropy inequality due to J. B.
Shearer that is proving to be a powerful tool in this direction. I
will give two applications. The first is a swift entropy proof of
an old result of Loomis and Watson, bounding the volume of a body
in $R^d$ in terms the volumes of its $(d-1)$-dimensional
projections. The second is a recent result (joint with P. Tetali
of Georgia Tech) giving a tight upper bound on the number of graph
homomorphisms from a regular bipartite graph to any fixed
constraint graph.
Time: Monday, January 26, 2004 at 2:30 pm.
Location: LOW 217
Speaker: D. Brian Walton (University of Washington)
Title: HIDDEN MARKOV MODELS AND SINGLE-MOLECULE
MOTOR PROTEIN EXPERIMENTS
Abstract:
Motor proteins convert chemical energy into directed mechanical
motion, often along filamentary tracks. This talk will introduce a
particular motor protein, kinesin, which converts the energy
released from ATP hydrolysis into moti