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Convergence of Nearest Neighbor Markov chains on discrete trees towards
Brownian Motion on real-trees
Siva Athreya
Indian Statistical Institute, Bangalore
We will begin with a review of the ν-Brownian motion on any locally compact R-tree (T, r)
equipped with a Radon measure. Then we will show that this ν-Brownian motion on (T, r) is
the scaling limit of nearest neighbor Markov chains on discrete metric graphs (Tn , rn ) which jump
from a vertex v ∈ Tn to a neighboring vertex v 0 ∼ v with rate
(νn ({v}) · rn (v, v 0 ))−1
provided that (Tn , rn , νn ) converges to (T, r, ν) in a Gromov-weak type of topology. We shall
explain the corresponding topology as well.
This is joint work with Wolfgang Lohr and Anita Winter
A stability theorem for the elliptic Harnack inequality
Richard Bass
University of Connecticut
Harnack inequalities are an important tool in probability theory, as well as for analysis and partial
differential equations. I will talk about a stability theorem for the elliptic Harnack inequality:
if two weighted graphs are equivalent, then the elliptic Harnack inequality holds for harmonic
functions with respect to one of the graphs if and only if it holds for harmonic functions with
respect to the other graph. I’ll also give a characterization of the elliptic Harnack inequality.
Limiting spectral distribution of patterned random matrices
Arup Bose
Indian Statistical Institute, Kolkata
We present a unified approach to establishing limiting spectral distribution (LSD) of patterned
matrices via the moment method. This allows us to demonstrate relatively short proofs for the
LSDs of common matrices (Wigner, Toeplitz, Hankel, Reverse Circulant, Symmetric Circulant)
and provide insight into the nature of different LSDs and their interrelations.
The method is applicable to matrices with appropriate dependent entries, banded matrices
(including triangular matrices) and matrices of the form Ap = n1 XX 0 where X is a p × n matrix
with real entries and p → ∞ with n = n(p) → ∞ and p/n → y with 0 ≤ y < ∞. The sample
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variance covariance matrix being a particular example of the latter. We can also establish the
existence of the LSD of the sample autocovariance matrix by this method.
Joint convergence of several copies of different patterned matrices may also be established by
this approach.
This approach raises interesting questions about the class of patterns for which LSD exists
and the nature of the possible limits. In many cases the LSDs are not known in any explicit
forms and so deriving probabilistic properties of the limit are also interesting issues.
Certain small noise limits for diffusions
Vivek Borkar
Indian Institute of Technology, Mumbai
This talk will describe some recent work (joint with K. Sureshkumar) on the small noise large
time asymptotics of the normalized Feynman-Kac semigroup using control theoretic techniques.
Pathwise uniqueness of stochastic partial differential equations in Banach spaces.
Sandra Cerrai
University of Maryland
We prove pathwise uniqueness for an abstract stochastic reaction-diffusion equation in Banach
spaces. The drift contains a bounded Holder term; in spite of this, due to the space-time white
noise it is possible to prove pathwise uniqueness. The proof is based on a detailed analysis of
the associated Kolmogorov equation. The model includes examples not covered by the previous
works based on Hilbert spaces or concrete SPDEs. This is a joint work with G. da Prato and F.
Flandoli.
Conditional ergodicity in infinite dimension
Ramon Van Handel
Princeton University
The classical ergodic theory of Markov chains has achieved a rather definitive form in the theory
of Harris chains, which characterizes total variation convergence to equilibrium. While this
theory is formulated in principle for any measurable state space, it is mainly applicable in finite
dimension: transition probabilities in infinite dimension are typically singular, so that the total
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variation convergence cannot hold. In models such as stochastic PDEs, ergodicity has therefore
largely been investigated by topological (rather than measure-theoretic) methods.
While many beautiful results can be obtained in this manner, topological methods are not
well suited to certain probabilistic questions. We are particularly motivated by the problem
of establishing ergodicity of nonlinear filters when the underlying model is a stochastic PDE:
as conditioning is a very singular measure-theoretic operation, topological properties are not
expected to be preserved. Nonetheless, I will argue that topological ergodicity can often be
encoded in a completely measure-theoretic fashion. This surprisingly simple idea provides a key
tool for establishing conditional ergodicity in infinite dimension. (Joint work with Xin Thomson
Tong.)
Nodal length of random eigenfunctions of the Laplacian on the 2-d torus
Manjunath Krishnapur
Indian Institute of Science, Bangalore
We consider Gaussian linear combinations of eigenfunctions of the Laplacian on the 2-dimensional
torus for a given eigenvalue. The mean of the length of the nodal set is easy to compute. As
the dimension of the eigenspace goes to infinity, we find asymptotics of the variance of the nodal
length. The proof has two main steps. First, the Kac-Rice formulas from probability give a
formula for the variance in terms of the covariance kernel of the Gaussian random function. The
covariance kernel is of an arithmetic nature and is analyzed using tools from additive combinatorics. This is joint work with Igor Wigman and Pr Kurlberg.
Time-change equations for diffusion processes
Tom Kurtz
University of Wisconsin
General notions of weak and strong solutions of stochastic equations will be described and a
general version of the Yamada-Watanabe-Engelbert theorem relating existence and uniqueness
of weak and strong solutions given. Time-change equations for diffusion processes provide an
interesting example. Such equations arise naturally as limits of analogous equations for Markov
chains. For one-dimensional diffusions they also are essentially given in the now-famous notebook of Doeblin. Although requiring nothing more than standard Brownian motions and the
Riemann integral to define, the question of strong uniqueness remains unresolved. To prove
weak uniqueness, the notion of compatible solution is introduced and the martingale properties
of compatible solutions used to reduce the uniqueness question to the corresponding question for
a martingale problem or an Ito equation.
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Rates of convergence of color count in balanced urn models
Krishanu Maulik
Indian Statistical Institute, Kolkata
We consider urn models containing balls of finitely many colors and with balanced replacement
matrices, that is, replacement matrices with common row sum. We study the rates of convergence
of each color count and provide relationship between the limiting random variables corresponding
to various colors thus obtained. We show that the convergence occurs both in almost sure and
in Lp sense, for any p > 0.
Parts of the work are jointly done with Amites Dasgupta and Gourab Ray.
Brownian Web in the Scaling Limit of Supercritical Oriented Percolation in
Dimension 1 + 1
Anish Sarkar
Indian Statistical Institute, Kolkata
In this talk, I will discuss the connection of oriented percolation with Brownian Web. In particular, we show that, after centreing and diffusively rescaling space and time, the collection of
rightmost infinite open paths in a supercritical oriented percolation configuration on the spacetime lattice Z2even := {(x, i) ∈ Z2 : x + i is even} converges in distribution to the Brownian
web. This proves a conjecture of Wu and Zhang (2008). Our key observation is that each rightmost infinite open path can be approximated by a percolation exploration cluster, and different
exploration clusters evolve independently before they intersect.
A metric between probability distributions of different sizes and applications to
order reduction
Mathukumalli Vidyasagar
University of Texas at Dallas
In this talk we define a metric distance between probability distributions when they are defined
over finite sets of different cardinalities. The metric is obtained by finding a joint distribution
whose marginals are the two given distributions, such that its entropy is minimized. This is
equivalent to maximizing the mutual information between two random variables having the two
given distributions. It turns out that in general actually computing the distance is NP-hard, so a
greedy algorithm is proposed instead, based on viewing this as a bin-packing problem. Next, the
problem of optimally approximating a high-order distribution by a lower-order distribution (in
this metric) is studied. It is shown that the optimal approximation is always an aggregation of the
original distribution that has maximum entropy (amongst all aggregations). Finding the optimal
aggregation also turns out to be NP-hard, so this problem is also formulated as a bin-packing
problem and a greedy algorithm is proposed for its solution.
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