# Past Probability Seminars Spring 2006

## UW Math Probability Seminar Spring 2006

Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted.

Organized by Jason Swanson

### Schedule and Abstracts

|| Thursday, January 19 || || *David Griffeath, * University of Wisconsin-Madison || || *MODELLING SNOW CRYSTAL GROWTH I: History, Morphology, and Remaining Riddles* ||

Six-sided ice crystals that fall to earth in ideal winter conditions, commonly known as snowflakes, have fascinated scientists for centuries. They exhibit a seemingly endless variety of shape and structure, often dendritic and strangely botanical, yet highly symmetric and mathematical in their designs. To this day, snowflake growth from molecular scales, with its tension between disorder and pattern formation, remains mysterious in many respects. This introductory lecture will first review the history of the scientific study of snowflakes, concentrating on the seminal contributions of Johannes Kepler, Wilson Bentley, and Ukichiro Nakaya. Then I will survey mathematical idealizations, beginning with one of the first fractals -- the Koch Snowflake introduced over a century ago -- and proceeding to more recent growth models such as diffusion limited aggregation and the phase field process. The talk will conclude by summarizing recent progress in our understanding of snow crystal dynamics and by highlighting some of the most intriguing open questions.

|| Thursday, January 26 || || *David Griffeath, * University of Wisconsin-Madison || || *MODELLING SNOW CRYSTAL GROWTH II: Rigorous Results for Packard's Digital Snowflakes* ||

Digital snowflakes are solidifying cellular automata on the triangular lattice with the property that a site having exactly one occupied neighbor always becomes occupied at the next time. They were introduced by Norman Packard in 1984 as prototypes for crystal solidification, and subsequently popularized by Stephen Wolfram in Scientific American, "A New Kind of Science" and elsewhere, to illustrate how simple mathematical algorithms can emulate complex natural phenomena. Wolfram argues that simulation by computer may be the only way to predict how systems such as this evolve, and that patterns grown from a single cell according to Packard's rules bear a close resemblance to real snowflakes. This talk will demonstrate that each such rule is in fact amenable to traditional mathematical analysis, filling the lattice with an asymptotic density that is independent of the initial finite set. There are some cases in which this density can be computed exactly, and others in which it can only be approximated. We will characterize when the final occupied set comes within a uniformly bounded distance of every lattice point. Other issues addressed include macroscopic dynamics and exact solvability.

|| Thursday, February 2 || || *David Griffeath, * University of Wisconsin-Madison || || *MODELLING SNOW CRYSTAL GROWTH III: A Mesoscopic Lattice Map with Plausible Attachment Kinetics* ||

I will report on current development of a realistic lattice algorithm for snow crystal formation based on physical principles. The model combines diffusion limited growth with anisotropic attachment kinetics based on local geometry and pressure. Aggregation is regulated by an idealized semi-liquid layer. Our scheme, motivated by a particle system representation of a suitable Stefan problem, attempts to mimic the snowflake's hydrodynamics. This is apparently the first approach that successfully captures the form of core and tip instabilities, branch faceting, and other aspects of real snow crystal growth. As its parameters are varied, our nearest neighbor system reproduces the basic features of most of the observed varieties of snowflakes, and offers new insights into their genesis.

|| Thursday, February 9 || || *Frederi G. Viens, * Purdue University || || *Feynman-Kac formulas with applications to random media and to financial mathematics* ||

The classical Feynman-Kac representation of a linear multiplicative parabolic PDE, with space-time dependent potential, is easily generalized to a random potential whose time behavior is of white-noise type.

When the potential is jointly Gaussian in time and space, the resulting formula yields the so-called Stochastic Parabolic Anderson model, which is popular as a toy model for more complex random medium physical systems such as stochastic magneto-hydrodynamics, and yet can be considered as a step up in complexity and physical relevance from some simpler random Gibbs measures for spin-glass models. We will present several new techniques, some using the Malliavin calculus, some simply performing a detailed analysis of paths under the Gibbs measure, which result in a sharp analysis of the Anderson model's large-time exponential behavior (Lyapunov exponent) or of its Gibbs paths' maximal reach (polymer wandering exponent)

In the framework of del Moral et al., we apply the particle filtering properties of Feynman-Kac formulas to provide an efficient dynamic estimation of stochastic volatility for high-frequency financial data. We combine this estimation, which is of "smart Monte-Carlo" type, with standard Monte-Carlo methods, to solve two applied problems: portfolio optimization and option pricing, both under partially observed mean-reverting stochastic volatility. For the former, a drastic decrease in dimensionality is achieved via a special approximation. For the latter, a quadrinomial tree is used, in which a compact set of martingale measures is identified.

|| Friday, February 10 in B239 Van Vleck Hall at 4:00 PM || || *MATH DEPARTMENT COLLOQUIUM* || || *Frederi G. Viens, * Purdue University || || *Gaussian and non-Gaussian fields, the Malliavin calculus, and applications* ||

The Malliavin calculus, a stochastic analysis tool typically constructed from a given Gaussian process such as Brownian motion, is most prolific in contemporary probability theory, including in many non-Gaussian situations. Popular areas of application include anticipating stochastic calculus, fractional Brownian motion, financial mathematics, infinite-dimensional stochastic analysis, and many others.

We will show how appropriate boundedness of a stochastic process's iterated Malliavin derivatives can be sharply related to sub-Gaussian and non-sub-Gaussian properties. We will explain how to derive non-sub-Gaussian concentration and supremum estimation properties as a consequence; these generalize the Borell-Sudakov inequality and the Dudley-Fernique theorem, and open new challenging problems in regularity theory.

In a sub-Gaussian context, we will summarize Malliavin derivative applications to existence and/or sharp estimation of Lyapunov exponents for stochastic PDEs or polymer measures, and to Maximum Likelihood Estimators for fraction-Brownian-driven stochastic differential equations. In a purely Gaussian framework, we will indicate how to use Malliavin derivatives to extend (Skorohod) stochastic integration -- including Ito's and Tanaka's formulas -- to the entire fractional Brownian scale and beyond. All these applications also open many new questions, including what may remain true for non-sub-Gaussian models.

|| Thursday, February 16 || || *Seminar cancelled due to weather* ||

|| Thursday, February 23 || || *Timo Seppäläinen, * University of Wisconsin-Madison || || *Fluctuations around characteristics for some interacting particle systems* ||

Some interacting particle systems in one dimension have fluctuations on the scale n^{1/4} when the evolution is observed around a characteristic curve of the macroscopic equation. Examples include the random average process and independent walks. In these cases the characteristic curves are parallel straight lines. In a space-time scaling limit the fluctuations converge to a family of Gaussian processes. If the particle system is in equilibrium, the time marginal of the limit process is fractional Brownian motion with Hurst parameter 1/4. There are related results for quenched mean processes of certain random walks in random environment. The n^{1/4} scaling picture contrasts with the n^{1/3} scaling with Tracy-Widom limits known for asymmetric exclusion and Hammersley processes.

|| Thursday, March 2 || || *Thomas G. Kurtz, * University of Wisconsin-Madison || || *The Yamada-Watanabe-Engelbert theorem for general stochastic equations* ||

In the study of stochastic equations, it is common to distinguish between “strong” solutions and “weak” or distributional solutions. Roughly, a strong solution is one that exists for a given probability space and given stochastic inputs while existence of a weak solution simply ensures that a solution exists on some probability space for some stochastic inputs having the specified distributional properties. Similarly, strong uniqueness asserts that two solutions on the same probability space with the same stochastic inputs agree almost surely while weak uniqueness asserts that two solutions agree in distribution.

For Ito equations, Yamada and Watanabe (1971) proved that weak existence and strong uniqueness imply strong existence and weak uniqueness. Engelbert (1991} extended this result to a somewhat more general class of equations and gave a converse in which the roles of existence and uniqueness are reversed, that is, strong existence and weak uniqueness, in the sense that the joint distribution of the solution and the stochastic inputs is uniquely determined, imply strong uniqueness.

The issues addressed in these results arise naturally for any stochastic equation and extensions to other settings occur frequently in the literature. The talk will give general results that cover all the cases in the literature, known to the speaker, as well as other settings in which these questions have not yet been addressed.

|| Thursday, March 9 || || *Márton Balázs, * University of Wisconsin-Madison || || *t^1/3^-order fluctuations in the simple exclusion process* ||

Fluctuations created by the dynamics of interacting particle systems show exotic behavior. They are in the order t^1/3^ or t^1/4^ depending on the model, and can be observed along the characteristics of the macroscopic hydrodynamic equation. This is the curve where the t^1/2^-order normal fluctuations, which basically come from the noise of the initial distribution of the process, disappear. The talk will concentrate on the totally asymmetric simple exclusion process, where combinatorial analysis was used before to show that the quantities of interest have Tracy-Widom distribution in the t^1/3^-scaling limit. We first map the particle system to a last-passage percolation with appropriate boundary conditions. Then we show results concerning the t^1/3^-scaling of fluctuations along the guidelines of a recent argument for the Hammersley's process. The novelty is that our methods do not rely on the combinatorial analysis mentioned above.

|| Thursday, March 16 || || *No seminar because of* || || *SPRING BREAK* ||

|| Thursday, March 23 || || *Benjamin Doerr, * Max-Planck-Institut für Informatik, Saarbrücken, Germany || || *The Rotor Router Model on the infinite path and grid* ||

The rotor router model is a deterministic process resembling a random walk. Instead of distributing particles ("chips") to randomly chosen neighbors, it serves the neighbors in a fixed order. To this aim, each vertex has a "rotor" pointing to one of its neighbors. Whenever a chip reaches a vertex, it moves on in the direction of the rotor; then the rotor direction is updated to the next direction in some cyclic ordering.

In this talk, we regard the rotor router model on the infinite path and the 2D grid. This is joint work with Josh Cooper, Tobias Friedrich, Joel Spencer and Gabor Tardos.

|| *Probability Seminar Double Feature!* || || Thursday, March 30 in 901 Van Vleck Hall at 2:25 PM || || *Esko Valkeila, * Helsinki University of Technology || || *Weighted quadratic variation and fractional Brownian motion* ||

It seems that weighted quadratic variation and related notions are important tools in studying the properties of fractional Brownian motion. We try to illustrate this by giving a direct proof that fractional Brownian motion is not a semimartingale. Weighted quadratic variation can also be used to characterize fractional Brownian motion. The talk is based on joint work with Yulia Mishura (Kiev).

|| Thursday, March 30 in B139 Van Vleck Hall at 4:00 PM || || *Yuval Peres, * University of California, Berkeley || || *Distortion bounds for metric embedding, using Markov chains and Martingales* ||

In 1968, P. Enflo proved that the standard embedding of the n-dimensional hypercube in Euclidean space has the lowest distortion of all embeddings, namely the square root of n. In cases without such symmetry, using the rate of escape of Markov chains is a more robust method. In 1993, K. Ball proved that for any finite state reversible Markov chain in a Hibert space, the distance from the starting point has at most linear mean-square growth. We extend this property to L^p spaces for p>2, to trees and Gromov-hyperbolic spaces. (This yields bounds on embedding into these spaces). The proof is based on the decomposition of a reversible, stationary chain into the sum of a forward and a backward Martingale.

(Joint work with Assaf Naor, Oded Schramm and Scott Sheffield).

As a digression from the main topic, I will also sketch a proof (obtained jointly with Lionel Levine) that rotor-router aggregation has a ball as a scaling limit, see http://stat-www.berkeley.edu/~peres/router/router.html

|| Friday, March 31 in 901 Van Vleck Hall at 2:25 PM || || *PDE SEMINAR* || || *Yuval Peres, * University of California, Berkeley || || *Tug of war and the infinity Laplacian* ||

The infinity Laplacian (informally, the "second derivative in the gradient direction") is a simple yet mysterious operator with many applications, in particular to optimal Lipschitz extensions. Classical analysis of this operator is hampered by nonsmoothness of solutions. "Tug of war" is a two player random turn game played as follows: Given disjoint target sets $T_1$ and $T_2$ in the plane, and a token at $x$, toss a fair coin; the player who wins the coin toss moves the particle up to distance $r$ in the direction of his/her choice. This is repeated until the token reaches a target set $T_i$; player $i$ is then declared the winner. Write $u_r(x)$ for the probability that player 1 wins when both players play optimally. We show that as $r \to 0$, the functions $u_r(x)$ converge to the infinity harmonic function with boundary conditions 1 on $T_1$ and 0 on $T_2$. Our analysis of tug of war leads yields new estimates, and significant generalizations of several classical results about infinity Laplacians. I will also describe our original motivation for studying random-turn games: A variant of the game of Hex with a conformally-invariant limit. (Talk based on joint work with Oded Schramm, Scott Sheffield and David Wilson.)

|| Friday, March 31 in B239 Van Vleck Hall at 4:00 PM || || *MATH DEPARTMENT COLLOQUIUM* || || *Yuval Peres, * University of California, Berkeley || || *Point processes, repulsion, and fair allocation* ||

A random collection of points in space is called a "point process". The simplest point process is the Poisson process, where the numbers of points in disjoint regions are independent. Recently, there has been increasing interest in processes that exhibit "repulsion", such as zeros of random polynomials, noncolliding particles and eigenvalues of random matrices. I will describe the class of determinantal point processes, which exhibit perfect repulsion, and discuss the dynamical meaning of repulsion; see the movie at http://stat-www.berkeley.edu/~peres/GAF/dynamics/dynamics.html.

In the second part of the talk I will discuss the problem of "fair allocation": allocating the same area to every point of an isometry-invariant point process. Given such a point process _M_ in the plane, the Voronoi tesselation assigns a polygon (of different area) to each point of _M_. The geometry of fair allocations is much richer: see http://stat-www.berkeley.edu/~peres/stable/stable.html. For any isometry-invariant point process, we show that there is a unique fair allocation that is "stable" in the sense of the Gale-Shapley stable marriage problem. It turns out that repelling point processes have allocations that are better localized than the Poisson process. In higher dimensions, it appears that "gravitational allocation" does better than "market forces".

|| Thursday, April 6 || || *Rodrigo Bañuelos, * Purdue University || || *Finite dimensional distributions of Brownian motion and stable processes* ||

We investigate properties of finite dimensional distributions of Brownian motion and other Levy processes, as a function of the starting point, which arose from our efforts to answer the following very "simple" question: What is the lowest eigenvalue for the symmetric stable process of order _α_, 0 < _α_ < 2, in the interval (-1, 1)? While we still don't know the answer to this question, techniques have been developed which have applications to several interesting open problems on eigenvalues and eigenfunctions for the Laplacian, Schrödinger operators, and for other processes subordinate to Brownian motion.

|| Friday, April 7 in B239 Van Vleck Hall at 4:00 PM || || *MATH DEPARTMENT COLLOQUIUM* || || *Gregory Lawler, * Cornell University || || *Conformal invariance and two-dimensional statistical physics* ||

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 explain what the previous sentence means at least in the case of three examples: simple random walk, self-avoiding walk, and 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 models. 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.

|| Thursday, April 13 || || *Jim Kuelbs, * University of Wisconsin-Madison || || *Some remarks on the CLT and the Compact LIL* ||

|| Thursday, April 20 || || *Alan Hammond, * University of British Columbia || || *Mass-conservation and gelation in reaction-diffusion PDE.* ||

Smoluchowski's coagulation equation is a PDE that models the changing densities of a system of mass bearing particles that diffuse and are liable to coagulate in pairs at close range. We analyse the behaviour of solutions of the PDE by monitoring the trajectory of a tracer particle, this particle diffusing in space and colliding with other particles in accordance with the densities provided by a given solution of the PDE. We prove uniform bounds on the total particle density under certain assumptions on the parameters that specify the PDE. These permit us to derive conditions under which the solutions conserve mass for all time. In the other case, a gelation phenomenon occurs, much like a phase transition. Time permitting, I will also discuss results proving that gelation occurs.

This is joint work with Fraydoun Rezakhanlou.

|| Thursday, April 27 || || *Jason Swanson, *University of Wisconsin-Madison || || *Variations of the Solution to a Stochastic Heat Equation* ||

We consider the solution to a 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, _F_(_t_), has a nontrivial quartic variation. This process, therefore, has infinite quadratic variation and is not a semimartingale. It follows that the classical Itô calculus does not apply. Motivated by heuristic ideas about a possible new calculus for this process, we are led to study modifications of the quadratic variation. Namely, we modify each term in the sum of the squares of the increments so that it has mean zero. We then show that these sums, as functions of _t_, converge weakly to a Brownian motion, _B_, which is independent of _F_. As a corollary, we find that a certain sequence of Riemann sum approximations of ∫_2F dF_ converges in law to _F_^2^ - _B_.

|| Thursday, May 4 || || *Sona Zaveri Swanson, *UW || || *The Second Eigenfunction of the Neumann Laplacian on Thin Regions* ||

The "Hot Spots Conjecture" is that, as heat propagates in a bounded region with Neumann boundary conditions, the hottest spot in the region tends to the boundary. When properly formulated, this is a conjecture about the extrema of the second eigenfunction of the Neumann Laplacian. Since the first eigenfunction is a constant, it is the second eigenfunction which determines the shape of the solution to the heat equation for large time _t_.

We consider the Neumann Laplacian on a "thin" region in two dimensions, which consists of the points between the graphs of two Lipschitz functions defined on the interval [0,1]. Of interest is the behavior of the second eigenfunction as the thickness of the region tends to zero. We show that the second eigenfunction converges, in several senses, to the second eigenfunction of a one-dimensional Sturm-Liouville problem on the interval [0,1].