Titles and Abstracts

Brownian motion speaks analysis and geometry in a beautiful connection with bounded variation (BV) measures and the perimeter of sets. This talk will review that relation in a tour (re)visiting Cacciopoli in the 20s, de Giorgi in the 50s, and Ledoux in the 90s to (re)discover their characterizations of perimeter and BV functions.

Moving beyond the Euclidean setting to more general metric measure spaces will bring us to recent developments in the context of fractals. In particular we will explain an oscillatory phenomenon that occurs when the underlying space is an unbounded planar nested fractal. This is joint work with Fabrice Baudoin.

Smooth random functions of many variables can be topologically very complex and hard to explore and minimize.

A first class of such functions arises in statistical physics of disordered media. A second important class comes from the natural loss landscapes of high-dimensional statistics and machine-learning.

In my first tutorial I will concentrate on the topological properties of these landscapes, using the only tool available to mathematicians, i.e the Kac-Rice formula, which establishes a direct link between this complexity question and Random Matrix Theory (RMT).

I will illustrate this by examples in physics (with spin glasses and more recent work on the so-called “Elastic Manifold” model introduced by Mézard and Parisi) and in high-dimensional statistics (with the Tensor PCA model and its variants).

In the second lecture, I will concentrate on the dynamics in these complex landscapes, with a special emphasis on stochastic gradient descent for natural neural network models, and the recent understanding of so-called summary statistics and effective dynamics.

All along, I will try to mention some of the numerous important questions remaining open.

This work relies on collaborations with A. Auffinger, G. Biroli, P. Bourgade, C. Cammarota, J. Cerny, A. Choromanska, Y. Fyodorov, R. Gheissari, J. Huang, D. Huang, A. Jagannath, B. Khoruzenko, Y. Le Cun, Z.Li, A. Maillard,  B. Mc Kenna, A. Montanari, S. Mei, M. Nica, V. Ros.

I will present an overview of recent results on the asymptotic behavior of large systems of weakly interacting controlled diffusion processes. These systems underlie recent theories of mean-field games and mean-field control depending on whether the controlled particles are competitive or cooperative. A very difficult point is to address the convergence, with a rate, to the limiting mean field system. One strategy in this perspective is to use the so-called "master equation" generated by the limiting dynamics. This master equation is a partial differential equation (PDE) set on the space of probability measures. The key idea is that the regularity of the solutions to the PDE governs the convergence rate of the particle system. If the mean field system were not controlled, the PDE would be driven by the generator of the mean field dynamics on the space of probability measures. In the presence of an additional control term, the PDE becomes nonlinear and addressing the regularity of the solutions becomes much more difficult. We provide two main examples that have been discussed in the literature: (i) the first example corresponds to mean-field games with an appropriate monotonicity condition that forces the master equation to have smooth solutions; (ii) the second one is more recent and corresponds to mean-field control without any convexity assumption, in which case the solutions of the master equation cannot be understood in a classical sense and must be regarded as viscosity solutions. The first example was treated in a work with P. Cardaliaguet, J.M. Lasry and P.L. Lions. The second one is handled in a work in progress with S. Daudin and J. Jackson. 

A basic goal for random graph matching is to recover the vertex correspondence between two correlated graphs from an observation of these two unlabeled graphs. Random graph matching is an important and active topic in combinatorial statistics: on the one hand, it arises from various applied fields such as social network analysis, computer vision, computational biology and natural language processing; on the other hand, there is also a deep and rich theory that is of interest to researchers in statistics, probability, combinatorics, optimization, algorithms and complexity theory.

Recently, extensive efforts have been devoted to the study for matching two correlated Erdős–Rényi graphs, which is arguably the most classic model for graph matching. In this talk, we will review some recent progress on this front, with emphasis on the intriguing phenomenon on (the presumed) information-computation gap. In particular, we will discuss progress on efficient algorithms thanks to the collective efforts from the community. We will also point out some important future directions, including developing robust algorithms that rely on minimal assumptions on graph models and developing efficient algorithms for more realistic random graph models. 

This is based on joint works with Hang Du, Shuyang Gong, Zhangsong Li, Zongming Ma, Yihong Wu and Jiaming Xu. 

In this talk, I will present an exclusion process with different types of particles, let us say types A, B, and C. Depending on the interaction rate between different types of particles, the limiting fluctuations end up in different universality classes: either the fluctuations are governed by energy solutions of the stochastic Burgers equation or by the Ornstein-Uhlenbeck equation. 

These results match the predictions from the non-linear fluctuating hydrodynamics of Spohn. 

(Joint with G. Cannizzaro; A. Occelli, R. Misturini.)

The Kardar-Parisi-Zhang (KPZ) universality class of models is characterized by non-Gaussian asymptotic fluctuations coming from random matrices, although the general mechanism for the appearance of these fluctuations is poorly understood. In this talk, I will introduce a number of models which are known or expected to be in the KPZ  universality class, including discrete and semi-discrete models of random polymers, as well as a family of interacting diffusions. 

I will explain how, using an idea of Emrah-Janjigian-Seppalainen, one derives for each of these models the analogue of a formula due to Rains in the context of Last Passage Percolation. From this formula, several results on fluctuations, including tail bound on the correct scale follow.

Joint work with Benjamin Landon

We consider maps which are constructed from plane trees by assigning marks to the corners of each vertex and then connecting each pair of consecutive marks on their contour by a single edge.  A measure is defined on the set of such maps by assigning Boltzmann weights to the faces. When every vertex has exactly one marked corner, these maps are dissections of a polygon which are bijectively related to non-crossing trees. When every vertex has at least one marked corner, the maps are outerplanar and each of its two-connected component is bijectively related to a non-crossing tree. We study the scaling limits of the maps under these conditions and establish that for certain choices of the weights the scaling limits are either the Brownian CRT or the $\alpha$-stable looptrees of Curien and Kortchemski.

Preferential attachment networks are a type of randomly growing network where vertices that already have many connections are more likely to continue to grow in the future. These have been studied by authors such as Antunovic, Mossel and Racz, who included competing types in their models. In these networks, all vertices have a type, and new vertices inherit a type from their parents, according to a preset rule (such as choosing the type that is most prevalent in their parents). 

For many sensible choices of rule, the types are able to co-exist. For example, if there are N types, and a "pick random visible type" inheritance rule is used, meaning that new vertices have equal probability of taking each type that is present in their parents at least once, then each type will represent a proportion 1/N of all vertices in the long term. This is a result of Haslegrave and Jordan.

An extension to preferential attachment is to specify fitnesses of vertices, and use the fitnesses when choosing the parents of new vertices. This models the possibility of factors other than popularity being important for nodes to grow. An example studied by Haslegrave, Jordan and Yarrow has new vertices choose three 'candidate' parents by preferential attachment, and then select the one with middling fitness. (Other choices, such as choosing the fittest vertex, have also been studied.) In the case of this middle-of-three choice, the graph has a behaviour wherein sometimes a single vertex, known as a 'hub', grows linearly with time to have a very large degree. This is an example of a behaviour known as 'condensation.'

We study the effects of this middle-of-three choice mechanism on competing types. If a hub emerges, this gives a large advantage to the type of the hub. This phenomenon allows for examples where 'weak' types, with an evolutionary disadvantage over stronger types, have the potential to dominate the network in spite of this fact. It also distorts the results of cases such as the ""pick random visible type"" rule. If a hub does emerge, its type will over time represent a proportion approaching 100% of all vertices, rather than the types coexisting.

In this work, we study the existence and uniqueness of Feller weak solutions in the case of degenerate diffusions. It is well known that under the assumption of Lipshitz coefficients, the corresponding degenerate diffusion has a unique weak solution which is also Feller. However upon relaxing this assumption, it is possible that there can be many weak solutions and a Feller weak solution may not even exist. Under an appropriate set of conditions (that is strictly weaker than Lipshitz continuity) on the coefficients, we establish unique existence of a Feller weak solution.  The proof of this result uses a crucial observation that every Feller weak solution corresponds to a viscosity solution of the associated Kolmogorov equation.  In fact, it is shown that if an appropriate comparison principle holds, then there is a one-one correspondence between viscosity solutions of the associated Kolmogorov equation and Feller weak solutions. In addition to this, we also give a procedure to select these Feller weak solutions. 

We design an importance sampling scheme for statistics related to rare events for the empirical measure on a system of N weakly interacting diffusions. We use the subsolution approach of Dupuis and Wang. It turns out that, due to the large deviation rate function's connection to mean-field control, the natural Hamilton-Jacobi-Bellman Equation to consider is posed on Wasserstein Space and involves derivatives in the sense of Lions . We identify conditions under which our scheme is provably asymptotically optimal in N in the sense of log-efficiency. We also provide evidence, both analytical and numerical, that our scheme can have vanishingly small relative error as N increases.

We describe the Loewner chains of the real locus of a class of real rational functions whose critical points are on the real line. Our main result is that the poles of the rational function lead to explicit formulas for the dynamical system that governs the driving functions. Our formulas give a simple method for mapping the class of rational functions into solutions to a non-trivial system of quadratic equations, and for directly showing that the curves in the real locus satisfy geometric commutation and have the geodesic multichord property. These results are entirely self-contained and have no reliance on probabilistic objects, but make use of an integral of motion for the Loewner chain that is motivated by ideas from conformal field theory. We also show that the dynamics of the driving functions are a special case of the Calogero-Moser integrable system, restricted to a particular submanifold of phase space carved out by the Lax matrix. Our approach complements a recent result of Peltola and Wang, who showed that the real locus is the deterministic kappa to 0 limit of the multiple SLE(kappa) curves.

The 15 puzzle is a sliding puzzle having 15 square tiles numbered 1–15 in a frame that is 4 tiles high and 4 tiles wide, leaving one unoccupied tile position. We answer a question about how long it takes to randomize a n^2-1 puzzle, as proposed by Diaconis decades ago. We show that a single numbered piece on the n^2-1 puzzle given periodic boundary conditions has a randomized location after order n^4 random moves, and that if the number of random moves tends to infinity compared to n^4, then the number of pieces left in their original position converges to a Poisson(1) distribution. The distribution of pieces on the board converges to uniform in O(n^4 log n) random moves.

The stochastic blockmodel (SBM) is a simple probabilistic model for graphs which exhibit clustering. Each vertex is assigned a type i = 1, 2, ..., m and edges are included independently with probability depending on the types of the two incident vertices. The degree corrected SBM (DCSBM) exhibits similar clustering behavior but allows for inhomogeneous degree distributions. We show that under fairly general conditions, the asymptotic sizes of connected components in the DCSBM can be precisely described in terms of a multiparameter and multidimensional random field allowing us to describe the asymptotic proportion of vertices of each type in each of the macroscopic connected components. Based on joint work with Vitalii Konarovskyi and Vlada Limic.

We consider the symmetric exclusion system on Z starting from an infinite particle ``step'' configuration in which there are no particles to the right of a maximal one. As shown in Arratia (1983), this same initial profile for a system of independent random walks yields a Gumbel law in the limit of the properly-scaled maximal particle position. We show that the same limit under the same scaling holds for the leading particle in the exclusion system, and moreover we provide related limit distributions for all order statistics of particle positions. In the case of nearest-neighbor interaction, this gives the asymptotic distribution of all tagged particle positions. We also consider a time-dependent finite-particle initial condition to probe the influence of the mass of particles on the behavior of the maximal one. This is joint work with Sunder Sethuraman. 

We consider an homogenization and large deviations problem in partial differential equations (PDE) with Neumann boundary condition, periodic coefficients and highly oscillating drift and nonlinear term. Our  method is entirely probabilistic and studies the behavior of the solution through a ratio delta/epsilon between small parameters (delta) and  (epsilon), both tending to zero.  When delta and epsilon go at the same rate, we establish a large deviation principle and we prove the existence of the homogenized limit of the PDE solution.

We consider a balanced, sparse directed SBM with 2 communities, in which we know the identities of a small, positive proportion of members of one community. Given this information, we propose a method to recover the rest of the community using personalized PageRank scores on the network. We do this by analyzing a system of distributional fixed point equations satisfied by the PageRank of a random vertex in each community to obtain a threshold for classification.

Liouville quantum gravity (LQG) surfaces are random fractal two-dimensional Riemannian manifolds which arise as scaling limits of probability measures on planar maps. Each LQG surface is equipped with a random area measure and a random Riemannian distance function (a metric), which are obtained by exponentiating some approximation of the Gaussian free field, then taking a limit. For the area measure, one has almost sure convergence in the sense of weak convergence of measures. For the metric, a series of papers established convergence in probability with respect to the topology of local uniform convergence, but almost sure convergence requires different techniques. We utilize an approximate bi-Lipschitz equivalence between the metric and its approximations, arguing that the Lipschitz constant can be made close to one and deducing the almost sure convergence from this. As a consequence, we show that the scaling invariance property of the LQG metric almost surely holds for all scales simultaneously. We also improve the known asymptotic bounds for the scaling constants used in the approximation of the metric.

We consider a diploid bi-parental Wright Fisher model with occasional highly reproductive pairs whose offspring replace a fraction $\psi$ of the population, see [2,3]. The pedigree (or parent-ship graph) is the set of all family relationships among  the members of the population for all generations.  We establish convergence in probability for the genealogy of $n$ genes conditional on the pedigree.

In this poster we will present the asymptotics of Yule's nonsense correlation for two independent Ornstein-Uhlenbeck processes, we will give the CLT that this statistic satisfies with the rate of the convergence in law for discrete and continuous time observations using Malliavin Calculus and Stein's method. Several numerical results will be presented as well using the software R.

I will present that in the supercritical case, a clssical loop model named infinite Fortuin-Kasteleyn maps, after appropriate re-scaling, will converge in law to the infinite continuum random tree as metric-measure spaces (i.e., in the local Gromov-Hausdorff-Prokhorov topology). We also show that the maps do not admit any FK loops of macroscopic graph distance diameter. The main techniques are based on Sheffield's hamburger-cheeseburger bijection.

In this paper,  we introduce and solve the problem of quickest detection of constant drift with false negatives and repeated detection attempts when we have linear penalty cost. The time at which drift(unobserved) appears in an otherwise drift free Brownian motion follows exponential distribution independent of Brownian motion. When the optimizer tries to detect the drift, the test can return a false negative i.e if the drift has occurred, it will not detect the drift due to measurement error with fixed positive probability ε, as a consequence of which the process continues. The optimization ends when the drift is detected. Our aim is to simultaneously optimize the cost of late detection and number of tests our optimizer does in order to detect the drift. The initial formulation leads to an optimal multiple stopping problem. We showed the equivalence of the above problem with a recursive optimal stopping problem. Exploiting such connection and free boundary methods we find explicit formulae for the expected cost and the optimal strategy.

Let $X$ be a $d$-dimensional Gaussian process on $[0,1]$, where the component are independents copies of a scalar Gaussian process $X_0$ on $[0,1]$ with a given general variance function $\gamma^2(r)=\operatorname{Var}\left(X_0(r)\right)$ and a canonical metric $\delta(t,s):=(\mathbb{E}\left(X_0(t)-X_0(s)\right)^2)^{1/2}$ which is commensurate with $\gamma(|t-s|)$. We prove that for any Borel set $E\subset [0,1]$ the Hausdorff dimension of the image $X(E)$ and the graph $Gr_E(X)$ are constants almost surely under some general conditions on $\gamma$.

Belief dynamics describes how an individual or a group of people’s collection of beliefs change over time. There are a great many facets to belief dynamics, such as convergence to shared belief (i.e. how long it takes a group of people to get to the same set of beliefs), belief interdependence (two beliefs are interdependent if one changes when the other does), and clustering (in this case people who share a common belief set, like a political party). It may be argued that belief dynamics are Markovian, thus the belief dynamics of a given population may be modeled as a set of probabilistic transitions between all possible belief states. My research is focused on the effectiveness of existing algorithms for estimating the    transitional probabilities in different situations that arise in belief dynamics.

We consider the incompressible 2D Navier-Stokes equations on the torus driven by a deterministic time quasi-periodic force and a noise that is white in time and extremely degenerate in Fourier space. We show that the asymptotic statistical behavior is characterized by a uniquely ergodic and exponentially mixing quasi-periodic invariant measure. The result is true for any value of the viscosity and does not depend on the strength of the external forces. By utilizing this quasi-periodic invariant measure, we are able to show the strong law of large numbers and central limit theorem for the continuous time inhomogeneous solution processes. Estimates of the corresponding rate of convergence are also obtained, which is the same as in the time homogeneous case for the strong law of large numbers, while the convergence rate in the central limit theorem depends on the Diophantine approximation property on the quasi-periodic frequency and the mixing rate of the quasi-periodic invariant measure. We also prove the existence of a stable quasi-periodic solution in the laminar case (when the viscosity is large).

We study logarithmic Sobolev inequalities with respect to a heat kernel measure on finite-dimensional and infinite-dimensional Heisenberg groups. First we consider logarithmic Sobolev inequalities on non-isotropic Heisenberg groups. These inequalities are considered with respect to the hypoelliptic heat kernel measure, and we show that the logarithmic Sobolev constants can be chosen to be independent of the dimension of the underlying space. In this setting, a natural Laplacian is not an elliptic but a hypoelliptic operator. Furthermore, we apply these results in an  infinite-dimensional setting and prove a logarithmic Sobolev inequality on an infinite-dimensional Heisenberg group modelled on an abstract Wiener space.

Schramm-Loewner Evolutions (SLE) were introduced in 2000 by Oded Schramm in order to give meaning to scaling limits of interfaces of some models of Planar Statistical Physics. In the last years, there were many models that were proven to have their interfaces in the scaling limit described by SLE. The SLE curves are studied through the Loewner Differential Equation with a Brownian motion driver. I will describe the model and present my results on the continuity of this model in a natural parameter. I will also present some recent work on extensions of this model to multiple SLE curves where the driver is Dyson Brownian motion. For the latter model, I will also emphasize its current and potential future connections with another pillar of modern Probability Theory, namely Random Matrix Theory.

Let G be a finite, combinatorial graph. We define a notion of curvature on the vertices via the inverse of the resistance distance matrix. We prove that this notion has a number of desirable properties. For instance, from a lower bound on the curvature one can deduce an upper bound on the diameter and a lower bound on the spectral gap.

In convex geometry and its probabilistic aspects, many fundamental inequalities can be reversed up to universal constants in the presence of geometric properties, for instance reverse holder inequality, reverse isoperimetric inequality for convex bodies.  Another example is the equivalence of distances between probability distributions established by Meckes and Meckes (2014), who explored how distances between probability measures are comparable under log-concavity assumption.

In the poster we will see that distances between probability measures are equivalent for convex measures, which extends a result of Meckes and Meckes (2014). The class of convex measures contains fundamental distributions in probability and statistics. Examples include Gaussian distributions, uniform distributions on a convex set and more general log-concave distributions, as well as heavy tailed distributions such as Cauchy distribution.

We will see that the convergence in total variation of an isotropic s-concave probability measures is equivalent to convergence in bounded Lipschitz, and is further equivalent to Renyi and Tsallis divergence with respect to Gaussian.

Gaussian elimination with partial pivoting (GEPP) remains the most common method to solve dense linear systems. Each GEPP step uses a row transposition pivot movement if needed to ensure the leading pivot entry is maximal in magnitude for the leading column of the remaining untriangularized subsystem. We will use theoretical and numerical approaches to study how often this pivot movement is needed. We provide full distributional descriptions for the number of pivot movements needed using GEPP using particular Haar random ensembles, as well as compare these models to other common transformations from randomized numerical linear algebra. Additionally, we introduce new random ensembles with fixed pivot movement counts and fixed sparsity, α. Experiments estimating the empirical spectral density (ESD) of these random ensembles lead to a new conjecture on a universality class of random matrices with fixed sparsity whose scaled ESD converges to a measure on the complex unit disk that depends on α and is an interpolation of the uniform measure on the unit disk and the Dirac measure at the origin.

In this poster presentation, I provide an introduction to a new regularity structure used for solving rough mean-field equations. The index set of this regularity structure is described by a collection of novel objects which we refer to as Lions trees. These arise in Taylor expansions involving the Lions derivatives and the resulting combinatorics capture many of the desirable properties of distribution dependent dynamics.

We consider line ensembles of non-intersecting random walks constrained by a hard wall, each tilted by the area underneath it with geometrically growing pre-factors b^i where b>1. This is a model for the level lines of the (2+1)D SOS model above a hard wall, which itself mimics the low-temperature 3D Ising interface. A similar model with b=1 and a fixed number of curves was studied by Ioffe, Velenik, and Wachtel (2018), who derived a scaling limit as the time interval [-N,N] tends to infinity. Line ensembles of Brownian bridges with geometric area tilts (b>1) were studied by Caputo, Ioffe, and Wachtel (2019), and later by Dembo, Lubetzky, and Zeitouni (2022+). Their results show that as the time interval and the number of curves n tend to infinity, the top k paths converge to a unique limiting law. We address the open problem of proving existence of a scaling limit for random walk ensembles with geometric area tilts. We prove that with mild assumptions on the jump distribution, under suitable scaling the top k paths converge to the same limiting law as N and n go to infinity.

A random walk in a random environment (RWRE) is a random Markov chain on the d-dimensional integer lattice where transition probability vectors at each site are chosen randomly according to some a priori distribution. In the nearest-neighbor, one-dimensional setting, directional transience is clean to analyze, although the result differs in general from the naïve guess. In higher dimensions, it is much harder to say anything. However, random walks in Dirichlet environments (RWDE) are an exactly solvable model where transition probabilities are drawn according to a Dirichlet distribution. In the nearest-neighbor case, directional transience has been characterized for RWDE in all dimensions in terms of the Dirichlet parameters (and in this special case, the naïve guess manages to be true). We extend this result to finite-range RWDE by (1) proving a zero-one law for directional transience in two dimensions, and (2) proving almost-sure recurrence for a fixed direction in the case where Dirichlet parameters are balanced.

Liouville quantum gravity (LQG) is a theory of surfaces with random geometry that is conjectured to describe the scaling limits of various random planar maps. Many of these models have a natural exploration procedure, which in the scaling limit corresponds to an independent space-filling SLE curve exploring an LQG surface. We show that, almost surely, the 𝛾-LQG diameter of a finite segment of the space-filling SLE(16/𝛾²) curve is finite if and only if 𝛾<sqrt(8/3). This resolves a conjecture of Gwynne, Holden, and Sun (2016) on the distance exponent for mated-CRT maps. This is joint work with Yuyang Feng.

Losses in the market traded asset classes are generally not normally distributed, they tend to have fat tails. The log returns can better be estimated using Levy Processes, in particular we will show how the Normal Inverse Gaussian distribution almost always estimates the tail losses in an accurate way for most of the asset classes.

We shed light on the functional Taylor expansion (FTE), a generalization of the stochastic Taylor expansion for path-dependent functionals and/or non-Markovian processes. The FTE connects the functional Itô calculus with the path signature (iterated Stratonovich integrals) to quantify the effect in a functional when a “perturbation” path is concatenated with the source path. In particular, it elegantly separates the functional from future trajectories. The notions of real analyticity and radius of convergence are also extended to the path space. Finally, we connect the FTE with the Wiener chaos expansion. 

This is joint work with Bruno Dupire (Bloomberg LP). Related paper: https://arxiv.org/abs/2212.13628

A Bernoulli Gibbsian line ensemble is the law of the trajectories of (n-1) independent Bernoulli random walks with possibly random initial and terminal locations that are conditioned to never cross each other or a given random up-right path. In this poster we investigate the asymptotic behavior of sequences of Bernoulli Gibbsian line ensembles when the number of walks tends to infinity. We show that if one has mild but uniform control of the one-point marginals of the top curves then the sequence is tight in the space of line ensembles. We can also show that if the top curves converge in the finite dimensional sense to the parabolic Airy2 process then the ensemble converges to the parabolic Airy line ensemble.

Let $A$ be a transition probability kernel on a finite state space $\Delta^o =\{1, \ldots , d\}$ such that $A(x,y)>0$ for all $x,y \in \Delta^o$. Consider a reinforced chain given as a sequence $\{X_n, \; n \in \NN_0\}$ of $\Delta^o$-valued random variables,  defined recursively according to,

$$L^n = \frac{1}{n}\sum_{i=0}^{n-1} \delta_{X_i}, \;\; P(X_{n} \in \cdot \mid X_0, \ldots, X_{n-1}) = L^n A(\cdot).$$

We establish a large deviation principle for $\{L^n, \; n \in \NN\}$. The rate function takes a strikingly different form than the Donsker-Varadhan rate function associated with the empirical measure of the Markov chain with transition kernel $A$ and is described in terms of a novel deterministic infinite horizon discounted cost control problem with an associated linear controlled dynamics and a nonlinear running cost involving the relative entropy function. Proofs are based on an analysis of time-reversal of controlled dynamics in  representations for log-transforms of exponential moments, and on weak convergence methods. This poster is based on joint work with Amarjit Budhiraja.

We prove a stochastic representation formula for the $B$-continuous viscosity solution of an infinite dimensional semilinear partial differential equation (PDE) in terms of a scalar-valued backward stochastic differential equation (BSDE). We also discuss possible extensions to fully nonlinear PDEs using second order BSDEs. These results are based on joint work with Andrzej Andrzej Święch.

We establish a seminorm estimates between random unitaries and random Gaussian matrices and we show how that estimates can be applied to quantum information theory.

Cluster expansion, as a powerful combinatorial scheme, has been widely used in many statistical mechanics problems. However, most of these focused on the non-disordered spin system. In disordered system, such as spin glasses, Aizenman-Lebowitz-Ruelle in 1987 first utilized the cluster expansion idea to obtain some rigorous results in the Sherrington-Kirkpatrick (SK) model at zero external field. Nevertheless, since then, it was believed in the literature that this approach does not work once the external is present. In this talk, we will show that under some “weak” external field, the cluster expansion still works, but a new cluster appears. This implies some transition results for the free energy in SK model. We will also discuss the extension of this framework to a very general setting, the mixed p-spin models, where a new multiple transition phenomenon was shown along with the discovery of a collection of fruitful cluster structures. This is based on joint works with Partha S. Dey.

We consider an homogenization and large deviations problem in partial differential equations (PDE) with Neumann boundary condition, periodic coefficients and highly oscillating drift and nonlinear term. Our  meth"Weak convergence theory for stochastic integrals on Skorokhod space, as developed by Jakubowski, Memin & Pages (PTRF, ’89) and Kurtz & Protter (AOP, ’91). While the theory appears both highly elegant and surprisingly powerful in the classical setting of Skorokhod’s J1-topology, things change drastically when looking at coarser topologies. Jakubowski (AOP, ’96) manages to give a strikingly elegant statement for a rather coarse non-Skorokhod topology, but, unlike the J1 setting, it seems that elegance and power no longer go hand-in-hand. Hoping to close part of this gap, I will present some recent results, both positive and negative, on what can be said for Skorokhod’s M1 topology. This topology is coarser than J1 and has recently seen a surge of interest in the applied probability literature. More generally, the aim is twofold: to make the general theory more transparent and to contribute concrete verifiable criteria for convergence in both the J1 and M1 topologies. At its core, we rely on a combination of ideas from Jakubowski (AOP, ’96) and Kurtz & Protter (AOP, ’91), but we make several new technical contributions and derive conditions that can be of practical interest. In particular, I will briefly discuss some illuminating applications to co-integrating regression in econometrics as well as pricing and hedging for sub-diffusive models in mathematical finance."od is entirely probabilistic and studies the behavior of the solution through a ratio delta/varepsilon between small parameters (delta) and  (varepsilon), both tending to zero.  When delta and varepsilon go at the same rate, we establish a large deviation principle and we prove the existence of the homogenized limit of the PDE solution.

SPDE on graphs is a flexible framework that bypasses the illposed issue on higher dimension and enables the study of spatial effect. We obtain an explicit formula for the extinction probability of the stochastic FKPP equation on general metric graphs in terms of the graph structure, spatially in-homogeneous diffusion coefficient and initial condition. This generalizes in several ways the known formula for the real line case heuristically obtained before.

We study countable-state semi-Markov locally interacting particle systems on sparse random graphs in which the state at a vertex evolves depending on its age and the states of the neighboring vertices. For certain sequences of random graphs, including Erdos-Renyi graphs G(n,c/n), we show that as the number of vertices becomes large, the family of neighborhood empirical measures satisfies a large deviation principle on the space of marked rooted graphs. The rate function is given in terms of certain relative entropies.

This poster is based on joint work with Kavita Ramanan.

We study two problems. First, we consider the large deviation behavior of empirical measures of certain diffusion processes as, simultaneously, the time horizon becomes large and noise becomes vanishingly small. The law of large numbers (LLN) of the empirical measure in this asymptotic regime is given by the unique equilibrium of the noiseless dynamics. Due to degeneracy of the noise in the limit, the methods of Donsker and Varadhan (1976) are not directly applicable and new ideas are needed. Second, we study a system of slow-fast diffusions where both the slow and the fast components have vanishing noise on their natural time scales. This time the LLN is governed by a degenerate averaging principle in which local equilibria of the noiseless system obtained from the fast dynamics describe the asymptotic evolution of the slow component. We establish a large deviation principle that describes probabilities of divergence from this behavior. On the one hand our methods require stronger assumptions than the nondegenerate settings, while on the other hand the  rate functions take  simple and  explicit forms that have striking differences from their nondegenerate counterparts.