Purdue Probability Seminar, Spring 2024

We construct an intrinsic family of Gaussian noises on compact Riemannian manifolds which we call the colored noise on manifolds. It consists of noises with a wide range of singularities. Using this family of noises, we study the parabolic Anderson model on compact manifolds. To begin with, we started our investigation on a flat torus and established existence and uniqueness of the solution, as well as some sharp bounds on the second moment of the solution. In particular, our methodology does not necessarily rely on Fourier analysis and can be applied to study the PAM on more general manifolds.

In this talk I will start by reviewing some classical results relating machine learning problems with control theory. I will mainly discuss some very basic notions of supervised learning as well as reinforcement learning. Then I will show how noisy environments lead to very natural equations involving rough paths. This will include a couple of motivating examples. In a second part of the talk I will try to explain the techniques used to solve reinforcement learning problems with a minimal amount of technicality. In particular, I will focus on rough HJB type equations and their respective viscosity solutions. If time allows it, I will give an overview of our current research program in this direction. This talk is based on an ongoing joint work with Prakash Chakraborty (Penn State) and Harsha Honnappa (Purdue, Industrial Engineering).

Branching Brownian motion is a random particle system that incorporates both the tree-like structure and the diffusion process. In this talk, we consider a slightly subcritical branching Brownian motion with absorption, where particles move as Brownian motion with drift, undergo dyadic fission at a constant rate, and are killed upon hitting the origin. We are interested in the asymptotic behaviors of the process conditioned on survival up to a large time t as the process approaches criticality. Results like this are called Yaglom type results. Specifically, we will discuss the existence of the Yaglom limit law, Yaglom-type limits for the number of particles and the maximal displacement. This is based on joint work with Julien Berestycki, Bastien Mallein and Jason Schweinsberg.

The Parisi formula is a fundamental result in spin glass theory. It gives a variational characterization of the asymptotic limit of the expected free energy. The upper bound is a consequence of an interpolation identity due to F. Guerra and the lower bound is a celebrated result of M. Talagrand. In this talk I will present a new approach to (an enhanced version of) Guerra's identity using stochastic analysis, more specifically Brownian motion and Ito’s calculus. This approach is suggested by the form of the Parisi formula in which the solution of a Hamilton-Jacobi equation is involved. It helps in many ways to illuminate the original method of Guerra (basically a discretized version of Girsanov’s transform) and suggests a possible similar approach to the significantly deeper lower bound, which has been intensively studied since Talagrand’s work. Among the techniques from stochastic analysis we will use include path space integration by parts for the Wiener measure, Girsanov’s transform (i.e., exponential martingales), and probabilistic representation of solutions to (linear) partial differential equations. The key observation is that the nonlinear Hamilton-Jacobi partial differentiation equation figuring in Parisi’s variation formula becomes linear after differentiating with respect to Guerra’s interpolation parameter, thus bringing the full strength of stochastic analysis based on Ito’s calculus into play. It is hoped that this approach will shed some lights on the much more difficult lower bound in the Parisi formula.

Excitable systems exhibit a wide range of emergent behaviors, including traveling waves and spirals, and may converge to synchronous or asynchronous equilibria or fluctuate in non-equilibrium with complex dynamic patterns. I will discuss two excitable cellular automata models: the cyclic cellular automaton and the Greenberg-Hastings model. I will discuss the long-time behavior for these models on trees, where the absence of cycles prevents formation of finite 'stable periodic objects' to drive the dynamics. Based on joint works with Jason Bello, Janko Gravner and Hanbaek Lyu.

In this talk, we will present the exact expression of the $L^2$-norm of the symmetric-Stratonovich stochastic integral driven by a multi-dimensional fractional Brownian motion with parameter $\frac{1}{4} < H < \frac{1}{2}$. Our main result is a complete description of a Hilbert space of integrand processes which realizes the $L^2$-isometry where no regularity condition in the sense of Malliavin calculus is imposed. The Hilbert space is characterized in terms of a random Radon $\sigma$-finite measure on $[0,T]^2$ off diagonal which can be characterized as a product of a non-Markovian version of the stochastic Nelson derivatives. As a by-product, we present the exact explicit expression of the $L^2$-norm of the pathwise rough integral in the sense of Gubinelli for $\frac{1}{4} < H < \frac{1}{2}$.

Cluster expansion is a powerful tool for understanding spin models and was used in the seminal work of Aizenman-Lebowitz-Ruelle (1987) to understand the distributional limit of the free energy in the high-temperature SK spin glass model with zero external field. The main idea is to convert the probabilistic problem into a counting problem and use CLT under local dependence for distributional approximation. We use this approach to understand the high-temperature behavior of a few mean-field disordered models in detail. Examples include -

a) matching, traveling salesman, spanning tree, and k-factor problems at finite temperature,
b) SK spin glass model under weak external field,
c) pure and mixed p-spin glass model with zero and weak external field,
d) heavy-tailed p-spin glass under zero external field.

Based on joint works with Grigory Terlov and Qiang Wu.

In the study of networks, one of the central tasks is to uncover structures hidden in noisy networks. To probe the statistical and computational limits of such problems, random graph models with planted structures have been proposed and studied extensively in the literature. In this talk, I will discuss some techniques and results for inferring planted structures in random graphs, using the planted dense cycle model as the main example. This model, a variant of the Watts-Strogatz small-world model, posits the existence of latent one-dimensional geometry in a random graph. We characterize the information-theoretic and computational thresholds for the detection and recovery problems in this model. The talk is primarily based on joint work with Alexander S. Wein and Shenduo Zhang.

Transition paths connecting metastable states in a stochastic model are rare events which are fundamental but appear with small probability. In this talk, I will present a stochastic optimal control formulation for transition path problems in an infinite time horizon, modeled by Markov jump processes on Polish spaces. An unbounded terminal cost at a stopping time, along with a controlled transition rate, regulates the transitions between metastable states. To maintain the original bridges after control, the running cost adopts an entropic form for the control velocity, contrasting with the quadratic form typically used for diffusion processes. Via the Girsanov transform, this optimal control problem can be framed within a unified approach - converting to an optimal change of measures in càdlàg path space. The unbounded terminal cost however leads to a singular optimal control and brought difficulties in the Girsanov transform. Gamma-convergence techniques and passing limit in the corresponding Martingale problem allow us to obtain a singular optimally controlled transition rate. We demonstrate that the committor function, which solves a backward equation with specific boundary conditions, provides an explicit formula for the optimal path measure. The optimally controlled process realizes the transition paths almost surely but without altering the bridges of the original process.

In modern machine learning and generative AI applications, it often involves a computational task of generating samples satisfying a given probability density function (PDF). One particular sampling scheme, called Langevin Monte Carlo scheme, can be obtained by approximating the Langevin stochastic differential equation (SDE), which is related to Brownian motion. For many matrix manifolds without explicit coordinates, efficient Riemannian optimization schemes have been well studied yet it is quite difficult to design efficient Langevin Monte Carlo schemes by approximating Brownian motion on manifolds, due to the fact that the Langevin SDE on manifolds are in Stratonovich form. An efficient scheme can be obtained once the SDE is converted from Stratonovich form to Ito form, which was rarely discussed or used for defining Brownian motion in probability literature, since the Ito stochastic term on manifolds are only local martingales. In 1980s, it was shown by J. Lewis that Stratonovich-Ito conversion on surfaces gives precisely the mean curvature. We are able to extend this result to a parallelizable submanifold, i.e., the Stratonovich-Ito conversion on a submanifold is equal to a mean curvature normal vector, which is a concept not very often used in geometry literature. The assumption of parallelizable manifolds is quite restricive but can be relaxed via partition of unity with each piece in the partition is parallelizable. In 2000s, Stroock also mentioned mean curvature normal vector when giving an extrinsic definition of Brownian motion on submanifolds but Stroock’s results are not easy to use in this context. For Riemannian submersion and quotient manifolds, we are also able to show a similar result: the Brownian motion on a quotient manifold is related to Brownian motion on its total space via the mean curvature normal vector of each fiber/orbit or equivalent class, which is also equal to the gradient of log of the volume of each fiber/orbit for compact manifolds. We apply these two main results on the manifold of positive semi-definite matrices of fixed rank, to obtain two efficient Riemannian Langevin Monte Carlo schemes: one for its embedded geometry, and the other one is for its quotient geometry with the Bures-Wasserstein metric. For studying the convergence rate to equilibrium, it is well known that the SDE solution is related to Fokker-Planck equation, for which the exponential convergence rate to equilibrium is related to Bakry-Emery-Ricci tensor. For a special PDF, we also give some preliminary estimates of the Bakry-Emery-Ricci tensor, which suggests a difference in convergence rate between the two schemes. Numerical examples (arXiv:2309.04072) including Monte Carlo numerical integration on manifolds will be shown. This is based on joint work with Tianmin Yu (student) and Govind Menon at Brown University, Jianfeng Lu at Duke University, and Shixin Zheng (student) at Purdue University.

We will study the existence, and non-existence, of scaling limits for the biased randomwalks on the supercritical percolation cluster in the zero-speed regime. This is joint work with Alan Hammond.