Past Stochastic Analysis Seminar

(Joint work with Siva Athreya & Leonid Mytnik).

It is well known from the literature that ordinary differential equations (ODEs) regularize in the presence of noise. Even if an ODE is “very bad” and has no solutions (or has multiple solutions), then the addition of a random noise leads almost surely to a “nice” ODE with a unique solution. The first part of the talk will be devoted to SDEs with distributional drift driven by alpha-stable noise. These equations are not well-posed in the classical sense. We define a natural notion of a solution to this equation and show its existence and uniqueness whenever the drift belongs to a certain negative Besov space. This generalizes results of E. Priola (2012) and extends to the context of stable processes the classical results of A. Zvonkin (1974) as well as the more recent results of R. Bass and Z.-Q. Chen (2001).

In the second part of the talk we investigate the same phenomenon for a 1D heat equation with an irregular drift. We prove existence and uniqueness of the flow of solutions and, as a byproduct of our proof, we also establish path-by-path uniqueness. This extends recent results of A. Davie (2007) to the context of stochastic partial differential equations.

[1] O. Butkovsky, L. Mytnik (2016). Regularization by noise and flows of solutions for a stochastic heat equation. arXiv 1610.02553. To appear in Annal. Probab.

[2] S. Athreya, O. Butkovsky, L. Mytnik (2018). Strong existence and uniqueness for stable stochastic differential equations with distributional drift. arXiv 1801.03473.

Numerical methods for SDEs typically use only the discretized increments of the driving Brownian motion. As one would expect, this approach is sensible and very well studied.

In addition to generating increments, it is also straightforward to generate time integrals of Brownian motion. These quantities give extra information about the Brownian path and are known to improve the strong convergence of methods for one-dimensional SDEs. Despite this, numerical methods that use time integrals alongside increments have received less attention in the literature.

In this talk, we will develop some underlying theory for these time integrals and introduce a new numerical approach to SDEs that does not require evaluating vector field derivatives. We shall also discuss the possible implications of this work for multi-dimensional SDEs.

Based on the notion of paracontrolled distributions, existence and uniqueness results are presented for rough convolution equations. In particular, this wide class of equations includes rough differential equations with possible delay, stochastic Volterra equations, and moving average equations driven by Lévy processes. The talk is based on a joint work with Mathias Trabs.

Let f be the planar Bargmann-Fock field, i.e. the analytic Gaussian field with covariance kernel exp(-|x-y|^2/2). We compute the critical point for the percolation model induced by the level sets of f. More precisely, we prove that there exists a.s. an unbounded component in {f>p} if and only if p<0. Such a percolation model has been studied recently by Beffara-Gayet and Beliaev-Muirhead. One important aspect of our work is a derivation of a (KKL-type) sharp threshold result for correlated Gaussian variables. The idea to use a KKL-type result to compute a critical point goes back to Bollobás-Riordan. This is joint work with Alejandro Rivera.

We shall explain, from variational point of view, why the  Laplaciian operator introduced by Ebin-Marsden using deformations is suitable to describe the fluid motion in a milieu with viscosity.

Smooth Gaussian functions appear naturally in many areas of mathematics. Most of the talk will be about two special cases: the random plane model and the Bargmann-Fock ensemble. Random plane wave are conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator in a generic domain. The Bargmann-Fock ensemble appears in quantum mechanics and is the scaling limit of the Kostlan ensemble, which is a good model for a `typical' projective variety. It is believed that these models, despite very different origins have something in common: they have scaling limits that are described be the critical percolation model. This ties together ideas and methods from many different areas of mathematics: probability, analysis on manifolds, partial differential equation, projective geometry, number theory and mathematical physics. In the talk I will introduce all these models, explain the conjectures relating them, and will talk about recent progress in understanding these conjectures.

TBC

In this talk I will present three families of differential equations (SDEs, BSDEs and PDEs) and their links to each other. The novel fact is that some of the coefficients are generalised functions living in a fractional Sobolev space of negative order. I will discuss the appropriate notion of solution for each type of equation and show existence and uniqueness results. To do so, I will use tools from analysis like semigroup theory, pointwise products, theory of function spaces, as well as classical tools from probability and stochastic analysis. The link between these equations will play a fundamental role, in particular the results on the PDE are used to give a meaning and solve both the forward and the backward stochastic differential equations.  

Stochastic differential equations have Taylor expansions in terms of iterated Wiener integrals. The convergence of such expansion depends on the limiting behavior of the order-N iterated integrals as N tends to infinity. Recently, there has been increased interests in processes stopped at a random time. A breakthrough in the study of the iterated integrals of Brownian motion up to the exit time of a domain was included in the work of Lyons-Ni (2012). The paper leaves open an interesting question: what is the sharp rate of decay for the expected iterated integrals up to the exit time. We will review the state of the art in this problem and report some recent progress. Joint work with Ni Hao (UCL).

Abstract: In this talk we describe an invariance principle for a class of non-homogeneous martingale random walks in $\RR^d$ that can be recurrent or transient for any dimension $d$. The scaling limit, which we construct, is a martingale diffusions with law determined uniquely by an SDE with discontinuous coefficients at the origin whose pathwise uniqueness may fail. The radial component of the diffusion is a Bessel process of dimension greater than 1. We characterize the law of the diffusion, which must start at the origin, via its excursions built around the Bessel process: each excursion has a generalized skew-product-type structure, in which the angular component spins at infinite speed at the start and finish of each excursion. Defining a Riemannian metric $g$ on the sphere $S^{d−1}$, different from the one induced by the ambient Euclidean space, allows us to give an explicit construction of the angular component (and hence of the entire skew-product decomposition) as a time-changed Browninan motion with drift on the Riemannian manifold $(S^{d−1}, g)$. In particular, this provides a multidimensional generalisation of the Pitman–Yor representation of the excursions of Bessel process with dimension between one and two. Furthermore, the density of the stationary law of the angular component with respect to the volume element of $g$ can be characterised by a linear PDE involving the Laplace–Beltrami operator and the divergence under the metric $g$. This is joint work with Nicholas Georgiou and Andrew Wade.

We propose a new splitting algorithm to solve a class of quasilinear PDEs with convex and quadratic growth gradients. 

By splitting the original equation into a linear parabolic equation and a Hamilton-Jacobi equation, we are able to solve both equations explicitly. 

In particular, we solve the associated Hamilton-Jacobi equation by the Hopf-Lax formula, 

and interpret the splitting algorithm as a stochastic Hopf-Lax approximation of the quasilinear PDE.  

We show that the numerical solution will converge to the viscosity solution of the equation.  

The upper bound of the convergence rate is proved based on Krylov's shaking coefficients technique, 

while the lower bound is proved based on Barles-Jakobsen's optimal switching approximation technique. 

Based on joint work with Shuo Huang and Thaleia Zariphopoulou.

Recurrent major mood episodes and subsyndromal mood instability cause substantial disability in patients with bipolar disorder. Early identification of mood episodes enabling timely mood stabilisation is an important clinical goal. The signature method is derived from stochastic analysis (rough paths theory) and has the ability to capture important properties of complex ordered time series data. To explore whether the onset of episodes of mania and depression can be identified using self-reported mood data.

In this talk, I am going to report on some on-going research at the interface between Rough Paths Theory and Schramm-Loewner evolutions (SLE). In this project, we try to adapt techniques from Rough Differential Equations to the study of the Loewner Differential Equation. The main ideas concern the restart of the backward Loewner differential equation from the singularity in the upper half plane. I am going to describe some general tools that we developed in the last months that lead to a better understanding of the dynamics in the closed upper half plane under the backward Loewner flow.
Joint work with Prof. Dmitry Belyaev and Prof. Terry Lyons

Abstract: Butcher’s B-series is a fundamental tool in analysis of numerical integration of differential equations. In the recent years algebraic and geometric understanding of B-series has developed dramatically. The interplay between geometry, algebra and computations reveals new mathematical landscapes with remarkable properties. 

The shuffle Hopf algebra,  which is fundamental in Lyons’s groundbreaking work on rough paths,  is based on Lie algebras without additional properties.  Pre-Lie algebras and the Connes-Kreimer Hopf algebra are providing algebraic descriptions of the geometry of Euclidean spaces. This is the foundation of B-series and was used elegantly in Gubinelli’s theory of Branched Rough Paths. 
Lie-Butcher theory combines Lie series with B-series in a unified algebraic structure based on post-Lie algebras and the MKW Hopf algebra, which is giving algebraic abstractions capturing the fundamental geometrical properties of Lie groups, homogeneous spaces and Klein geometries. 

In these talks we will give an introduction to these new algebraic structures. Building upon the works of Lyons, Gubinelli and Hairer-Kelly, we will present a new theory for rough paths on homogeneous spaces built upon the MKW Hopf algebra.

Joint work with: Charles Curry and Dominique Manchon

Abstract: Butcher’s B-series is a fundamental tool in analysis of numerical integration of differential equations. In the recent years algebraic and geometric understanding of B-series has developed dramatically. The interplay between geometry, algebra and computations reveals new mathematical landscapes with remarkable properties. 

The shuffle Hopf algebra,  which is fundamental in Lyons’s groundbreaking work on rough paths,  is based on Lie algebras without additional properties.  Pre-Lie algebras and the Connes-Kreimer Hopf algebra are providing algebraic descriptions of the geometry of Euclidean spaces. This is the foundation of B-series and was used elegantly in Gubinelli’s theory of Branched Rough Paths. 
Lie-Butcher theory combines Lie series with B-series in a unified algebraic structure based on post-Lie algebras and the MKW Hopf algebra, which is giving algebraic abstractions capturing the fundamental geometrical properties of Lie groups, homogeneous spaces and Klein geometries. 

In these talks we will give an introduction to these new algebraic structures. Building upon the works of Lyons, Gubinelli and Hairer-Kelly, we will present a new theory for rough paths on homogeneous spaces built upon the MKW Hopf algebra.

Joint work with: Charles Curry and Dominique Manchon

We consider L^2-approximations of white noise within the framework of regularity structures. Possible applications include support theorems for SPDEs driven by degenerate noises and numerics. Joint work with Ilya Chevyrev, Peter Friz and Tom Klose. 

This talk is based on a joint work with Dmitry Beliaev.

We study the volume distribution of nodal domains of families of naturally arising Gaussian random field on generic manifolds, namely random band-limited functions. It is found that in the high energy limit a typical instance obeys a deterministic universal law, independent of the manifold. Some of the basic qualitative properties of this law, such as its support, monotonicity and continuity of the cumulative probability function, are established.

Having made sense of differential equations driven by rough paths, we now have a new set of models available but when it comes to calibrating them to data, the tools are still underdeveloped. I will present some results and discuss some challenges related to building these tools.

Consider the Loewner equation associated to the upper-half plane. This is an equation originated from an extremal problem in complex analysis. Nowadays, it attracts a lot of attention due to its connection to probability. Normally this equation is driven by a real-valued function. In this talk, we will show that the equation still makes sense when being driven by a complex-valued function. We will relate this situation to the classical situation and also to complex dynamics. 

In the talk, we discuss how to combine the recurrent neural network with the signature feature set to tackle the supervised learning problem where the input is a data stream. We will apply this method to different datasets, including the synthetic datasets( learning the solution to SDEs ) and empirical datasets(action recognition) and demonstrate the effectiveness of this method.

In this talk, we will revisit the proof of the large deviations principle of Wiener chaoses partially given by Borell, and then by Ledoux in its full form. We show that some heavy-tail phenomena observed in large deviations can be explained by the same mechanism as for the Wiener chaoses, meaning that the deviations are created, in a sense, by translations. More precisely, we prove a general large deviations principle for a certain class of functionals $f_n : \mathbb{R}^n \to \mathcal{X}$, where $\mathcal{X}$ is some metric space, under the probability measure $\nu_{\alpha}^n$, where $\nu_{\alpha} =Z_{\alpha}^{-1}e^{-|x|^{\alpha}}dx$, $\alpha \in (0,2]$, for which the large deviations are due to translations. We retrieve, as an application, the large deviations principles known for the so-called Wigner matrices without Gaussian tails of the empirical spectral measure, the largest eigenvalue, and traces of polynomials. We also apply our large deviations result to the last-passage time which yields a large deviations principle when the weight matrix has law $\mu_{\alpha}^{n^2}$, where $\mu_{\alpha}$ is the probability measure on $\mathbb{R}^+$ with density $2Z_{\alpha}^{-1}e^{-x^{\alpha}}$ when $\alpha \in (0,1)$.

The signature of a path has many properties that make it an excellent feature to be used in machine learning. We exploit this properties to analyse a stream of data that arises from a psychiatric study whose objective is to analyse bipolar and borderline personality disorders. We build a machine learning model based on signatures that tries to answer two clinically relevant questions, based on observations of their reported state over a short period of time: is it possible to predict if a person is healthy, has bipolar disorder or has borderline personality disorder? And given a person or borderline personality disorder, it is possible to predict his or her future mood? Signatures proved to be very effective to tackle these two problems.

Abstract. As the first  step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multi-plicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution (Xt; Yt) of the equations dXt = Ytdt, dYt = jXtj_dBt, (X0; Y0) = (x0; y0). In particular, we prove that solutions arenonunique if 0 < _ < 1 and (x0; y0) = (0; 0) and unique if 1=2 < _ and (x0; y0) 6= (0; 0). We also show that blowup in _nite time holds if _ > 1 and (x0; y0) 6= (0; 0).

This is a joint work with A. Gomez, J.J. Lee, C. Mueller and M. Salins.

Inverting the signature of a path with ideas from linear algebra with implementations.