Past Stochastic Analysis Seminar

Malliavin calculus provides a framework to differentiate functionals defined on a Gaussian probability space with respect to the underlying noise. This allows to develop analysis on path space with infinite-dimensional generalisations of Fourier analysis, Sobolev spaces, etc from R^d. In this talk, we attempt to build a Lipschitz à la E. M. Stein (as opposed to Sobolev) function theory on rough path space. This framework allows to pathwise differentiate functionals on rough paths with respect to the underlying rough path. Time permitting, we show how to obtain Feynman-Kac-type representations for solutions to some high-order (>2) linear parabolic equations on R^d.

We consider families of fast-slow skew product maps of the form \begin{align*}x_{n+1}   = x_n+\eps^2 a_\eps(x_n,y_n)+\eps b_\eps(x_n)v_\eps(y_n), \quad

y_{n+1}   = T_\eps y_n, \end{align*} where $T_\eps$ is a family of nonuniformly expanding maps, $v_\eps$ is of mean zero and the slow variables $x_n$ lie in $\R^d$.  Under an exactness assumption on $b_\eps$ (automatically satisfied in the cases $d=1$ and $b_\eps\equiv I_d$), we prove convergence of the slow variables to a limiting stochastic differential equation (SDE) as $\eps\to0$.   Our results include cases where the family of fast dynamical systems

$T_\eps$ consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters) and Viana maps.Similar results are obtained also for continuous time systems  \begin{align*} \dot x   =  \eps^2 a_\eps(x,y,\eps)+\eps b_\eps(x)v_\eps(y), \quad \dot y   =  g_\eps(y). \end{align*}

Here, as in classical Wong-Zakai approximation, the limiting SDE is of Stratonovich type $dX=\bar a(X)\,dt+b_0(X)\circ\,dW$ where $\bar a$ is the average of $a_0$

and $W$ is a $d$-dimensional Brownian motion.

This talk will be about  Lévy processes on compact groups - discrete or continuous - and  two-dimensional analogues called pure Yang-Mills fields. The latter are indexed by  reduced loops of finite length in the plane and satisfy properties analogue to independence and stationarity of increments.     There is a one-to-one correspondance between Lévy processes invariant by adjunction and pure Yang-Mills fields. For Brownian motions, Yang-Mills fields stand for a rigorous version of the Euclidean Yang-Mills measure in two dimension.  I shall first sketch this correspondance for  Lévy processes with large jumps. Then, I will discuss two applications of an extension theorem, due to Thierry Lévy, similar to Kolmogorov extension theorem. On the one hand, it allows to construct pure Yang-Mills fields for any invariant Lévy process. On the other hand, when the group acts on vector spaces of large dimension, this theorem also allows to study the asymptotic behavior  of traces. The limiting objects yield a natural family of states on the group algebra of reduced loops.  We characterize among them the master field defined by Thierry Lévy by a continuity property.   This is  a joint work with Guillaume Cébron and Franck Gabriel.

Singular stochastic control problems ae largely studied in literature.The standard approach is to study the associated Hamilton-Jacobi-Bellman equation (with gradient constraint). In this work, we use a different approach (BSDE:Backward stochastic differntial equation approach) to show that the optimal value is a solution to BSDE.

The advantage of our approach is that we can study this kind of singular stochastic control with path-dependent coefficients

We consider fully nonlinear parabolic equations of the form $du = F(t,x,u,Du,D^2 u) dt + H(x,Du) \circ dB_t,$ which can be made sense of by the Lions-Souganidis theory of stochastic viscosity solutions. I will first recall the ideas of this theory, and will discuss more recent developments (including the use of rough path theory in this context). In the second part of my talk, I will explain how in the case where $H(x,Du)=|Du|^2$, the solution $u$ may enjoy better regularity properties than the solution to the unperturbed equation, which can be measured by (a pair of) solutions to a reflected SDE. Based on joint works with P. Friz, B. Gess, P.L. Lions and P. Souganidis.

The planar Ising model is one of the simplest and most studied models in Statistical Mechanics. On one hand, it has a rich and interesting phase transition behaviour. On the other hand, it is "solvable" enough to allow for many rigorous and exact results. This, in particular, makes it one of the prime examples in Conformal Field Theory (CFT). In this talk, I will review my joint work with C. Hongler and D. Chelkak on the scaling limits of correlations in the planar Ising model at criticality. We prove that these limits exist, are conformally covariant and given by explicit formulae consistent with the CFT predictions. This may be viewed as a step towards a rigorous understanding of CFT in the case of the Ising model.TBC

We consider the random conductance model: random walk among iid, uniformly elliptic conductnace on the d-dimensional lattice. We state,and explain, the Einstein relation for this model:It says that the derivative of the velocity of a biased walk as a function of the bias equals the diffusivity in equilibrium. For fixed bias, we show that there is an invariant measure for the environment seen from the particle.These invariant measures are often called steady states.

The Einstein relation follows, at least for dimensions three and larger, from an expansion of the steady states as a function of the bias.

The talk is gase on joint work with Jan Nagel and Xiaoqin Guo

The extension of standard stochastic models (SDEs, SPDEs) to general fractional noises is known to be a tricky issue, which cannot be studied within the classical martingale setting. We will see how the recently-introduced theory of regularity structures allows us to overcome these difficulties, in the case of a heat equation model with non-linear perturbation driven by a space-time fractional Brownian motion.

The analysis relies in particular on the exhibition of an explicit process at the core of the dynamics, the so-called K-rough path, the definition of which shows strong similarities with that of a classical rough path.

In this talk we will give an overview of the recent research on global quantizations on spaces of different types: compact and nilpotent Lie groups, general locally compact groups, compact manifolds with boundary.

The talk will focus on continuous time random walk with unbounded i.i.d. random conductances on the grid $\mathbb{Z}^d$  In the first place, in a joint work with Kumagai and Mathieu, we obtain Gaussian heat kernel bounds and also local CLT for bounded from above and not bounded from below conductances. The proof is given at first in a general framework, then it is specified in the case of plynomial lower tail conductances. It is essentially based on percolation and spectral analysis arguments, and Harnack inequalities. Then we will discuss the same questions for the same model with i.i.d. random conductances, bounded from below and with finite expectation.

Using Vovk's game-theoretic approach to mathematical finance and probability, it is possible to obtain new results in both areas.We first prove that one can make an arbitrarily large profit by investing in those one-dimensional paths which do not possess a local time of finite p-variation.  Additionally, we provide pathwise Tanaka formulas suitable for our local times and for absolutely continuous functions with sufficient regular derivatives. In the second part we derive a model-independent super-replication theorem in continuous time. Our result covers a broad range of exotic derivatives, including look-back options, discretely monitored Asian options, and options on realized variance.
 This talk is based on joint works with M. Beiglböck, A.M.G. Cox, M. Huesmann and N. Perkowski.

 

We show how the theories of paracontrolled distributions and regularity structures can be implemented on manifolds, to solve singular SPDEs like the parabolic Anderson model.

This is ongoing work with Bruce Driver (UCSD) and Antoine Dahlqvist (Cambridge)

Malliavin calculus is implemented in the context of [M. Hairer, A theory of regularity structures, Invent. Math. 2014]. This involves some constructions of independent interest, notably an extension of the structure which accommodates a robust and purely deterministic translation operator in L^2-directions between models. In the concrete context of the generalized parabolic Anderson model in 2D -one of the singular SPDEs discussed in the afore-mentioned article - we establish existence of a density at positive times.

In this talk, I will present recent results that give the necessary mathematical foundation for the study of rough path driven PDEs in the framework of weak solutions. The main tool is a new rough Gronwall Lemma argument whose application is rather wide: among others, it allows to derive the basic energy estimates leading to the proof of existence for e.g. parabolic RPDEs. The talk is based on a joint work with Aurelien Deya, Massimiliano Gubinelli and Samy Tindel.

In this talk we want to present a detailed study of the algebraic objects appearing in the theory of regularity structures. In particular we aim at introducing a class of co-algebras on labelled forests and trees and show that these allow to describe in an unified setting the structure group and the renormalisation group. Based on joint work with Yvain Bruned and Martin Hairer

We aim to study the long time behaviour of the solution to a rough differential equation (in the sense of Lyons) driven by a random rough path. To do so, we use the theory of random dynamical systems. In a first step, we show that rough differential equations naturally induce random dynamical systems, provided the driving rough path has stationary increments. If the equation satisfies a strong form of stability, we can show that the solution admits an invariant measure.

This is joint work with I. Bailleul (Rennes) and M. Scheutzow (Berlin).    

We derive a Stratonovich-to-Skorohod integral conversion formula for a class of integrands which are path-level solutions to RDEs driven by Gaussian rough paths. This is done firstly by showing that this class lies in the domain of the Skorohod integral, and secondly, by appending the Riemann-sum approximants of the Skorohod integral with a suitable compensation term. To show the convergence of the Riemann-sum approximants, we utilize a novel characterization of the Cameron-Martin norm using higher dimensional Young-Stieltjes integrals. Moreover, in the case where complementary regularity is absent, i.e. when the integrand has finite p-variation and the integrator has finite q-variation but 1/p + 1/q <= 1, we give new and sufficient conditions for the convergence these Young integrals.

The Hilbert transform is a central operator in harmonic analysis as it gives access to the harmonic conjugate function. The link between pairs of martingales (X,Y) under differential subordination and the pair (f,Hf) of a function and its Hilbert transform have been known at least since the work of Burkholder and Bourgain in the UMD setting.

During the last 20 years, new and more exact probabilistic interpretations of operators such as the Hilbert transform have been studied extensively. The motivation for this was in part the study of optimal weighted estimates in harmonic analysis. It has been known since the 70s that H:L^2(w dx) to L^2(w dx) if and only if w is a Muckenhoupt weight with its finite Muckenhoupt characteristic. By a sharp estimate we mean the correct growth of the weighted norm in terms of this characteristic. In one particular case, such an estimate solved a long standing borderline regularity problem in complex PDE.

In this lecture, we present the historic development of the probabilistic interpretation in this area, as well as recent results and open questions.

Abstract: Kolmogorov backward equations related to stochastic evolution equations (SEE) in Hilbert space, driven by trace class Gaussian noise have been intensively studied in the literature. In this talk I discuss the extension to non trace class Gaussian noise in the particular case when the leading linear operator generates an analytic semigroup. This natural generalization leads to several complications, requiring new existence and uniqueness results for SEE with initial singularities and a new notion of an extended transition semigroup. This is joint work with Arnulf Jentzen and Ryan Kurniawan (ETH).

We consider the non-linear equation $T^{-1} u+\partial_tu-\partial_x^2\pi(u)=\xi$

driven by space-time white noise $\xi$, which is uniformly parabolic because we assume that $\pi'$ is bounded away from zero and infinity. Under the further assumption of Lipschitz continuity of $\pi'$ we show that the stationary solution is - as for the linear case - almost surely Hölder continuous with exponent $\alpha$ for any $\alpha<\frac{1}{2}$ w. r. t. the parabolic metric. More precisely, we show that the corresponding local Hölder norm has stretched exponential moments.

On the stochastic side, we use a combination of martingale arguments to get second moment estimates with concentration of measure arguments to upgrade to Gaussian moments. On the deterministic side, we first perform a Campanato iteration based on the De Giorgi-Nash Theorem as well as finite and infinitesimal versions of the $H^{-1}$-contraction principle, which yields Gaussian moments for a weaker Hölder norm. In a second step this estimate is improved to the optimal

Hölder exponent at the expense of weakening the integrability to stretched exponential.

This is joint work with Felix Otto.

First order asymptotic of scalar renewal sequences with infinite mean characterized by regular variation has been classified in the 60's (Garsia and Lamperti). In the recent years, the question of higher order asymptotic for renewal sequences with infinite mean was motivated by obtaining 'mixing rates' for dynamical systems with infinite measure. In this talk I will present the recent results we have obtained on higher order expansion for renewal sequences with infinite mean (not necessarily generated by independent processes) in the regime of slow regular variation (with small exponents).  I will also discuss some consequences of these results for error rates in certain limit theorems (such as arcsine law for null recurrent Markov processes).

 

The first reaction-diffusion equation developed and studied is the Fisher-KPP equation.  Introduced in 1937, it accounts for the spatial spreading and growth of a species.  Understanding this population-dynamics model is equivalent to understanding the distribution of the maximum particle in a branching Brownian motion.  Various generalizations of this model have been studied in the eighty years since its introduction, including a model with non-local reaction for the cane toads of Australia introduced by Benichou et. al.  I will begin the talk by giving an extended introduction on the Fisher-KPP equation and the typical behavior of its solutions.  Afterwards, I will describe the model for the cane toads equations and give new results regarding this model.  In particular, I will show how the model may be viewed as a perturbation of a local equation using a new Harnack-type inequality and I will discuss the super-linear in time propagation of the toads.  The talk is based on a joint work with Bouin and Ryzhik.

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We consider several classes of sequences of random variables whose Laplace transform presents the same type of \textit{splitting phenomenon} when suitably rescaled. Answering a question of Kowalski-Nikeghbali, we explain the apparition of a universal term, the \textit{Gamma factor}, by a common feature of each model, the existence of an auxiliary randomisation that reveals an independence structure.
The class of examples that belong to this framework includes random uniform permutations, random polynomials or random matrices with values in a finite field and the classical Sathe-Selberg theorems in probabilistic number theory. We moreover speculate on potential similarities in the Gaussian setting of the celebrated Keating and Snaith's moments conjecture. (Joint work with R. Chhaibi)
 

Motivated by a theorem of Barbour, we revisit some of the classical limit theorems in probability from the viewpoint of the Stein method. We setup the framework to bound Wasserstein distances between some distributions on infinite dimensional spaces. We show that the convergence rate for
the Poisson approximation of the Brownian motion is as expected proportional to λ −1/2 where λ is the intensity of the Poisson process. We also exhibit the speed of convergence for the Donsker Theorem and extend this result to enhanced Brownian motion.

 

TBC