Theoretical and Applied Stochastic Analysis (18w5129)

Arriving in Oaxaca, Mexico Sunday, September 9 and departing Friday September 14, 2018


Cheng Ouyang (University of Illinois at Chicago)

Fabrice Baudoin (University of Connecticut)

(Purdue University)


We have defined several key topics on which we wish to give a comprehensive account.

(1) Stochastic PDEs. A growing attention has been devoted recently to track physically relevant phenomenon displayed by certain classes of stochastic PDEs. Among the most exciting and challenging situations is the one involving the so-called parabolic Anderson model. This model can be described by a simple linear PDE which can be written formally as follows: \begin{equation}\label{eq:spde-pam} \frac{\partial u}{\partial t} = \Delta u + u \, \dot{W}\,, \quad t\ge 0, \quad x\in\mathbb{R}^{d}\, , \end{equation} where $\dot{W}$ is a noisy term, usually described as a Gaussian field. In spite of its simple form, equation \eqref{eq:spde-pam} exhibits all sorts of non classical behaviors, in terms of moment estimates, growth rate in time and space, or energy landscape. Among the speakers, D.~Khoshnevisan, Y.~Hu and D.~Nualart will be able to give an account on recent developments in the area. Other exciting aspects of the theory, such as the geometry of the fields for solutions to stochastic PDEs or multiple points problems, will be addressed by R.~Dalang and C.~Mueller.

(2) Rough paths techniques. The theory of rough paths has been originally developed in the mid-nineties by T. Lyons. It is based on the profound insight that stochastic differential equations can be solved pathwise and that the solution map is continuous in suitable rough path metrics. The topic has now grown into a mature and widely applicable mathematical theory. Its scope of applications is twofold: (i) By establishing a continuous relation between noise and solutions to differential equations, it sheds a new light on diffusion processes and other differential systems driven by a Brownian motion. (ii) It extends widely the class of processes which can be considered as driving noises of differential equations. One particularly popular example of application is fractional Brownian motion, and the rough paths method is the only one allowing to properly define and solve general equations driven by this noise. A lot of the current effort of the community is devoted to an in-depth study of equations driven by fractional Brownian motions $B^{1},\ldots,B^{d}$, of the form \begin{equation*} Y_t= a+ \int_{0}^{t} V_0(Y_s) \, ds + \sum_{j=1}^{d} \int_{0}^{t} V_{j}(Y_s) \, dB_s^{j}, \end{equation*} for a given initial condition $a\in\R^{n}$ and some smooth vector fields $V_{1},\ldots,V_{d}$ on $\R^{n}$. This is precisely one of the instances where stochastic analysis enters into the play, and has to be mixed with rough paths in order to produce fruitful results. E.~Nualart, P. Friz and F. Viens will give lectures on this topic.

(3) Regularity structures. The theory of regularity structures has opened a new space of investigation to the stochastic analysis community. Introduced by M. Hairer (Fields medal 2014) a few years ago, it can be seen a wide extension of the rough paths analysis, which encompasses the definition of rough paths indexed by $\R^{n}$, a richer rough paths structure indexed by trees, products of distributions, an additional group structure for renormalizations, and evaluation of singularities. Thanks to regularity structures, a meaning to inextricable systems, such as the KPZ equation, has eventually been given on a rigorous basis. As in the case of rough paths, there is now a need for a better understanding of the new objects created (or rigorously defined) by the theory. This will be explained by M. Gubinelli, M.~Hairer and L.~Zambotti in our conference.

(4) Dirichlet forms. Dirichlet form theory is the most convenient and useful tool to construct diffusion processes in non smooth settings, like graphs, fractals, or Gromov-Hausdorff limits of Riemannian manifolds. In the last few years, there have been tremendous efforts to use Dirichlet forms theory to prove geometric or analytic theorems on singular spaces, that before only were available in smooth spaces. A crowning achievement in non-smooth geometry is the Lott-Villani-Sturm theory of synthetic Ricci lower bounds on metric measure spaces and the most recent Riemannian curvature dimension theory by Ambrosio-Gigli-Savar\'e. Both of those theories are nowadays the object of intensive studies by geometers, analysts and probabilists, showing the ubiquity of Dirichlet forms techniques. World leading experts on the probabilistic side of the story will be invited in our conference to expose the most recent developments of the theory. This includes M. Hino, K.T.~Sturm and K.~Kuwada.

On a more analytic side, Dirichlet forms have also been a powerful tool in the analysis of partial differential equations, where intrinsic properties of the Dirichlet form (like the volume doubling property or the scale invariant Poincar\'e inequality) have been proven equivalent to Harnack type inequalities for solutions of relevant parabolic partial differential equations. Recently, the range of those techniques has been extended to cover more and more singular situations like graphs or fractals. The world leading expert in this area is L. Saloff-Coste, who will be invited to the conference. Young promising researchers in this field like D. Kelleher and J. Lierl will also be invited.

(5) Stochastic and rough differential geometry. Despite the tremendous interest in applying techniques from stochastic analysis to non smooth spaces, stochastic analysis on smooth manifolds is still a rich area in which many open and challenging questions are being addressed. A recent trend has been the stochastic analysis of sub-Riemannian manifolds. Despite the smoothness of the underlying manifold, in sub-Riemannian geometry, the space of geodesics is extremely singular and poorly understood. Classical geometric methods generally fail and stochastic analysis provide a set of robust tools that have successfully applied in the last few years. We mention for instance the generalized curvature dimension inequalities by Baudoin-Garofalo. Among our participants, M. Gordina will present an overview of the present state of the art and J. Wang some of the most recent developments.

Another topic covered by the conference is the study of rough paths on manifolds. As mentioned before, rough paths theory techniques are the only tools available to make sense to solutions of differential equations driven by very irregular paths. It turns out that those techniques extend to the framework of differential manifolds. This opens the door to a theory of rough signals on manifolds. The ongoing efforts in this direction will be presented by B. Driver and T. Cass.

(6) L\'evy processes. L\'evy processes have been studied for a long time and their theory is rich in applications (finance, telecommunications). More recently, deep relationship between L\'evy processes and the theory of singular integrals have been uncovered and exploited by R. Ba\~nuelos. R. Ba\~nuelos will give us a survey of the exciting interplay between L\'evy processes and sharp constants in some functional inequalities.

(7) Limit theorems. The recent discovery by Nourdin and Peccati of the connection between Malliavin calculus and Stein's method, has led to a burst of new research in stochastic analysis. Subjects like dimension free estimates for the rate of convergence in limit theorems, Sudakov-Fernique theorem or Slepian inequality can now be generalized to Wiener chaos by methods of Malliavin calculus. They represent important tools when investigating a wide number of demanding contexts, such as central and non-central limit theorems, stochastic PDEs, spin glasses or polymer measures. In another direction, estimates of Malliavin-Stein type can be successfully applied to geometric problems, involving in particular random graphs defined on the points of a homogeneous Poisson process. A number of new applications include Poisson-Voronoi approximation and Boolean models. We should also mention a deep and abstract work by Ledoux, generalizing certain bounds of Nourdin and Peccati to random variables living in the chaos of a general Markov operator. Among our participants, I. Nourdin and G. Peccati will give us a survey on several aspects of those promising developments.

(8) Applications. In spite of being inspired by theoretical considerations, stochastic analysis tools have been applied in a wide range of situations, and have also generated a large array of interdisciplinary research topics. We have already mentioned some links with challenging models of theoretical physics, such as KPZ or $\Phi^{4}$ equations, spin glasses or polymer measures. Finance is also an effervescent area where rough paths techniques are fruitfully applied, either for stochastic volatility models or for numerical methods such as cubature on Wiener's space. We should also mention that signatures (that is iterated integrals of paths or processes) are central objects in the rough paths theory. These objects have been recently exploited to characterize written (and in particular Chinese) characters, which paves the way for new real world applications. An overview of this trend will be discussed by T. Lyons.

Conclusion.\\ In conclusion, we hope to bring together some of the top worldwide experts in stochastic analysis. Through expository talks and research presentations, they will give a vibrant overview of the most recent developments. One of our primary goals is to bring in together world renowned specialists and more junior researchers. The speakers selected have been chosen to cover a wide array of trendy topics. The scientific committee of the conference is well connected to the experts in the covered areas. We are therefore confident in organizing a high-level event. Several of our proposed list of speakers have already confirmed their interest in participating in the event. This includes prestigious researchers such as: Martin Hairer (Fields medal 2014), David Nualart (Black-Babcock Distinguished Professor), and Karl Theodor Sturm. We also extended our invitations to several female professors (10 already accepted) and researchers form Central and South America.