Stochastic Analysis

In this talk we present some recent results on the stochastic heat equation on R^d driven by a Gaussian noise which is a fractional Brownian motion in the time variable with Hurst parameter H, where H is in (0,1/2). We derive a Feynman-Kac formula for the solution and we use this representation to establish matching lower and upper bounds for the moments of the solution that lead to intermittency properties.

We will look at an SPDE whose solution is known to exist and is unique. The main focus of this talk is the qualitative behaviour of the solution. It is quite surprising that even though the SPDE is quite easy to make sense of, many of its basic properties are still unknown. We will describe how the boundedness properties of the solution depends on the initial condition. The proof requires several relatively new ideas, one of which is a moment comparison theorem.

We study deterministic and stochastic Navier-Stokes equations with a constraint on $L^2$ energy of the solution. We prove the existence and uniqueness of local strong solutions and the existence of a global solutions for the constrained 2D Navier-Stokes equations on the torus on the whole Euclidean space. This is based on joint works with Gaurav Dhariwal (York) and Mauro Mariani (Roma I).

The main purpose of the talk is to show a new approach, based on the so called weak vorticity formulation, to the 2D deterministic Euler equations with white noise initial conditions. Besides that, comments on the stochastic case and the 3D case will be added.

In this talk, I will present some recent results on global existence of solutions of stochastic heat equations with superlinear drifts and multiplicative space-time white noise.

It will be shown that the value function, corresponding to a control problem of an evolution equation with Levy noise, is the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. The result was obtained in collaboration with A. Święch.

Stochastic partial differential equations arise naturally as limits of finite systems of weighted interacting particles. For a variety of purposes, it is useful to keep the particles in the limit obtaining an infinite exchangeable system of stochastic differential equations for the particle locations and weights. The corresponding de Finetti measure then gives the solution of the SPDE. These representations frequently simplify existence, uniqueness and convergence results. Beginning with the classical McKean-Vlasov limit, the basic results on exchangeable systems along with several examples will be discussed.

We prove existence and uniqueness of solutions to continuity equation in a separable Hilbert space. We look for solutions which are absolutely continuous with respect to a reference measure which is the invariant measure of a reaction-diffusion equation. We exploit that the gradient operator $D_x$ is closable with respect to $L^p(H;\gamma)$ and a recent formula for the commutator $D_xP_t - P_tD_x$ where $P_t$ is the transition semi-group corresponding to the reaction-diffusion equation, [DaDe14]. We stress that $P_t$ is not necessarily symmetric. Our paper is an extension of [DaFlRo14] where $\gamma$ was the invariant measure of a suitable Ornstein-Uhlenbeck process. This is joint work with Giuseppe da Prato.

In this talk I will give an introduction to Multilevel Monte Carlo methods which can reduce the cost of Monte Carlo simulation in a wide variety of contexts through the use of a sequence of different approximations of the quantity of interest, with cheaper, less accurate approximations being used as a control variate for more expensive, more accurate approximations.

We prove an a priori bound for the dynamic $\Phi^4_3$ model on the torus which is independent of the initial condition. In particular, this bound rules out the possibility of finite time blow-up of the solution. It also gives a uniform control over solutions at large times, and thus allows to construct invariant measures via the Krylov-Bogoliubov method. It thereby provides a new dynamic construction of the Euclidean $\Phi^4_3$ field theory on finite volume. Our method is based on the local-in-time solution theory developed recently by Gubinelli, Imkeller, Perkowski and Catellier, Chouk. The argument relies entirely on deterministic PDE arguments (such as embeddings of Besov spaces and interpolation), which are combined to derive energy inequalities. This is joint work with J.-C. Mourrat (Lyon).

We study a class of mean-field stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H in(1/2,1) and a related stochastic control problem. We derive a Pontryagin type maximum principle and the associated adjoint mean-field backward stochastic differential equation driven by a classical Brownian motion, and we prove that under certain assumptions, which generalise the classical ones, the necessary condition for the optimality of an admissible control is also sufficient.

We show that a class of two-dimensional diffusion processes are described as images by some mappings of time-changed Brownian motions, similarly to one-dimensional diffusion processes. We use quasiconformal mappings in place of scale functions.

This talk will begin with a review on the significant contributions of the paper by Higham, D.J., Mao, X. and Stuart, A.M. (Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM Journal on Numerical Analysis 40(3) (2002), 1041-1063.) This was the first to study the strong convergence of numerical solutions of SDEs under a local Lipschitz condition. Prior to this, all positive results were based on a much more restrictive global Lipschitz assumption, which rules out most realistic models. The field of numerical analysis of SDEs now has a very active research profile, much of which builds on the techniques developed in that paper, which has attracted 385 Google Scholar Citations. In particular, the theory developed there has formed the foundation for several recent very popular methods, including tamed Euler-Maruyama method and truncated Euler-Maruyama. The talk will show more details on the development of the truncated Euler-Maruyama.

In structural models for default of firms the firm value is modelled as a diffusion and default occurs at the first hitting time of a lower barrier. We will consider such models in a large portfolio setting where the diffusions are correlated through global factors. In the large portfolio limit these models lead naturally to some SPDEs and stochastic McKean-Vlasov equations. We will discuss the existence, uniqueness and properties of these equations.

To compute Greeks, it is common to use Eulrr-Maruyama Approximation and its difference as approximation for differentials. We discuss about this method and give its justification.

(Joint with Giacomo Zanella and Mylene Bédard) Abstract: I will discuss the use of Dirichlet forms to deliver proofs of optimal scaling results for Markov chain Monte Carlo algorithms (specifically, Metropolis-Hastings random walk samplers) under regularity conditions which are substantially weaker than those required by the original approach (based on the use of infinitesimal generators). The Dirichlet form method has the added advantage of providing an explicit construction of the underlying infinite-dimensional context. In particular, this enables us directly to establish weak convergence to the relevant infinite-dimensional diffusion. Reference: Zanella, G., Bédard, M., & Kendall, W. S. (2016). A Dirichlet Form approach to MCMC Optimal Scaling. To appear in Stochastic Processes and Their Applications. See also arXiv, 1606.01528, 22pp. URL: arxiv.org/abs/1606.01528.

We will discuss perturbation to conservation laws and stochastic averaging of slow-fast stochastic differential equations.

We consider a semilinear parabolic PDE, with random highly oscillating coefficients, where the spatial coordinate lives in a bounded subset the d dimensional Euclidean space, d=1, 2 or 3. We prove a law of large number (homogenization) and a central limit theorem, by using the methodology of regularity structures. This is work in progress, joint with M. Hairer.

We revisit some classical results of Kurtz, Protter, Pardoux concerning stability of stochastic differential equations and put them in perspective with latest results on (cadlag) rough paths. Joint work with A. Shekhar, I. Chevyrev and H. Zhang.

We consider a class of nonlinear population models on a two-dimensional lattice which are influenced by a small random potential, and we show that on large temporal and spatial scales the population density is well described by the continuous parabolic Anderson model, a linear but singular stochastic PDE. The proof is based on a discrete formulation of paracontrolled distributions on unbounded lattices which is of independent interest because it can be applied to prove the convergence of a wide range of lattice models. This is joint work with Jörg Martin.

A general signal and observation model driven by Wiener processes is considered, and new conditions for an affirmative answer for the innovations problem are presented. The talk is based on a joint work with Nicolai Krylov.

We present stochastic variational principles with advected quantities whose configuration space is a Lie group. They yeld deterministic as well as stochastic differential equations of motion (resp. pde's and spde's when the Lie group is infinite-dimensional). We give some examples. This is a joint work with X. Chen and T. Ratiu (Jiao Tong Univ. Shanghai)

The Yang-Mills measure over the sphere is considered is the case where the structure group is that of unitary NxN matrices, in the limit as N goes to infinity. We show a law of large numbers. The limit is the master field on the sphere. The development has similarities but also differences with the whole plane case studied by Thierry Levy. This is joint work with Antoine Dahlqvist.

We study the convergence, as N tends to infinity, of a system of N stochastic differential equations, coupled through N Hamilton-Jacobi equations, when the coupling between the equations becomes increasingly singular. Such a system appears in the description of control problems with N players. The limit equation turns out to be a Mean Field Game system with a local coupling.

We relate the large noise limit of linear SDEs to certain asymptotics of high order iterated integrals, which we will study in depth. Our main result is a limit theorem for the high order iterated integrals of Brownian motion. We will discuss its implication to the construction of an additive variation norm on rough paths. The talk will be based on the joint work arXiv:1609.08111 with Xi Geng.

\newcommand{\R}{\mathbb{R}} \noindent{\bf Abstract.} Let $W = (W(s),\, s\in \R^2_+)$ be a standard Brownian sheet indexed by the nonnegative quadrant. We show that with probability one, the Hausdorff dimension of the boundary of any connected component of the random open set $\{(s_1,s_2)\in \R^2_+: W(s_1,s_2) >0\}$ is equal to $$ \frac{1}{4}\left(1 + \sqrt{13 + 4 \sqrt{5}}\right) \simeq 1.421\, . $$ This result is first established for additive Brownian motion, which provides good local approximations to the Brownian sheet, and then extended, with some technical effort, to the Brownian sheet itself. This is joint work with T.~Mountford (Ecole Polytechnique F\'ed\'erale de Lausanne). \noindent A preprint is available at http://arxiv.org/abs/1702.08183.

We introduce a Weyl functional calculus for the Ornstein-Uhlenbeck operator $L = -\Delta + x\cdot \nabla$, and give a simple criterion for $L^p$-$L^q$ boundedness of operators in this functional calculus. It allows us to recover, unify, and extend, old and new results concerning the boundedness of $\exp(-zL)$ as an operator from $L^p(\mathbb{R}^d,\gamma_{\alpha})$ to $L^q(\mathbb{R}^d,\gamma_{\beta})$ for suitable values of $z\in \mathbb{C}$ with $\Re z>0$, $p,q\in (1,\infty)$, and $\alpha,\beta>0$. Here, $\gamma_\tau$ denotes the centred Gaussian measure on $\mathbb{R}^d$ with density $(2\pi\tau)^{-d/2}\exp(-|x|^2/2\tau)$. This is joint work with Pierre Portal.

Joint work with: Anna Ananova (Imperial College London) We define a functional calculus for controlled rough paths and show that several concepts in the theory of controlled rough paths may be interpreted in terms of the recently developed non-anticipative functional calculus (NAFC) [3]. We show that regular functionals in the sense of NAFC define controlled rough paths and, conversely, that a family of controlled rough paths may be represented as such a regular functional. This leads to a change of variable formula and a strictly pathwise calculus for controlled rough paths. Finally, we establish a pathwise Norris-type lemma for regular functionals of paths with strictly increasing quadratic variation. [1] Anna Ananova, Rama Cont, Pathwise integration with respect to paths of finite quadratic variation, Journal de Mathématiques Pures et Appliquées, Volume 107, Issue 6, 2017, Pages 737-757. http://dx.doi.org/10.1016/j.matpur.2016.10.004. [2] A Ananova, R Cont (2017) Functional calculus for controlled rough paths, Working Paper. [3] R Cont: Functional Ito calculus and functional Kolmogorov equations, in: V Bally et al: Stochastic integration by parts and Functional Ito calculus ( Barcelona Summer School on Stochastic Analysis, July 2012), Springer.

Stochastic Interacting Particle System (SIPS) and they limiting stochastic McKean-Vlasov equations offer a very rich and versatile modelling framework. On one hand interactions allow us to capture complex dependent structure, on the other provide a great challenge for Monte Carlo simulations. The non-linear dependence of the approximation bias on the statistical error makes classical variance reduction techniques fail in this setting. In this talk, we will devise a iterative MLMC that will allow to overcome this difficulty. Obtained method allow to reduce computational cost of simulating SIPS by the the order of magnitude.

We will present various probabilistic representations based on the parametrix method which may be used for Monte Carlo simulations. In particular, we concentrate on a second order method. This is joint work with P. Andersson (Uppsala) and T. Yuasa (Ritsumeikan Univ.) This research is supported by KAKENHI 24340022, 16H03642, 16K05215

We show how knowing Schauder or Sobolev-space estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs for equations with coefficients depending only on time variable with the same constants as in the case of the one-dimensional heat equation. The method is quite general and is based on using the Poisson stochastic process. We will also present other applications of the method. It looks like no other method is available at this time and it is a very challenging problem to find a purely analytic approach to proving such results. This is a joint work with N.V. Krylov.

In this talk we focus on a series of results concerning p-variation limits, as well as Itô type formulas in law for Gaussian processes. This line of research has been quite active in the recent past in the stochastic analysis community. Most of the techniques involve integration by parts, Stein’s method, and other Malliavin calculus tools. This yields a series of limitations on the nature of the results, as well as the dimension of the Gaussian process at stake. Our aim is to show how those questions can possibly be handled in a more natural way thanks to rough path type techniques. More specifically we will show how to transfer limits taken on a Gaussian signature to limits involving controlled processes, by means of the typical expansions of the rough paths theory. Applications of this rather simple trick include the aforementioned p-variations and Itô type formulas, as well as central limit theorems for numerical schemes.

We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss function given by the error between the prescribed terminal condition and the solution of the BSDE. The policy function is then approximated by a neural network, as is done in deep reinforcement learning. Numerical results using TensorFlow illustrate the efficiency and accuracy of the proposed algorithms for several 100-dimensional nonlinear PDEs from physics and finance such as the Allen-Cahn equation, the Hamilton-Jacobi-Bellman equation, and a nonlinear pricing model for financial derivatives.

I will talk random periodic processes, periodic measures and their “equivalence”. I will also talk about their existence, ergodic theory and pure imaginary eigenvalues of the infinitesimal generator of the corresponding Markovian semigroup. This is a joint work with C Feng.

A continuous-state branching process is the mathematical model for the random evolution of a large population of small individuals. The trajectory of the process was constructed in Dawson and Li (2006, 2012) as the strong solution to a stochastic integral equation driven by Gaussian and Poisson time-space noises. The genealogical structure of population is represented by a continuum random tree, which was uniquely characterized by its height process. The later was constructed by Le Gall and Le Jan (1998) as a functional of some spectrally positive Levy process. A different construction of the height process can be given in terms of a stochastic integral equation driven by Poisson point processes. More general population models may include the impact of immigration, competition, environment and so on. In this talk, we present a number of stochastic integral equations in the theory of continuous-state branching processes and continuum random trees. We also explain how those stochastic equations can be used in the study the structural properties of the models.

Analysis of MFG with common noise leads to remarkable new problems in nonlinear SPDEs, for instance McKean-Vlasov SPDE, and interacting particle systems, and to the certain class of infinite-dimensional second order equations on measures with some peculiare degeneracy making them difficult to analyse. The main objective of the talk will be to demonstrate that the solutions to these problems yield the $\epsilon$-Nash equilibria for the approximating games of finite number of players.

In my talk, I will first talk an ergodic theory of an expectation-preserving map on a sublinear expectation space. We also study the ergodicity of invariant sublinear expectation of sublinear Markovian semigroup. As an example we show that G-Brownian motion on the unit circle has an invariant expectation and is ergodic. This is a joint work with Huaizhong Zhao.

We build a connection between rough path theory and a noncommutative algebra, and interpret the integration of geometric rough paths as a non-abelian version of Young integration.

In this talk, we present a brief overview of the algebraic structures used for renormalising SPDEs such as Hopf algebras and pre-Lie algebras. They both appear in the framework of Regularity Structures and allow us to have a good description of the renormalisation group and its action on the equation.

We establish the existence and uniqueness of strong solutions for a one-dimensional stochastic differential equation driven by a Brownian motion and a pure jump Levy process. It is shown that under fairly general conditions on the coefficients, pathwise uniqueness holds based on the methods of weak uniqueness and local time technique.

Poster presenters:

Tomasz Kosmala
Oana Lang
Yu Liu
Yujia Liu
Le Ngan Trang
Martin Weidner

We develop a new technique for establishing quantitative propagation of chaos for systems of interacting particles. Using this technique we prove propagation of chaos for diffusing particles whose interaction kernel is merely Holder continuous, even at long ranges. Moreover, we do not require specially prepared initial data. On the way, we establish a law of large numbers for SDEs that holds over a class of vector fields simultaneously. The proofs bring together ideas from empirical process theory and stochastic flows.

We study the small-time fluctuations for diffusion processes which are conditioned by their initial and final positions and whose diffusivity has a sub-Riemannian structure. In the case where the endpoints agree, we discuss the convergence of the suitably rescaled fluctuations to a limiting diffusion loop, which is equal in law to the loop we obtain by taking the limiting process of the unconditioned rescaled diffusion processes and condition it to return to its starting point. The generator of the unconditioned limiting rescaled diffusion process can be described in terms of the original generator.

Approximate schedule:

9:00 bus leaves Grey College
10:30 arrive at Alnwick Castle
12:00 lunch (please collect at breakfast)
16:00 bus leaves Alnwick Castle
17:30 arrive at Grey College

Several approaches to the problem of the regularity of the law of the solution of an SDE with jumps, based on some Malliavin type calculus, have been given and developed since 30 years. Our work represents a new approach based on the following idea: we replace the small jumps by a Brownian motion and we use the standard Malliavin calculus based on this Brownian motion in order to obtain a smooth density and to estimate the short time behavior of this density. Our approach works for general intensity measures (including pure atomic ones) and in this sense it represents an alternative to Jean Picard's method. We succeed to improve his results. The delicate point in our method concerns the approximation procedure that we use: one has to obtain a good equilibrium between the error of the approximation and the quantity of Gaussian noise in the approximating process (the two quantities are in concurrence). This is the point where an interpolation criterion comes on.

We show a Freidlin-Wentzell large deviations principle for McKean-Vlasov equations in certain path space topologies. The equations have a drift of polynomial growth and an existence/uniqueness result is provided. Lastly, we characterize the topological support of the law of the solution to these equations.

Stochastic integral operators defined in the stochastic integral of convolution manner appeared naturally in the mild formulation of SPDEs. In this talk, we derive BMO and Morrey-Campanato estimates for stochastic singular integral operators, We then apply our results to discuss various estimates (including the Schauder estimates) of the solutions of SPDEs with additive noises. Joint work with Guangying Lv, Hongjun Gao and Jinlong Wei.

In recent years the observation that "irreversible processes converge to equilibrium faster than their reversible counterparts" has sparked a significant amount of research to exploit irreversibility within sampling schemes, thereby accelerating convergence of the resulting Markov Chains. It is now understood how to design irreversible continuous time dynamics with prescribed invariant measure. However, for sampling/simulation purposes, such dynamics still need to undergo discretization and, as it is well known, naive discretizations can completely destroy all the good properties of the continuous-time process. In this talk we will i) give some background on irreversibility and briefly review the progress made so far on the study of non-reversible processes; ii) make further considerations on how to use (or not to use) irreversibility for algorithmic purposes. As a measure of efficiency for the algorithms at hand we will use the number of steps taken, for the MCMC chain, to explore state space. Theoretical results in this direction can be obtained through the use of diffusion limits for Markov chains.

Signature motivated constructions, combined with classic results from stochastic analysis, can provide new approaches to many machine learning tasks. On the other hand, this point of view leads to new theoretical insights and questions in rough path theory itself. For example, signature motivated constructions quickly run into computational issues when the underlying path gets only moderately high dimensional; further, it becomes somewhat unclear what to do if the path does not take values in a linear space which is a common situation in structured learning tasks. I will talk about joint works with I. Chevyrev, T. Lyons, F. Kiraly that address such issues.

The aim of this talk consists in introducing a new formalism for the deterministic analysis associated with backward stochastic differential equations driven by general càdlàg martingales, coupled with a forward Markov process. When the martingale is a standard Brownian motion, the natural deterministic analysis is provided by the solution u of a semilinear PDE of parabolic type coupled with a function v which is associated with the ∇u, when u is of class C1 in space. When u is only a viscosity solution of the PDE, the link associating v to u is not completely clear: sometimes in the literature it is called the identification problem. The idea is to introduce a suitable analysis to investigate the equivalent of the identification problem in a general Markovian setting with a class of examples. An interesting application concerns the hedging problem under basis risk of a contingent claim g(X_T , S_T ), where S (resp. X) is an underlying price of a traded (resp. non-traded but observable) asset, via the celebrated Föllmer-Schweizer decomposition. We revisit the case when the couple of price processes (X, S) is a diffusion and we provide explicit expressions when (X, S) is an exponential of additive processes.

We present sharp gradient estimates for the solution of the filtering equation and briefly report on some work in progress on a high order cubature method for the nonlinear filtering problem. Joint work with Terry Lyons and Dan Crisan.

We consider the problem of phase transitions for the McKean-Vlasov equation that appears in the mean field limit of a system of weakly interacting diffusions in a multiscale potential. We construct the bifurcation diagram and we study the situation when the confining potential is characterized by infinitely many local minima.

I will set up the framework for developing statistical inference methods for discretely observes Rough Differential Equations. I will discuss in detail how to construct an approximation to the likelihood and how this can be used in either a Bayesian or a frequentists context.

This talk is concerned win this talk, I will present a survey of some SPDEs I studied which can be represented in terms of interacting particle systems. These include those from stochastic filtering and from continuous state branching systems.

In this talk we consider the stochastic Cauchy problem driven by a cylindrical Lévy process. Here, a cylindrical Lévy process is understood in the classical framework of cylindrical random variables and cylindrical measures, and thus, it can be considered as a natural generalisation of cylindrical Wiener processes or Gaussian space-time white noises.

For many applications (especially, in molecular dynamics and statistics), it is of interest to compute the mean of a given function with respect to the invariant law of the diffusion, i.e. the ergodic limit. To evaluate these mean values, one usually has to integrate a large dimensional system of stochastic differential equations over long time intervals. In solving this computationally challenging problem, stochastic geometric integrators play an important role. We will illustrate geometric integration ideas using examples of stochastic thermostats for rigid body dynamics including those with hydrodynamic interactions as well as discuss some related theoretical aspects. The talk will be based on joint works with Ruslan Davidchack, Tom Ouldridge, Ben Leimkuhler and Charlie Matthews.

Recent work in regularity structures has provided a robust solution theory for a wide class of singular SPDEs. While much progress has been made on understanding the analytic and algebraic aspects of renormalisation of the driving signal, the action of renormalisation on the equation still needed to be performed by hand. In this talk, we aim to give a systematic description of the renormalisation procedure directly on the level of the PDE. Based on joint work in progress with Y. Bruned, A. Chandra, and M. Hairer.

We present a limit theorem for moments in space of the increments of Brownian local time. As special cases for the second and third moments, previous results by Chen et al. and Rosen, which were later reproven by Hu and Nualart and Rosen are included and a conjecture of Rosen for the fourth moment is settled. In comparison to the previous methods of proof, we follow a fundamentally different approach by exclusively working in the space variable of the Brownian local time, which allows to give a unified argument for arbitrary orders. The main ingredients are Perkins' semimartingale decomposition, the Kailath-Segall identity and an asymptotic Ray-Knight Theorem by Pitman and Yor.

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.

We introduce adaptive time stepping techniques to control growth in the numerical solution of SPDEs. This can be thought of as an alternative to proving moment bounds for the numerical method and to using a fixed step taming method. Ideas and the convergence result will be illustrated with some numerical experiments that also show that the adaptivity leads to more accurate solutions. If time permits we will introduce a new exponential based method for time stepping for SPDEs with multiplicative noise which have an improved rate of convergence in specific circumstances.

Planar random growth processes occur widely in the physical world. Examples include diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. One approach to mathematically modelling such processes is to represent the randomly growing clusters as compositions of conformal mappings. In 1998, Hastings and Levitov proposed a family of such models, which includes versions of the physical processes described above. In earlier work, Norris and I showed that the scaling limit of the simplest of the Hastings-Levitov models is a growing disk. Recently, Silvestri showed that the fluctuations can be described in terms of the solution to a stochastic fractional heat equation. In this talk, I will discuss on-going work in which we establish scaling limits and fluctuation results for a natural generalisation of the Hastings-Levitov family.

In this talk I will present a class of FBSDEs with singular coefficients, in particular some of the coefficients are elements of a fractional Sobolev space with order larger that -1/2. I will introduce a suitable notion of solution using the so-called Ito trick, and investigate existence and uniqueness of such a solution. This whole construction is based on results on the associated Kolmogorov PDE with singular coefficients, which I will present and study using mild solutions and properties of fractional Sobolev spaces and semigroups.

A major motivation for Ramer's development of his non-linear change of variable formula for Gaussian measures in his thesis ``Integration on infinite dimensional manifolds" was his construction of a theory of ``finite co-dimensional differential forms". I'll briefly describe these, mention how they could relate to certain non-linear but smooth elliptic pdes F(u)=f where f is random, and sketch their application to a quick, easily understood, but interesting proof of the Gauss-Bonnet-Chern theorem for vector bundles based on an approach by Liviu Nicolaescu.

Pseudodifferential operators come up in a natural way in the context of L\'evy processes. The Markovian semigroup of a stochastic differential equation driven by a L\'evy process has as generator a pseudodifferential operator. These operator are nonlocal, and exhibit a different dynamical behaviour. E.g.\ in the classical theory of thermodynamics, thermal signals propagate with infinite speed, local actions and cumulative behavior are neglected. However, this infinite speed of propagation of thermal signals is not realistic. Modelling heat conduction with pseudodifferential operator one can model finite speed of propagation, or even wave like behaviour. To treat this kind of pseudodifferential operators, one has mainly two approaches on the disposal. One can interpret the pseudodifferential operator as the infinitesimal generator of the Markovian semigroup of an stochastic differential equation driven by a L\'evy process an get estimates by probabilistic techniques, like the Malliavin calculus, or one uses analytic techniques coming from partial differential equations and harmonic analysis. The properties of these operators are not only important in technical science but also in many applications coming from stochastics. E.g. if one is interested in the numerical approximation of stochastic differential equations driven by L\'evy noise, one wants often to determine the weak error of an approximation. This means calculating the difference of the semigroup of the original solution and the approximated scheme. Another example, where the analytic properties of the pseudodifferential operators comes up is in nonlinear filtering with L\'evy noise. However, this analytic properties are important also in purely deterministic context, namely, if one is interested in the solvability of the filtering equation. In the talk we would like to introduce Hoh's symbols, explain its relation to pseudodifferential operators and to present some result using harmonic analysis.

The classical Kalman-Bucy filter is a beautiful result of practical significance. Using it, within a model for the dynamics of a signal and observation process, we can recursively find the posterior distribution of the signal given observations. In practice however, we also need to estimate the dynamics, and this introduces an additional source of uncertainty into our assessments. In this talk, we will consider a model for this uncertainty, and how it quickly leads to interesting problems in optimal stochastic control.