Abstract: Nonlinear white noise calculus generalizes usual stochastic analysis
by considering equations driven by higher (i.e. $\geq 2$) powers of white noise:
this is an old open problem in engineering and in physics in which recently some
important steps forward have been realized.
These results were motivated by a new approach to classical stochastic calculus
which emerged from the stochastic limit of quantum theory. The crucial step
in this new approach was the identification of both classical and quantum
stochastic differential equations (SDE) with a certain class of white noise
{\bf Hamiltonian } equations (WNHE). This identification follows from the
"time consecutive principle", one of the main results of the stochastic limit
of quantum theory, combined with some of the main new ideas of quantum probability,
namely the identification of classical SDE with a class of unitary quantum SDE
and the quantum decomposition of a classical random variable.
Wider classes of WNHE, with respect to those arising in classical SDE, were
considered in the physical and engineering literature, but no rigorous mathematical theory,
comparable for elegance and power to Ito calculus, was available for such equations.
In fact all available results were based on cut--off--and--take--limit procedures
which are far from being transparent.
One of the difficulties of this development relies on the renormalization problem,
well known in physics. The new developments, mentioned above, throw a new light
on this problem by connecting it to the impossibility of lifting certain unitary
representations of Lie algebras to the associated current algebras.
While the intrinsic connections between current algebras and infinitely divisible
classical processes were well known since the 1960's, this phenomenon and its
connection to renormalization theory seem to have been unnoticed until recently.
(preprint)
Abstract: We introduce a new method to discuss random walk in random environments in dimensions above 3. The main result is that under a symmetry assumption on the distribution of the random environment, and a small disorder assumption, the exit distributions are close to the exit distributions of ordinary random walk. This is measured in a slightly smooted total variation norm. The proof is by a multiscale analysis. (Joint work with Ofer Zeitouni, Minneapolis)
A. Bovier: Ageing and spectral properties of generatorsAbstract: Ageing is a common characterisation of slow dynmaics that is observed in numerous sochastic dynamics of complex systems, such as spin glasses. Here we adress the question how such properties manifest themselves in the spectra of the corresponding Markov generators. We consider two simple examples with very different properties: Bouchaud's REM-like trap model and Sinai's random walk in a random environment. The result reported on are joint work with Alessandra Faggionato (ROM I).
G. Chistyakov: Free convolutions and Limit TheoremsAstract: We establish a probabilistic representation for a wide class of semi-linear deterministic p.d.e.'s with potential term, including the wave equation in spatial dimensions 1 to 3. This representation is related to the classical Feynman-Kac formula for the heat equation. Our representation also applies to this parabolic equation, as well as to the telegraph and beam equations. If the potential is a (random) spatially homogeneous Gaussian noise, then this formula leads to an expression for the moments of the solution of this semi-linear stochastic p.d.e. This is joint work with C. Mueller (University of Rochester) and R. Tribe (University of Warwick).
Abstract: We construct two types of equilibrium dynamics of infinite particle systems in a Riemannian manifold $X$. These dynamics are analogs of the Glauber, respectively Kawasaki dynamics of lattice spin systems. The Glauber dynamics now is a process where interacting particles randomly appear and disappear, i.e., it is a birth-and-death process in $X$, while in the Kawasaki dynamics interacting particles randomly jump over $X$. We establish conditions on a prior explicitly given symmetrizing measures and generators of both dynamics under which corresponding conservative Markov processes exist. We also discuss a spectral gap for one type of the Glauber dynamics.
F. Martinelli: Glauber dynamics for spin models on trees: boundary conditons and mixing times (preprint1) (preprint2)Abstract: We consider a system of non-linear stochastic heat equations driven by a space-time white noise with Newmann or Dirichlet boundary conditions. We prove upper and lower bounds for the probability that the solution of this system of equations hits a given set, in terms of a capacity of the set. The proof uses tools of Malliavin calculus.
Abstract: We study the Navier-Stokes equations in dimension $3$ (NS3D) driven by a noise which is white in time. We establish that if the noise is at same time sufficiently smooth and non degenerate in space, then the weak solutions converge exponentially fast to equilibrium. We use a coupling method. The proof is much more difficult than in dimension two since, as is well-known, uniqueness is an open problem for NS3D. However, many simplifications appears since we work with non degenerate noises.
M. Ondrejat: Stochastic wave equationsAbstract: We consider the homogenization of a PDE with stationary random coefficients, in the case of one dimensional space parameter, with a highly oscillating random stationary potential. The limiting equation is either a SPDE with spatial white noise, or a SPDE with time white noise, or a deterministic PDE, depending on whether only the space parameter is accelerated, only the time parameter is accelerated, or both parameters are accelerated.
O. Popovych: Desychronization of coupled oscillators by nonlinear delayed feedbackAbstract: We propose a method for control of synchronization in ensembles of interacting oscillators. We suggest to use a nonlinear delayed feedback, where a synchronized population of oscillators is stimulated with a signal constructed by using the delayed mean field of the ensemble nonlinearly combined with its instantaneous mean field. The stimulation results in complete desynchronization of the oscillators and restores their natural frequencies, so that the oscillators rotate as if they were uncoupled. Moreover, the amplitude of the stimulation signal practically vanishes when a desynchronized state is achieved. We also discuss some other properties of the impact of nonlinear delayed feedback on stimulated ensembles of coupled oscillators. (Joint work with Christian Hauptmann and Peter A. Tass)
Abstract: Uniform gradient estimates are derived for diffusion semigroups, possibly with potential, generated by second order elliptic operators having irregular and unbounded coefficients. We first consider the $R^d$-case, by using the coupling method. Due to the singularity of the coefficients, the coupling process we construct is not strongly Markovian, so that additional difficulties arise in the study. Then, more generally, we treat the case of a possibly unbounded smooth domain of $R^d$ with Dirichlet boundary conditions. We stress that the resulting estimates are new even in the $R^d$-case and that the coefficients can be Holder continuous. Our results also imply a new Liouville theorem for space-time bounded harmonic functions w.r.t. the underlying diffusion semigroup. This is a joint work with Feng-Yu Wang (Bejing).
M. Sanz-Solé: A lattice scheme for an elliptic SPDEAbstract: We study a stochastic boundary value problem on $(0,1)^d$ of elliptic type in dimension $d\ge 4$, driven by a coloured noise. An approximation scheme based on a suitable discretization of the Laplacian on a lattice of $(0,1)^d$ is presented; we also give the rate of convergence to the original SPDE in $L^p(\Omega;L^{2}(D))$--norm, for some values of $p$. (slides)
B. Schmalfuss: Stochastic differential equations with dynamical boundary conditionsAbstract: We consider systems of non-linear parabolic stochastic partial differential equations with dynamical boundary conditions. These boundary conditions are qualitatively different from the standard, like Dirichlet, or Neumann, or Robin boundary conditions. Such conditions contain a time derivative and can be used to describe mathematical models with a dynamics on the boundary. In our model the noise is acting in the domain but also on the boundary and is presented as the temporal generalized derivative of an infinite dimensional Wiener processes. In addition, we have coefficients for the spatial differential operators depending on space and time. We prove existence and uniqueness of mild solutions to these stochastic partial differential equations and study the properties of these solutions. The stochastic equation on the boundary contains a parameter. If this parameter becomes small, then the equation on the boundary has an interpretation in a fast time scale. We prove the relative compactness of the distribution of the solution if the parameter mentioned tends to 0.
A. Stuart: SPDEs for sampling conditioned SDEsJoint work with Martin Hairer and Jochen Voss
(Warwick University).
Abstract: Sampling from the distribution of SDEs, conditional on observations, arises in
a wide range of applications including data assimilation, econometrics,
transition path sampling and signal processing. We study an approach to
sampling such distributions, using SPDEs. The SPDEs sample from the desired
distribution in their invariant measure, and are the infinite dimensional
analogue of Langevin equations in finite dimensions. The subject area gives
rise to a wealth of interesting questions in stochastic analysis and in
computational SPDEs. In this talk we will overview the subject area.
Abstract: The stochastic Anderson model is a classical object of study in the field of random medium. When this linear parabolic stochastic PDE has a multiplicative white noise potential, it is expected to grow exponentially fast in large time. We show that the rate of increase is closely related to the modulus of continuity of the potential in its space parameter. Our results improve all existing estimations of this "almost-sure Lyapunov exponent", while using new techniques based on the Malliavin calculus and Gaussian supremum estimates, which result in streamlined, more efficient proofs. (preprint)
J. Voss: Constructing SPDEs for sampling from a distribution on path spaceAbstract: A well known sampling technique in finite dimensions is Langevin Sampling where one constructs a stochastic differential equation whose stationary distribution coincides with the target distribution. We show how to transfer this idea to the infinite dimensional case by constructing Banach-space valued stochastic differential equations with a prescribed stationary distribution. Our motivation is to sample from the distribution of SDEs conditional on various types of observations. (Joint work with Martin Hairer and Andrew Stuart.)