The workshop aims to bring together the expertise of the research groups in probability theory and stochastic analysis from the area Halle-Jena-Leipzig. Topics will include the analysis of (nonlinear) stochastic partial differential equations, dynamics of random systems, Levy driven S(P)DE, numerical aspects of S(P)DE among others.

There will be invited talks only. The talks will be around 30 minutes.

The conference will start on Friday at 9 am and close on Saturday at 3 pm. There is no registration, participation is free for academics and practitioners. Please contact Katja Heid to reserve accomodation for you if needed until end of August 2017. Travel reimbursement is possible.

We consider a variant of the basic problem of the calculus of variations, where the Lagrangian is convex and subject to randomness adapted to a Brownian filtration. We solve the problem by reducing it, via a limiting argument, to an unconstrained control problem that consists in finding an absolutely continuous process minimizing the expected sum of the Lagrangian and the deviation of the terminal state from a given target position. Using the Pontryagin maximum principle we characterize a solution of the unconstrained control problem in terms of a fully coupled forward-backward stochastic differential equation (FBSDE). We use the method of decoupling fields for proving that the FBSDE has a unique solution.

This talk is concerned with regularization by noise for the continuity equation. We will review the classical deterministic theory of Di-Perna/Lions and the regularization by linear, multiplicative noise initiated by Flandoli/Gubinelli/Priola. Finally, I will present a new result, joint with Benjamin Gess, which allows for non-linear, partially degenerate noise in conservative form.

A drift parameter estimation problem is studied for a linear parabolic stochastic partial diﬀerential equation driven by a multiplicative cylindrical fractional Brownian motion with Hurst index h ∈]1/2, 1[ and a multiplicative Poisson process with values in a Hilbert space. Equations are introduced for the Galerkin approximations of the mild solution process. A mean square estimation criterion is used for these equations. It is proved that the estimate is unbiased and weakly consistent for the original problem.

We formulate conditions for existence and uniqueness for SDE's driven by an fbm with Hurst parameter in $(1/3,1/2]$. We then prove that under particular conditions on the coefficients of this equation for initial conditions in a random neighborhood of zero the solution converges exponentially fast to the trivial solution.

In this talk, which is based on joint work with Benjamin Gess, I will present a pathwise well-posedness theory for porous medium equations driven by conservative, nonlinear noise. The solution theory is based upon the kinetic formulation of the equation. On this level, the noise enters linearly, and the corresponding system of stochastic characteristics may be understood as rough paths. This yields a well-defined, pathwise notion of solution, for which test functions are transported along the inverse system of characteristics. I will discuss the existence and uniqueness of such solutions, and I will mention some estimates proving that, locally in time, pathwise solutions possess the basic spacial regularity of solutions to the deterministic porous medium equation.

The talk is devoted to a model of interacting diffusion particles on the real line. We propose a family of reversible fragmentating-coagulating processes of particles of varying size-scaled diffusivity with strictly local interaction. The construction is based on a new family of measures on the set of real increasing functions as reference measures for naturally associated Dirichlet forms. The processes are infinite dimensional versions of sticky reflecting dynamics on a simplex. We also identify the intrinsic metric leading to a Varadhan formula for the short time asymptotics with the Wasserstein metric for the associated measure valued diffusion. Joint work with Max von Renesse.

with Giovanni Conforti
We establish quantitative results about the bridges of the Langevin dynamics and the associated reciprocal processes. They include an equivalence between gradient estimates for bridge semigroups and couplings, comparison principles, bounds of the distance between bridges of different Langevin dynamics, and a Logarithmic Sobolev inequality for bridge measures.

We prove a path-by-path regularization by noise result for scalar conservation laws. In particular, this proves regularizing properties for scalar conservation laws driven by fractional Brownian motion and generalizes the respective results obtained in [G., Souganidis; Comm. Pure Appl. Math. (2017)]. We show that $(\rho,\gamma)$-regularity is a sufficient path-by-path condition implying such regularizing effects. In addition, we introduce a new path-by-path scaling property which is also shown to be sufficient to imply regularizing effects.

We discuss an initial boundary value problem of stochastic viscoelastic wave equation driven by multiplicative noise involving the nonlinear damping term $|u_t|^{q-2}u_t$ and a source term of the type $|u|^{p-2}u$. We firstly establish the local existence and uniqueness of solution by the iterative technique truncation function method. Moreover, we also show that the solution is global for $q\geq p$. Lastly, by modifying the energy functional, we give sufficient conditions such that the local solution of the stochastic equations will blow up with positive probability or explode in energy sense for $p>q$.

We consider a backward stochastic Volterra integral equation [BSVIE] in the Banach space $E=L^q(\mathbf{S}, \Sigma, \mu)$, where $\mu$ is $\sigma$-finte measure. The stochastic integral is defined with respect to an infinite dimensional Wiener process. Under appropriate assumptions, the existence and uniqueness of an adapted solution of BSVIE are being proofed by using martingale representation theorem in Banach space $E$ and Banach fixed-point theorem. Some properties of the solution are also discussed. We prove a duality principle between BSVIEs and forward It\^o Volterra stochastic integral equations with respect to a cylindrical Wiener process.

1. fractional Brownian motion (fBm) and an integral with respect to the fBm2. Properties of the solution of an Ornstein-Uhlenbeck process3. relation to partial differential equations4. Estimation of the solution with Monte-Carlo methods5. Introduction of Importance sampling to reduce the variance of the estimator

We consider an SPDE with a Lipschitz continuous nonlinearity driven by a multiplicative noise. Its mild solution generates a random dynamical system and we discuss the existence of a random attractor. Applying a cut-oﬀ function to the nonlinearity we obtain a local unstable manifold for the random dynamical system. The local unstable manifold contains an open subset which lies also in the related random attractor. This subset is used to obtain a lower bound on the Hausdorﬀ dimension of the random attractor.

We consider an optimal control problem for incompressible random ﬂows governed by the stochastic Navier-Stokes equations in a multidimensional domain. The control problem introduced in this talk is motivated by common strategies, such as tracking a desired velocity ﬁeld and minimizing the enstrophy energy. Since existence and uniqueness results of mild solutions are provided only locally in time, the cost functional has to incorporate stopping times dependent on the controls. We state an existence and uniqueness result for the optimal control and using a stochastic maximum principle, we derive necessary optimality conditions. As a consequence, we get an explicit formula of the optimal control based on the adjoint equation, which is given by a backward SPDE. Moreover, we show that suﬃcient optimality conditions are satisﬁed. This enables us to obtain unique solutions of control problems constrained by the stochastic Navier-Stokes equations for two-dimensional as well as for three-dimensional domains.

Iterative scaling is a widely used method to solve maximum entropy problems.
There are many variant iterative scaling algorithms. In order to reconnect two often-used iterative scaling algorithms, my bachelor thesis categorizes the different types of algorithms and analyses their relations. Four different implemented algorithms are used to study and compare their convergence behaviour.

In 1966, E. Nelson established a new interpretation of quantum mechanics, whereby the particles follow some conservative diffusion process, i.e. forward-backward stochastic differential equations (FBSDEs), which are equivalent to the Schrödinger equation (1). Until now, this equivalence has been applied in such a way that a known solution to the Schrödinger equation is used to integrate the stochastic differential equations numerically and analyze the statistical properties of the sample paths. Compared to the options available to treat classical systems this is limited, both in methods and in scope.However, in analogy to classical mechanics, we show that finding the Nash equilibrium for a stochastic optimal control problem, which is the quantum equivalent to Hamilton's principle of least action, allows to derive two aspects (2): i) the Schrödinger equation as the Hamilton-Jacobi-Bellman equation of this optimal control problem and ii) a set of quantum dynamical equations which are the generalization of Hamilton's equations of motion to the quantum world. We derive their general form for the $n$-dimensional, non stationary and the stationary case.The resulting forward-backward stochastic differential equations can be solved numerically without using the solution of the Schrödinger equation, which is done for many different systems, e.g. one- and two-dimensional harmonic oscillator, one-dimensional double-well potential or hydrogen atom.E. Nelson (1966). Derivation to the Schrödinger Equation from Newtonian Mechanics. Phys. Rev. 150(4), 1079--1085 J. Köppe, W. Grecksch, W. Paul (2017). Derivation and application of quantum Hamilton equations of motion. Ann. Phys. 529(3), 1600251

Participants

Stefan Ankirchner

FSU Jena

Mahdi Azimi

Martin-Luther-Universität Halle

Stefan Bachmann

Universität Leipzig

Michael Beyer

Martin-Luther-Universität Halle

Markus Böhm

FSU Jena

Maximilian Büttner

Martin-Luther-Universität Halle

Joscha Diehl

MPI MIS

Benjamin Fehrman

MPI MIS

Liang Fei

Xi'an + FSU Jena

Alexander Fromm

FSU Jena

Benjamin Gess

Max-Planck-Institut für Mathematik in den Naturwissenschaften

Wilfried Grecksch

Martin-Luther-Universität Halle

Gero Hillebrandt

Martin-Luther-Universität Halle

Vitalii Konarovskyi

Universität Leipzig

Jeanette Köppe

Martin-Luther-Universität Halle

Carlotta Langer

Martin-Luther-Universität Halle

Tobias Lehmann

Universität Leipzig

Hua Li

Duc Luu

MPI MIS

Marius Neuß

MPI MIS

Markus Patzold

Martin-Luther-Universität Halle

Björn Schmalfuss

FSU Jena

Scott Smith

MPI MIS

Dat Tran

MPI MIS

Christoph Trautwein

MPI Magdeburg

Max von Renesse

Universität Leipzig

Tobias Weihrauch

Universität Leipzig

Scientific Organizers

Benjamin Gess

Max-Planck-Institut für Mathematik in den Naturwissenschaften

Administrative Contact

Katja Heid

Max Planck Institute for Mathematics in the Sciences
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