International Workshop

Infinite Dimensional Stochastic Systems

13–16 January 2014Wittenberg, Germany

Topics:

• Stochastic analysis in Banach spaces

• Optimal control of infinitely dimensional systems

• SPDEs with fractal and Levy noise

• Random dynamical systems

• Applications of infinite dimensional stochastic systems in natural sciences and finance

Organizers:

• Wilfried Grecksch (Martin–Luther–Universitat Halle–Wittenberg)

• Ilya Pavlyukevich (Friedrich–Schiller–Universitat Jena)

• Max von Renesse (Universitat Leipzig)

• Bjorn Schmalfuß (Friedrich–Schiller–Universitat Jena)

Venue:Leucorea, Martin–Luther–Universitat Halle–Wittenberg

Financial support:

Deutsche Forschungsgemeinschaft(Grant PA 2123/2-1)

MLU Halle–Wittenberg FSU Jena U Leipzig

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Titles and abstracts

1. AHN, Vo (Queensland University of Technology, Australia)

Least-squares estimation of infinite-dimensional multifractional random fields

Data in many fields of applications display scaling or multiscaling, as in multifractional Brownianmotion, or even multifractal scaling as in random processes and fields whose sample paths possess non-trivial singularity spectrum. A smoother source of heterogeneity can be introduced through the theoryof pseudodifferential operators of variable order, defining multifractional second-order random fields.The theory of random field estimation has been extensively developed from many angles. We considerthe problem of estimation of an output random field from an input random field whose observation isaffected by additive noise with its reproducing kernel Hilbert space (RKHS) isomorphic to a fractionalSobolev space of variable order. A unique stable solution to the estimation problems is obtained underthe regularity conditions assumed on the RKHS norms involved. The results derived are applied to theclass of multifractional random fields with covariance operator defined by a con- tinuous function of aself-adjoint and elliptic pseudodifferential operator of variable order. Two numerical projection meth-ods based on the covariance eigenfunction expansion and a wavelet-based expansion of multifractionalrandom fields are employed to obtain finite-dimensional approximations of the estimation problem. Asimulation study is presented to illustrate the results.

(In collaboration with M. D. Ruiz–Medina, R. M. Espejo, J. M. Angulo and M. P. Frias (University ofGranada))

2. BESSAIH, Hakima (University of Wyoming, USA)

On a stochastic Leray-alpha model of Euler equations

We deal with the 3D inviscid Leray-alpha model. The well posedness for this problem is not known;by adding a random perturbation we prove that there exists a unique (in law) global solution.The random forcing term formally preserves conservation of energy. The result holds for initial velocityof finite energy and the solution has finite energy a.s.These results continue to hold in the 2D case.

3. BIRNIR, Bjorn (University of California in Santa Barbara, USA)

The Kolmogorov-Obukhov-She-Leveque scaling in turbulence and the resulting probability distributionsfunctions

We construct the 1962 Kolmogorov–Obukhov statistical theory of turbulence from the stochasticNavier–Stokes equations driven by generic noise. The intermittency corrections to the scaling expo-nents of the structure functions of turbulence are given by the She–Leveque intermittency corrections.We show how they are produced by She–Waymire log-Poisson processes, that are generated by theFeynmann–Kac formula from the stochastic Navier–Stokes equation. We find the Kolmogorov–Hopfequations and compute the invariant measures of turbulence for 1-point and 2-point statistics. Thenprojecting these measures we find the formulas for the probability distribution functions (PDFs) of thevelocity differences in the structure functions. In the limit of zero intermittency, these PDFs reduceto the Generalized Hyperbolic Distributions of Barndorff–Nilsen. The PDF for all the moments of thevelocity differences can be written as a convolution of a continuous and a discrete distribution.

4. BLOMKER, Dirk (University of Augsburg, Germany)

Front motion in the one-dimensional stochastic Cahn–Hilliard equation

We describe the joint motion of multiple kinks or interfaces for the one-dimensional Cahn–Hilliardequation on a bounded interval perturbed by small additive noise. The approach is based on anapproximate slow manifold, where the motion of interfaces are described by the projection onto themanifold. This was used by Bates and Xun (1994/95) to verify metastable behaviour of solutions withmultiple kinks.

Here we derive a stochastic differential equation for the motion of the interfaces. The approximationis (with high probability) valid until an interface breaks down, or until times that are large comparedto any negative power of the small interaction length.

(A joint work with Dimitra Antonopoulou and Georgia Karali)

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5. CARABALLO, Tomas (University of Sevilla, Spain)

Recent results on stability of delay evolution equations with stochastic perturbations

We will report on some recent results concerning the stability of solutions of delay stochastic evolutionequations. Almost sure stability as well as mean square stability will be analyzed. Also stability withgeneral decay rate will be shown in some particular cases. We will emphasize that, using differentLyapunov functionals, we will be able to show how one can obtain different stability regions for somestochastic partial differential equations

6. DOROGOVTSEV, Andrey (National Academy of Sciences, Kiev, Ukraine)

Stochastic flows and random maps

In this talk we present a conditions under which the stochastic flow with the singular interaction can berepresented as a solution of SDE with the space-time martingale. The properties of the correspondingrandom maps in the functional spaces are discussed. We consider the fractional step method for thestochastic flows and the possibility to recover the noise from the flow in the singular case

7. GABIH, Abdelali (Cadi Ayyad University, Marrakesh, Morocco)

Dynamic programming equations for portfolio optimization under partial information with expert opin-ions

This paper investigates optimal portfolio strategies in a market where the drift is driven by an un-observed Markov chain. Information on the state of this chain is obtained from stock prices andexpert opinions in the form of signals at random discrete time points. As in Frey et al. (2012),Int. J. Theor. Appl. Finance, 15, No. 1, we use stochastic filtering to transform the original prob-lem into an optimization problem under full information where the state variable is the filter for theMarkov chain. This problem is studied with dynamic programming techniques and with regularizationarguments. Using results from the recent literature we obtain the existence of classical solutions to thedynamic programming equation in a regularized version of the model. From this the optimal strategyin the regularized model is straightforward to compute. We give convergence results which show thatthis strategy is ε-optimal in the original model.

(A joint work with Rudiger Frey and Ralf Wunderlich)

8. GEISS, Stefan (University of Jyvaskyla)

Decoupling on the Wiener space and applications to BSDE’s

We consider a decoupling technique on the Wiener space to define anisotropic Besov spaces, i.e. Banachspaces that describe the fractional smoothness of a random variable in certain directions. Our class ofspaces contains the classical Besov spaces, but also new spaces that are needed to investigate the Lp-variation of the solution of a Backward Stochastic Differential Equation (BSDE) where the generatorf is of Lipschitz or quadratic type, and the BSDE is driven by a d-dimensional Brownian motionW . The point is, that there are no structural assumptions on the terminal condition ξ imposed. Bymeasuring the anisotropic singularities of ξ and f we obtain assertions about the Lp-variation of thesolution (Y ;Z) which are the basis for approximation schemes to our BSDE. As an intermediate stepwe derive that our anisotropic Besov spaces are stable with respect to non-linear expectations generatedby BSDEs

(A joint work with Juha Ylinen)

9. GESS, Benjamin (TU Berlin, Germany)

Finite time extinction for stochastic sign fast diffusion and self-organized criticality

We will first shortly review the informal derivation of a continuum limit for the Bak–Tang–Wiesenfeldmodel of self-organized criticality. This will lead to the stochastic sign fast diffusion equation. Akey property of models exhibiting self-organized criticality is the relaxation of supercritical states intocritical ones in finite time. However, it has remained an open question for several years whether thecontinuum limit — the stochastic sign fast diffusion — satisfies this relaxation in finite time. We willpresent a proof of this.

3

10. HAN, Xiaoying (Auburn University, USA)

Numerical formulation of particle filtering for nonlinear stochastic systems

As an approximation of the optimal stochastic filter, particle filter is a widely used tool for numericalprediction of complex systems when observation data are available. In this work we conduct an erroranalysis from a numerical analysis perspective. That is, we investigate the numerical error, which isdefined as the difference between the numerical implementation of particle filter and its continuouscounterpart, and demonstrate that the error consists of discretization errors for solving the dynamicequations numerically and sampling errors for generating the random particles. We then establish convergence of the numerical particle filter to the continuous optimal filter and provide bounds for theconvergence rate. Remarkably, our analysis suggests that more frequent data assimilation may lead tolarger numerical errors of the particle filter. Numerical examples are provided to verify the theoreticalfindings.

11. HAUSENBLAS, Erika (MU Leoben, Austria)

Controllability and qualitative properties of the solutions to SPDEs driven by boundary Levy noise

Let u be the solution to the following stochastic evolution equation{du(t, x) = Au(t, x) dt+B σ(u(t, x)) dL(t), t > 0;

u(0, x) = x(1)

taking values in an Hilbert space H, where L is a R valued Levy process, A : H → H an infinitesimalgenerator of a strongly continuous semigroup, σ : H → R bounded from below and Lipschitz continuous,and B : R→ H a possible unbounded operator. A typical example of such an equation is a stochasticPartial differential equation with boundary Levy noise. Let P = (Pt)t≥0 the corresponding Markoviansemigroup.

We show that, if the system {du(t) = Au(t) dt+B v(t) dt, t > 0;

u(0) = x

is approximate controllable in time T > 0 with control v, then under some additional conditions onB and A, for any x ∈ H the probability measure P?

T δx is positive on open sets of H. Secondly weinvestigate under which condition on the Levy process L and on the operators A and B the solution ofEquation (1) is asymptotically strong Feller, respectively, has a unique invariant measure. We applythese results to the damped wave equation driven by Levy boundary noise.

12. HINZ, Michael (Bielefeld University, Germany)

Stochastic analysis for magnetic Schrodinger operators on fractals

The objectives we discuss are (1) how to generalize a known connection between differential 1-forms andstochastic integrals to the case of processes on fractals and (2) how to establish a related Feynman–Kac–Ito formula for magnetic Schrodinger operators. We will use stochastic analysis for additivefunctionals of Markov processes as discussed by Fukushima, Ikeda, Manabe, Nakao, Fitzsimmons,Kuwae and others. This allows to isomorphically identify differential 1-forms and martingale additivefunctionals of finite energy and to verify a Feynman–Kac–Ito theorem. Additional remarks will addressinvariant measures for perturbations of Markov processes.

13. HOGELE, Michael (Postdam University, Germany)

Metastability of a SPDE driven by α-stable Levy noise for small intensity

After a short introduction we will explain the first exit problem for a non-linear reaction-diffusionequation with finitely many stable states perturbed by infinite-dimensional-stable Levy noise at smallintensity ε→ 0. After that, it will be shown that on the characteristic time scale the solution convergesto a Markov chain switching between the (deterministic) stable states. This phenomenon ist sometimesreferred to as metastability.

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14. JENTZEN, Arnulf (ETH Zurich, Switzerland)

Approximations and regularities for nonlinear stochastic ordinary and partial differential equations

In this talk we analyze how smooth the solution of a nonlinear stochastic differential equation (SDE;by which we mean a stochastic ordinary or partial differential equation) depends in the strong Lp-senseon the initial value as well as the related question of convergence rates for numerical approximationsof the considered nonlinear SDE.

In the first part of this talk we give an example of an SDE with a globally bounded and smooth driftcoefficient and a constant diffusion coefficient whose solution does in the strong Lp-sense not dependsmoothly on the initial value. In addition, we demonstrate that the Euler–Maruyama approximationscheme converges without any arbitrarily small polynomial rate of convergence to the solution processof this SDE.

In the second part of this talk we present a sufficient condition that ensures that the solutions ofnonlinear SDEs depend smoothly on the initial values. Based on this regularity analysis, we thenestablish strong convergence rates for suitable numerical approximations of the considered nonlinearSDEs. We illustrate these results by a few example SDEs from finance, physics and biology.

The first part of this talk is based on a joint work with Martin Hairer and Martin Hutzenthaler. Thesecond part of this talk is based on joint works with Sonja Cox, Martin Hutzenthaler and XiaojieWang.

15. KUEHN, Christian (Vienna University of Technology, Austria)

Stochastic operator perturbations of travelling waves

In this talk, I shall discuss a problem at the interface of probability, functional analysis, dynamicalsystems and partial differential equations. In particular, I shall outline the study of travelling waves forNagumo-type equations driven by nonlocal operators arising as macroscopic limits of jump-diffusionprocesses. The technique merges several tools such as fundamental properties of semigroups associatedto Levy processes and a priori estimates to obtain sub- and super- solutions which yield the existence,stability and uniqueness of travelling waves; this is joint work with Franz Achleitner (Vienna).

16. LISEI, Hannelore (Babes–Bolyai University, Cluj–Napoca, Romania)

Stochastic Schrodinger equation with a cubic nonlinearity

Stochastic Schrodinger equations with cubic nonlinearity perturbed by multiplicative Gaussian noiseare investigated. In order to prove the existence and uniqueness of the variational solution, a furtherprocess will be introduced which allows to transform the multiplicative Schrodinger problem into apathwise one. We use Galerkin approximations and compact embedding results.

17. LU, Kening (Brigham Young University, USA)

Entropy, chaos and weak horseshoe for infinite dimensional random dynamical systems

In this talk, we present an answer to the long standing problem on the implication of positive entropyof a random dynamical system. We study C0 infinite dimensional random dynamical systems in aPolish space, do not assume any hyperbolicity, and prove that chaos and weak horseshoe exist insidethe random invariant set when its entropy is positive. This result is new even for finite dimensionalrandom dynamical systems and infinite dimensional deterministic dynamical systems generated byeither parabolic PDEs or hyperbolic PDEs. We mention that in general one does not expect to havea horseshoe without assuming hyperbolicity. For example, consider the product system of a circlediffeomorphism with an irrational rotation number and a system with positive entropy. This productsystem has positive entropy and a weak horseshoe, but has no horseshoe. This is a joint work withWen Huang.

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18. MILLET, Annie (University Paris 1, France)

On the splitting up method for the 2D stochastic Navier–Stokes equations on the torus

We deal with the convergence of an iterative scheme for the 2-D stochastic Navier–Stokes Equationson the torus and prove an estimate of the approximation error.

The stochastic system is split into two problems which are simpler for numerical computations. Inparticular, we prove that the strong speed of the convergence in probabilit y is almost 1/2. This isshown by means of an L2 convergence localized on a set of arbitrary large probability. The assumptionson the diffusion coefficient, which may depend on the gradient of the solution or not, depend on thefact that some multiple of the Laplace operator is present or not with the multiplicative stochasticterm.

This is a joint work with H. Bessaih and Z. Brzezniak.

19. PESZAT, Szymon (Polish Academy of Sciences and AGH University of Science and Technology,Krakow, Poland)

Second order PDEs with Dirichlet white noise boundary conditions

The talk will be based on a joint work with Zdzislaw Brzezniak (York), Ben Goldys (Sydney), FrancescoRusso (Paris) and Jerzy Zabczyk (Warsaw). The paper will appear in J. Evol. Equ.

We study the Poisson and heat equations on bounded and unbounded domains with smooth boundarywith random Dirichlet boundary conditions. The main novelty of this work is a convenient frameworkfor the analysis of such equations excited by the white in time and/or space noise on the boundary. Ourapproach allows us to show the existence and uniqueness of weak solutions in the space of distributions.Then we prove that the solutions can be identified as smooth functions inside the domain, and finallythe rate of their blow up at the boundary is estimated. A large class of noises including Wiener andfractional Wiener space time white noise, homogeneous noise and Levy noise is considered

20. RIEDLE, Markus (King’s College London, UK)

Stochastic integration with respect to cylindrical Levy processes in Hilbert spaces

by Based on the classical theory of cylindrical processes and cylindrical measures we introduce cylin-drical Levy processes as a natural generalisation of cylindrical Wiener processes. We continue to char-acterise the distribution of cylindrical Levy processes by a cylindrical version of the Levy–Khintchineformula.

In Hilbert spaces we introduce a stochastic integral for operator-valued stochastic processes with respectto cylindrical Levy processes. We apply the developed theory to derive the existence of a solution fora Cauchy problem and to consider spatial and temporal regularity and irregularity properties of thesolution. (parts of this talk are based on joint works with D. Applebaum or A. Jakubowski)

21. RUFFINO, Paulo (Universidade Estadual de Campinas, Brazil)

Averaging principle on product space: application on the topology of submanifolds

Consider an SDE on a product manifold whose trajectories are constant on the second space. Weinvestigate the effective behaviour of a perturbation of order ε in the second coordinate. An averageprinciple is shown to hold such that the second coordinate converges to the solution of a deterministicODE, according to the average of the perturbing vector field with respect to invariant measures, as εgoes to zero. An estimate of the rate of convergence is given. These results generalize the geometricalscope of previous approaches, including completely integrable stochastic Hamiltonian system (X. M. Li,Nonlinearity, 2008). We apply this result to Brownian motion on compact submanifolds of Euclideanspaces, such that the coefficients of the corresponding deterministic ODEs in the vertical coordinateare given by the Euler characteristics of the submanifolds.

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22. SCHEUTZOW, Michael (TU Berlin, Germany)

Exponential growth rate for a singular linear stochastic delay differential equation

We study the very simple one-dimensional delay equation dX(t) = X(t − 1)dW (t) and establish theexistence of a deterministic exponential growth rate of a suitable norm of the solution via a Furstenberg–Hasminskii-type formula.

23. SHEVCHENKO, Georgiy (Taras Shevchenko National University of Kyiv, Ukraine)

Mixed stochastic equations with delay

The talk will be devoted to mixed stochastic differential equation with delay of the following form:

X(t) = X0 +

∫ t

0

a(s,Xs)ds+

∫ t

0

b(s,Xs)dWs +

∫ t

0

c(s,Xs)dBHs .

This equation is driven by both standard Wiener process W and fractional Brownian motion BH withthe Hurst parameter H > 1/2 The coefficients a, b, c depend on the past Xs = {Xs−t, t ∈ [0, r]} ofthe process X. I will give results concerning unique solvability of such equations, integrability of theirsolutions, and convergence of the solutions under convergence of coefficients.

24. STANNAT, Wilhelm (TU Berlin, Germany)

Stochastic stability of traveling waves in nerve axon equations

We consider stochastic partial differential equations modeling the propagation of the action potentialalong the nerve axon of a single neuron subject to channel noise fluctuations, including stochasticFitzHugh–Nagumo systems. Stochastic stability of the action potential is proven using functionalinequalities and an implicitly defined phase adaption. Our approach is new even for the deterministiccase. A stochastic differential equation for the speed of the action potential is derived that allows todecompose the stochastic dynamics into the propagating action potential and noise uctuations. Ourapproach also allows to calculate the probability of a propagation failure w.r.t. the underlying channelnoise fluctuations.

25. STARKLOFF, Hans–Jorg (University of Applied Sciences Zwickau, Germany)

On some approximate methods for Bayesian inversion for pdes with random parameters

Random coeffcients and other random parameters of partial differential equations occurring in realworld models and investigated in natural or engineering sciences are often not known and also notdirectly observable. So one has to solve an inverse problem in order to identify the parameters, basedon corrupted by noise measurements of related quantities. One recent approach to solve this problemuses appropriate stochastic and statistical models for this task, especially Bayesian techniques. Inthe talk some approximate methods for the Bayesian inversion problem, like Ensemble Kalman Filteror Polynomial Chaos Expansion Kalman Filter are investigated. It is given the stochastic modelunderlying these methods and it is shown, that they in principle cannot solve the problem of fullBayesian inversion.

26. TAPPE, Stefan (Leibnitz Universitat Hannover, Germany)

A decomposition result for Lvy driven SPDEs with affine realizations

We investigate when a stochastic partial differential equation (SPDE) driven by Levy processes admitsan affine realization; that is, its mild solutions stay on a finite dimensional submanifold, which is affinewith respect to one dimension representing the time.

For this purpose, we decompose the state space into three subspaces; the so-called quasi-exponentialpart, which is invariant with respect to the generator of the SPDE, the so-called Riccati part, whichgives rise to more general systems of ordinary differential equations, and the remaining part whichcomplements the state space.

We will provide geometric conditions on the characteristics of the SPDE which are necessary andsufficient for the existence of an affine realization. In particular, we will show that every SPDE with anaffine realization can be reduced to the quasi-exponential case. Our results are illustrated by examplesfrom natural sciences and economics.

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27. TUDOR, Ciprian (University of Lille, France)

Variations of the solution to the stochastic heat equation with fractional-colored noise

We will discuss recent results on the existence of the solution to the heat equation driven by a Gaussiannoise which behaves as a fractional Brownian motion in time and it has correlated spatial structure.We will present various properties of this solution: sharp regularity of the sample paths, its law andits scaling properties. Using Malliavin calculus, we analyze the asymptotic behavior of the quadraticvariations of the solution.

28. VALERO, Jose (Universidad Miguel Hernandez de Elche, Alicante, Spain)

Stabilization of reaction-diffusion equations without uniqueness by multiplicative Ito noise

We prove that the asymptotic behavior of partial differential inclusions and partial differential equationswithout uniqueness of solutions can be stabilized by adding some suitable Ito noise as an externalperturbation.

We consider first a reaction-diffusion equation in which the uniqueness of solutions of the Cauchyproblem is not guaranteed. Moreover, the autonomous equations (without noise) is not dissipativeand a global attractor does not exists. After adding a suitable multiplicative Ito noise we define amultivalued random dynamical system and prove that a random attractor exists. Therefore, the noisestabilizes the equations in the sense that a non-dissipative autonomous equations is transformed intoa dissipative random one. Also, with extra assumptions on the noise we can obtain that the attractorconsists of a single stationary point.

On the other hand, we study also a deterministic partial differential inclusion, proving that a simplemultiplicative Ito noise produces and stabilization effect in the solutions.

29. VERAAR, Mark (TU Delft, Holland)

A new approach to stochastic evolution equations with adapted drift

In this talk I will explain a new approach to stochastic evolution equations with an unbounded drift Awhich is dependent on time and the underlying probability space in an adapted way. It is well-knownthat the semigroup approach to equations with random drift leads to adaptedness problems for thestochastic convolution term. I will explain a new representation formula for the stochastic convolutionwhich avoids integration of nonadapted processes. Connections with other solution concepts such asweak solutions will be given and the usual regularity properties will be shown. The approach can beapplied in the study of semilinear problems with random drift.

This is based on joint work with Matthijs Pronk.

30. ZABCZYK, Jerzy (Polish Academy of Sciences, Warsaw, Poland)

On linear stochastic Volterra equations

The talk is devoted to time regularity of the solutions to linear stochastic Volterra equations on aHilbert space. The main tool applied is the dilation theorem of Nagy. The presentation is based onthe paper

S. Peszat and J. Zabczyk, Time regularity for stochastic Volterra equations by the dilation theorem.J. Math. Anal. Appl. 409 (2014), 676–683.

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31. ZADOROZHNIY, Vladimir (Voronezh State University, Russia)

On linear chaotic resonance in vortex motion

For vortex motion the linear mathematical model with random coefficients is considered:

dx

dt= ε1(t)x+ f(t),

dr

dt= ε2(t)x,

dz

dt= ε3(t),

with initial condtions x(t0) = x0, r(t0) = r0, z(t0) = z0. Here x is an angular velocity, r a distance tothe z-axis, ε1(t), ε2(t), ε3(t) are independent stochastic processes, f another stochastic process.

We find the formulas for the first two moment functions of solutions und the condition∫ t

t0

M(ε1(s)) ds+1

2

∫ t

t0

∫ t

t0

(M(ε1(s1)ε2(s2))−M(ε1(s1))M(ε2(s2))

)ds1ds2 → +∞

as t→ +∞ for linear chaotic resonance. In this case mean angular velocity M(x(t)) increases.

32. ZAHLE, Martina (FSU Jena, Germany)

SDE with mixed driving — a pathwise approach

We consider SDE in Rn with time dependent (not necessarily adapted) nonlinear random coeffcients,where one continuous driving process Z0 admits a generalized quadratic variation. The other drivingprocesses Z1, . . . , Zm may be vector-valued and possess sample paths in fractional Sobolev spaces oforder > 1/2. In particular, one Brownian motion and m multifractional Brownian motions can betreated. The corresponding stochastic integrals are determined as generalized forward integrals. Ahigher-dimensional version of the pathwise Doss–Sussman approach to global solutions is developedwhich combines the corresponding Ito formula with fractional calculus via two auxiliary differentialequations, one ordinary and the other fractional. This extends our former local approach. For the caseZ0 = 0 we also indicate some extensions to Banach spaces.

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Posters

1. da COSTA, Paulo Henrique (Universidade Estadual de Campinas, Brazil)

Degenerate semigroups and stochastic flows of mappings in foliated manifolds

2. GOMES, Andre Oliveira (HU Berlin, Germany)

First exit times for Levy driven diffusions with exponentially light tails

3. HOFMANOVA, Martina (Leipzig University, Germany)

Well-posedness for stochastic conservation laws

4. IZYUMTSEVA, Olga, (National Academy of Sciences, Kiev, Ukraine)

The geometry of a covariance of Gaussian integrators and local times

5. KONAROVSKYI, Vitalii (Chernivtsi National University, Ukraine / FSU Jena, Germany)

Asymptotic properties of heavy diffusion particles system

6. OLIVERA, Christian (Universidade Estadual de Campinas, Brazil)

Well-posedness of first order semilinear PDEs by stochastic perturbation

We show that first order semilinear PDEs by stochastic perturbation are well-posed for globally Holdercontinuous and bounded vector field, with an integrability condition on the divergence. The proof isbased on in the stochastic characteristics method and a version of the commuting Lemma.

References:

[1] Flandoli F., Gubinelli M., Priola, E. 2010. Well-posedness of the transport equation by stochasticperturbation, Invent. Math., 180(1): 1–53.

[2] Kunita, H. 1984. First order stochastic partial differential equations, in Proceedings of the TaniguchiInternational Symposium on Stochastic Analysis, North-Holland Mathematical Library, 249–269.

[3] Kunita, H. 1982. Stochastic differential equations and stochastic flows of diffeomorphisms, LectureNotes in Mathematics, 1097: 143–303.

[4] Kunita, H. 1990. Stochastic flows and stochastic differential equations, Cambridge University Press.

[5] Olivera, C. 2014, Well-posedness of first order semilinear PDEs by stochas- tic perturbation. Non-linear Analysis.

This research is partially supported by FAPESP 2012/18739-0.

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