Research

Research interests:

Since 2010, I have been publishing in the field of Theory of Stochastic Partial Differential equations which is at the interface of Probability theory (or Stochastic Analysis) and Mathematical Analysis (or Theory of Partial Differential Equations). I have worked in the analysis of SDEs/SPDEs with pure jump and Gaussian noise which find their application in the mathematical investigation of the turbulence or Mathematics of Finance, Filtering theory,....

Brief description of recent results

A full list of my articles either still in preparation, or accepted or in print can be found under the List of Publication link.

Below I will outline some ongoing projects, published or accepted mathematical results during 2014-2015.

(1) Numerical approximation of Wiener driven SPDEs: Recently, I started to have interest in the numerical treatment of SPDEs.

(a) Along with Bessaih, Hausenblas, Randrianasolo I also studied the discretization of stochastic evolution equation modeling the shell models and Ginzburg-Landau equations driven by multiplicative noise in the recent preprint [PR4]. Experience obtained from the aforementioned publications altogether with [PR11].

(b) In [PR11] I studied SPDEs with multiplicative Wiener noise for the second grade fluids and Lagrangian averaged Euler equations in non-smooth domain. Solutions of a time-discretization of the mixed formulation in term of linearized system of stochastic Stokes-like and transport systems is shown to converge in distribution to the solution of the stochastic models for for the second grade fluids and Lagrangian averaged Euler equations.

(2) Ergodicity of SPDEs driven by pure jump noise: From the 1st of July 2013 to the 31st July 2015, I was the Principal Investigator of the FWF Lise Meitner fellowship M1487 Ergodic Properties of the Lévy Stochastic Shell Models. Here are the mathematical results from this fellowship.

(a) Dr Fernando, Prof Hausenblas and I have recently proved several important results related to a large class of evolution equations with pure jump noise perturbation in [PR9]. In fact, we have proved support theorem and irreducibility of finite and infinite dimensional Ornstein-Uhlenbeck process driven by tempered stable processes. These results is used in the same paper to prove the irreducibility of a class of tempered stable process driven SPDEs which can describe, for instance, the stochastic 2-dimensional Navier-Stokes, Magnetohydrodynamics equations, 3-dimensional Leray-α model of turbulence, the GOY and Sabra shell models. We also prove the exponential mixing of the GOY and Sabra shell models. These results are all new and a great progress in the investigation of ergodicity of stochastic hydrodynamical systems driven by pure jump noise. This paper will be submitted for publication in September the latest.

(b) In the submitted [PR1], Bessaih, Hausenblas and I studied a stochastic evolution equation for Shell (SABRA & GOY) models with pure jump levy noise \sum_{k=1}^\infty l_k*e_k. Here {l_k; k \in \mathbb{N} } is a family of i.i.d real-valued pure jump Lévy processes and { e_k; k \in \mathbb{N} } is an orthonormal basis of H. To deal with this difficult question we established a Bismut-Elworthy-Li lemma for stochastic Differential Equations driven by Lévy noise with non-smooth Lévy measure. This lemma is new for SDEs with pure jump noise and our paper seems to be the first work to establish the uniqueness of invariant of Navier-Stokes like equations driven by external forcing of pure jump noise.

(c) Prof. Hausenblas and I also worked on a paper about irreducibility and uniqueness of invariant measure associated to the solution of Lévy driven Heat and Wave equations. In the paper [PR10] we use the some concepts of deterministic control theory to show the irreducibility and the asymptotic strong Feller property (a notion introduced by Hairer & Mattingly) of the Markovian semigroup associated to the unique solution of the Heat and Wave equations with nonlinear stochastic perturbation. This is the first paper to discuss such relation in the context of SPDEs driven by Lévy noise.

(3) Existence of Solution of SPDEs driven by Lévy noise:

(a) In the context of SPDEs in Banach spaces but with Lévy noise, Brzeźniak, Hausenblas and I established in [PR6] general concepts such as the modified Skorokhod Embedding Theorem and a Martingale Representation like theorem for Poisson random measure. As a by-product of these results we proved the existence of martingale solution for Lévy driven SPDEs without non-Lipschitz coefficients in general Banach spaces.

(b) In [PR3], Bessaih, Hausenblas and I designed a new general framework enabling to treat several hydrodynamical systems driven by pure jump noise. By this general framework, we established the existence and uniqueness of global strong (in PDE sense) solution to two dimensional stochastic the Navier-Stokes equations, MHD equations, Magnetic Bénard problems, Boussinesq model of the Bénard convection, Shell models of turbulence and maximal local solution to the three dimensional Leray-α model driven by pure jump noise in unbounded and bounded domains.

(4) Study of the Markov semigroup of the solutions of SDEs driven by stable-like processes:

By using the theory of pseudo-differential operators and the Hoh symbol I along with Dr Fernando, Prof Hausenblas proved in [PR8] that the Markov- Feller semi-group associated to the unique solution of a stochastic differential equations with smooth and bounded coefficients and driven by a stable Lévy noise is smoothing. This result is new as the only known result in this direction is the smoothness of the semi-group associated to stable Lévy process.

(5) Wiener driven SPDEs: This section is devoted to the summary of results obtained with my collaborators during our investigation of Wiener driven SPDEs.

(a) Wiener driven SPDEs in Banach setting: Brzeźniak and I wrote a paper about uniqueness of invariant measure for stochastic evolution equations in Banach spaces, see [PR5]. In [PR5] we generalized the celebrated Bismut-Elworthy-Li formula to SPDEs in Banach spaces of Martingale type 2 and with C²-smooth norm. Note that in contrast to the SPDEs in Hilbert spaces, there are not so many papers treating the uniqueness of invariant measure of stochastic evolution equations (SEEs) in infinite dimensional Banach spaces. The paper [PR5] seems to be one of the few to give a general result to check the uniqueness of an invariant measure of a SPDEs in Banach space with C² norm.

(b) Nematic liquid crystals under stochastic external controls: In [PR7] I altogether with Profs Brzeźniak and Hausenblas studied the existence and uniqueness of local maximal and global strong solution to a Ginzburg-Landau approximation of the stochastic version of the Leslie-Eriscksen equations describing the motion of nematic liquid crystals under random perturbation. Here strong solution is understood in the sense of stochastic calculus and PDEs. By a fixed point argument we firstly prove a general result which enables us to establish the existence of local and maximal solution to an abstract nonlinear stochastic evolution equations. Secondly, we consider some functional spaces with more regularity than in the case of weak and martingale solution and we show that our problem falls within previous general framework. Therefore, we are able to establish the existence of local and maximal strong solution. In the 2-D case we prove non-explosion of the maximal solution by a method based on a choice of an appropriate energy functionals (or Lyapunov function). Thus the existence of a unique global strong solution in the 2-D case. We also established a maximum principle type and a path-wise uniqueness results. This is the first paper which treats this stochastic evolution equations.

(c) Results related to some stochastic models of turbulence: Numerical studies showed that the two dimensional Leray-α model is a good (in term of computational cost) approximation of the two dimensional Navier-Stokes equations at high Reynolds number. Prominent mathematicians proved that the laws of the solution of the stochastic Leray-α model converges to the law of the stochastic Navier-Stokes equations. However, this is a quite weak result compared to the one Bessaih and I obtained in [PR2] where we established a convergence in probability of the path solutions. We proved that the rate of convergence for the stochastic equations are better that the rate of convergence obtained by Titi and his collaborators for the deterministic systems.

References

[PR1] H. Bessaih, E. Hausenblas and P. Razafimandimby. Ergodicity of stochastic shell models driven by pure jump noise, preprint, 32 pages, 2014.

[PR2] H. Bessaih and P. Razafimandimby. On the rate of convergence of the 2-D stochastic Leray-α model to the 2-D stochastic Navier-Stokes equations with multiplicative noise. To appear in Applied Mathematics and Optimization, 25 pages, 2015.

http://link.springer.com/article/10.1007/s00245-015-9303-7

[PR3] H. Bessaih, E. Hausenblas and P. A. Razafimandimby. Local and global strong solution for hydrodynamical systems driven by multiplicative jump noise. To appear in Nonlinear Differential Equations and Applications, 31 pages, 2015. http://link.springer.com/article10.1007/s00030-015-0339-9.

[PR4] H. Bessaih, E. Hausenblas, T. Randrianasolo and P. A. Razafimandimby. Numerical approximation of stochastic evolution equations: Convergence in scale of Hilbert spaces. Preprint, ready for submission, 26 pages, 2015.

[PR5] Z. Brzeźniak and P. A. Razafimandimby. Uniqueness of invariant measure for stochastic evolution equations in Banach spaces. Submitted, 15 pages, 2014.

[PR6] Z. Brzeźniak, E. Hausenblas, and P. Razafimandimby, Martingale solutions for Stochastic Equation of Reaction Diffusion Type driven by Lévy noise or Poisson random measure, Submitted, http://arxiv.org/abs/1010.5933., 2015.

[PR7] Z. Brzeźniak, E. Hausenblas, and P. Razafimandimby. Stochastic Non-parabolic dissipative systems modeling the flow of Liquid Crystals: Strong solution. RIMS Symposium on Mathematical Analysis of Incompressible Flow, February 2013. RIMS Kôkyûroku 1875: 41–73, 2014.

[PR8] P. Fernando, E. Hausenblas and P. Razafimandimby. Analytic properties of Markov semigroup generated by Stochastic Differential Equations driven by Lévy processes. To appear in Potential Analysis, http://arxiv.org/abs/1412.1453 2015.

[PR9] P. Fernando, E. Hausenblas and P. Razafimandimby. Irreducibility and Exponential mixing of some stochastic hydrodynamical systems driven by pure jump noise. preprint, will be submitted soon, 2015.

[PR10] E. Hausenblas and P. A. Razafimandimby. Controllability and Qualitative properties of the solutions to SPDEs driven by boundary Lévy noise. Stochastic Partial Differential Equations: Analysis and Computations. 3(2):221–271, 2015.

[PR11] P. A. Razafimandimby. SPDEs with multiplicative noise for grade-two fluids and Lagrangian Averaged Euler equations on non-smooth domain: Existence, uniqueness and regularity. Preprint, ready for submission, 50 pages, 2015.