Torstein Kastberg Nilssen

Associate Professor

Department of Mathematical Sciences

Email: torstein.nilssen@uia.no
Phone: +47 38141320

Office J2044 (Universitetsveien 25, 4630 Kristiansand, Norway)

Research groups

Pure and Applied Mathematical Analysis Research group (PAMAR)

Research

Academic interests

Nonlinear stochastic partial differential equations and variational methods. Pathwise solutions and rough paths. Stochastic (Partial) Differential Equations with irregular coeffcients. The (Stochastic) Transport Equation. Regularizing effect of noise. Malliavin calculus.

Selected publications

Antoine Hocquet, Torstein Nilssen: An Itô Formula for rough partial differential equations and some applications.
Potential Analysis (2020) https://doi.org/10.1007/s11118-020-09830-y
David Banos, Torstein Nilssen, Frank Proske: Strong existence and higher order Frechet differentiability of stochastic flows of fractional Brownian motion driven SDE's with singular drift.
Journal of Dynamics and Differential Equations (2019). https://doi.org/10.1007/s10884-019-09789-4
Preprint: http://arxiv.org/abs/1511.02717
Torstein Nilssen: Rough linear PDE's with discontinuous coeffcients - existence of solutions via regularization by fractional Brownian motion.
Electron. J. Probab. Volume 25 (2020), paper no. 34, 33 pp.
Preprint: http://arxiv.org/abs/1509.01154
Peter Friz, Torstein Nilssen Wilhelm Stannat: Existence, uniqueness and stability of semilinear rough partial differential equations.
Journal of Differential Equations, 2019, ISSN 0022-0396, https://doi.org/10.1016/j.jde.2019.09.033.Antoine Hocquet, Torstein Nilssen, Wilhelm Stannat: Generalized Burgers equation with rough transport noise.
Stochastic Processes and their Applications, 2019, ISSN 0304-4149, https://doi.org/10.1016/j.spa.2019.06.014.
Martina Hofmanova, James-Michael Leahy, Torstein Nilssen: On the Navier-Stokes equation perturbed by rough transport noise.
Journal of Evolution Equations (2018). https://doi.org/10.1007/s00028-018-0473-z
Torstein Nilssen: Quasi-linear Stochastic Partial Differential Equations with irregular Coeffcients - Malliavin Regularity of the Solution.
Stochastic Partial Differential Equations: Analysis and Computations. September 2015, Volume 3, Issue 3, pp 339-359.
Salah-Eldin A. Mohammed, Torstein Nilssen, Frank Proske: Sobolev Differentiable Stochastic Flows of SDE`s with Measurable Drift and Applications.
Ann. Probab. Volume 43, Number 3 (2015), 1535-1576.
David Banos, Torstein Nilssen: Malliavin regularity and regularity of densities of SDE's. A classical solution to the stochastic transport equation. Stochastics, (2015), DOI: 10.1080/17442508.2015.1102265.
Torstein Nilssen: Regularity of Strong Solutions of one-dimensional SDE's with discontinuous and unbounded drift.
Stochastics, An International Journal of Probability and Stochastic Processes, Volume 88, 2016 -
Issue 5
Thilo Meyer-Brandis, Torstein Nilssen, Olivier Menoukeu-Pamen, Frank Norbert Proske, Tusheng Zhang: A variational approach to the construction and Malliavin differentiability of strong solutions of SDE's.
Mathematische Annalen, Springer-Verlag Berlin Heidelberg 201310.1007/s00208-013-0916-3, 2013.

Short biography

October 2019. Associate Professor, Institute of Mathematics, University of Agder, Norway.
February 2017 - Oktober 2019. Post-doc at Technische Universität Berlin, Insitut für Mathematik.
November 2015 - July 2016. Guest researcher at the University of Southern California, Los Angeles.
March 2014 - June 2014. Guest researcher at the University of Manchester, Department of Mathematics.
February 2014 - Januar 2017. Post-doc at University of Oslo, Institute of Mathematics, Norway.
January 2013 - July 2013. Guest researcher at Humbold Universität zu Berlin, Institut für Mathematik.
January 2010 - December 2013. PhD candidate at the Universitety of Oslo, Institute of Mathematics.

Coghi, Michele; Nilssen, Torstein; Nüsken, Nikolas & Reich, Sebastian (2023). Rough mckean vlasov dynamics for robust ensemble kalman filtering. The Annals of Applied Probability. ISSN 1050-5164. 33(6B), p. 5693–5752. doi: 10.1214/23-AAP1957.
Crisan, Dan; Holm, Darryl D.; Leahy, James-Michael & Nilssen, Torstein (2022). Solution properties of the incompressible Euler system with rough path advection. Journal of Functional Analysis. ISSN 0022-1236. 283(9). doi: 10.1016/j.jfa.2022.109632. Full text in Research Archive Show summary
The present paper aims to establish the local well-posedness of Euler's fluid equations on geometric rough paths. In particular, we consider the Euler equations for the incompressible flow of an ideal fluid whose Lagrangian transport velocity possesses an additional rough-in-time, divergence-free vector field. In recent work, we have demonstrated that this system can be derived from Clebsch and Hamilton-Pontryagin variational principles that possess a perturbative geometric rough path Lie-advection constraint. In this paper, we prove the local well-posedness of the system in -Sobolev spaces with integer regularity and establish a Beale-Kato-Majda (BKM) blow-up criterion in terms of the -norm of the vorticity. In dimension two, we show that the -norms of the vorticity are conserved, which yields global well-posedness and a Wong-Zakai approximation theorem for the stochastic version of the equation.
Crisan, Dan; Holm, Darryl D.; Leahy, James-Michael & Nilssen, Torstein (2022). Variational principles for fluid dynamics on rough paths. Advances in Mathematics. ISSN 0001-8708. 404(A). doi: 10.1016/j.aim.2022.108409. Full text in Research Archive
Flandoli, Franco; Hofmanová, Martina; Luo, Dejun & Nilssen, Torstein (2022). GLOBAL WELL-POSEDNESS OF THE 3D NAVIER-STOKES EQUATIONS PERTURBED BY A DETERMINISTIC VECTOR FIELD. The Annals of Applied Probability. ISSN 1050-5164. 32(4), p. 2568–2586. doi: 10.1214/21-AAP1740.
Grong, Erlend; Nilssen, Torstein & Schmeding, Alexander (2022). Geometric rough paths on infinite dimensional spaces. Journal of Differential Equations. ISSN 0022-0396. 340, p. 151–178. doi: 10.1016/j.jde.2022.08.034. Full text in Research Archive Show summary
Similar to ordinary differential equations, rough paths and rough differential equations can be formulated in a Banach space setting. For α ∈ (1/3,1/2), we give criteria for when we can approximate Banach space-valued weakly geometric α-rough paths by signatures of curves of bounded variation, given some tuning of the Hölder parameter. We show that these criteria are satisfied for weakly geometric rough paths on Hilbert spaces. As an application, we obtain Wong-Zakai type result for function space valued martingales using the notion of (unbounded) rough drivers.
Cardona, Jorge; Hofmanová, Martina; Nilssen, Torstein & Rana, Nimit (2022). Random dynamical system generated by the 3D Navier-Stokes equation with rough transport noise. Electronic Journal of Probability (EJP). ISSN 1083-6489. 27. doi: 10.1214/22-EJP813. Full text in Research Archive Show summary
We consider the Navier-Stokes system in three dimensions perturbed by a transport noise which is sufficiently smooth in space and rough in time. The existence of a weak solution was proved in [26], however, as in the deterministic setting the question of uniqueness remains a major open problem. An important feature of systems with uniqueness is the semigroup property satisfied by their solutions. Without uniqueness, this property cannot hold generally. We select a system of solutions satisfying the semigroup property with appropriately shifted rough path. In addition, the selected solutions respect the well accepted admissibility criterium for physical solutions, namely, maximization of the energy dissipation. Finally, under suitable assumptions on the driving rough path, we show that the Navier-Stokes system generates a measurable random dynamical system. To the best of our knowledge, this is the first construction of a measurable single-valued random dynamical system in the state space for an SPDE without uniqueness.
Harang, Fabian Andsem; Nilssen, Torstein & Proske, Frank Norbert (2022). GIRSANOV THEOREM FOR MULTIFRACTIONAL BROWNIAN PROCESSES. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. doi: 10.1080/17442508.2022.2027948. Full text in Research Archive
Gerasimovičs, Andris; Hocquet, Antoine & Nilssen, Torstein (2021). Non-autonomous rough semilinear PDEs and the multiplicative Sewing lemma. Journal of Functional Analysis. ISSN 0022-1236. 281(10). doi: 10.1016/j.jfa.2021.109200.
Coghi, Michele & Nilssen, Torstein (2021). Rough nonlocal diffusions. Stochastic Processes and their Applications. ISSN 0304-4149. 141, p. 1–56. doi: 10.1016/j.spa.2021.07.002.
Hofmanova, Martina; Leahy, James-Michael & Nilssen, Torstein (2021). On a rough perturbation of the Navier–Stokes system and its vorticity formulation. The Annals of Applied Probability. ISSN 1050-5164. 31(2), p. 736–777. doi: 10.1214/20-AAP1603. Full text in Research Archive
Hocquet, Antoine & Nilssen, Torstein (2020). An Itô Formula for rough partial differential equations and some applications. Potential Analysis. ISSN 0926-2601. 54, p. 331–386. doi: 10.1007/s11118-020-09830-y. Full text in Research Archive
Nilssen, Torstein (2020). Rough linear PDEs with discontinuous coefficients – existence of solutions via regularization by fractional Brownian motion. Electronic Journal of Probability (EJP). ISSN 1083-6489. 25, p. 1–33. doi: 10.1214/20-EJP437. Full text in Research Archive
Hofmanova, Martina; Leahy, James-Michael & Nilssen, Torstein Kastberg (2019). On the Navier–Stokes equation perturbed by rough transport noise. Journal of evolution equations (Printed ed.). ISSN 1424-3199. 19(1), p. 203–247. doi: 10.1007/s00028-018-0473-z. Full text in Research Archive
Baños, David Ruiz; Nilssen, Torstein Kastberg & Proske, Frank Norbert (2019). Strong Existence and Higher Order Fréchet Differentiability of Stochastic Flows of Fractional Brownian Motion Driven SDEs with Singular Drift. Journal of Dynamics and Differential Equations. ISSN 1040-7294. 32, p. 1819–1866. doi: 10.1007/s10884-019-09789-4. Full text in Research Archive Show summary
In this paper we present a new method for the construction of strong solutions of SDE’s with merely integrable drift coefficients driven by a multidimensional fractional Brownian motion with Hurst parameter H<1/2. Furthermore, we prove the rather surprising result of the higher order Fréchet differentiability of stochastic flows of such SDE’s in the case of a small Hurst parameter. In establishing these results we use techniques from Malliavin calculus combined with new ideas based on a “local time variational calculus”. We expect that our general approach can be also applied to the study of certain types of stochastic partial differential equations as e.g. stochastic conservation laws driven by rough paths.
Nilssen, Torstein Kastberg; Leahy, James & Hofmanova, Martina (2017). Navier Stokes Equation with Rough Transport Noise. Preprint series (Universitetet i Oslo. Matematisk institutt). ISSN 0806-2439. Full text in Research Archive Show summary
We study a Navier Stokes equation on the d-dimensional torus with a transport type noise term. The noise is smooth in space but rough in time, so we can use the recently developed theory of "Unbounded Rough Drivers" to study the equation. We prove existence and uniqeness of a solution for the 2-d case and existence of a solution for the 3-d case.
Nilssen, Torstein Kastberg (2016). Regularity of strong solutions of one-dimensional SDE’s with discontinuous and unbounded drift. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. 88(5), p. 779–802. doi: 10.1080/17442508.2016.1144755.
Baños, David Ruiz & Nilssen, Torstein Kastberg (2016). Malliavin and flow regularity of SDEs. Application to the study of densities and the stochastic transport equation. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. 88(4), p. 540–566. doi: 10.1080/17442508.2015.1102265.
Nilssen, Torstein Kastberg (2015). Rough path transport equation with discontinuous coefficient - regularization by fractional Brownian motion. arXiv.org. ISSN 2331-8422.
Nilssen, Torstein Kastberg (2015). Rough path continuity equations with discontinuous coefficients - regularization by fractional Brownian motion. arXiv.org. ISSN 2331-8422. doi: 10.1214/20-ejp437.
Banos, David Ruiz; Nilssen, Torstein Kastberg & Proske, Frank Norbert (2015). Strong existence and higher order Fréchet differentiability of stochastic flows of fractional Brownian motion driven SDE's with singular drift. arXiv.org. ISSN 2331-8422. doi: 10.1007/s10884-019-09789-4.
Nilssen, Torstein Kastberg (2015). Quasi-linear stochastic partial differential equations with irregular coefficients - Malliavin regularity of the solutions. Stochastics and Partial Differential Equations: Analysis and Computations. ISSN 2194-0401. doi: 10.1007/s40072-015-0053-y. Full text in Research Archive Show summary
We study quasi-linear stochastic partial differential equations with discontinuous drift coefficients. Existence and uniqueness of a solution is already known under weaker conditions on the drift, but we are interested in the regularity of the solution in terms of Malliavin calculus. We prove that when the drift is bounded and measurable the solution is directional Malliavin differentiable.
Mohammed, Salah-Eldin; Nilssen, Torstein Kastberg & Proske, Frank Norbert (2015). Sobolev differentiable stochastic flows for sdes with singular coefficients: Applications to the transport equation. Annals of Probability. ISSN 0091-1798. 43(3), p. 1535–1576. doi: 10.1214/14-AOP909.
Nilssen, Torstein Kastberg & Zhang, Tusheng (2014). A Note on the Malliavin Differentiability of One-Dimensional Reflected Stochastic Differential Equations with Discontinuous Drift. arXiv.org. ISSN 2331-8422.
Flandoli, Franco; Nilssen, Torstein Kastberg & Proske, Frank Norbert (2013). Malliavin differentiability and strong solutions for a class of SDE in Hilbert spaces. Preprint series (Universitetet i Oslo. Matematisk institutt). ISSN 0806-2439. Full text in Research Archive
Menoukeu, Olivier Pamen; Meyer-Brandis, Thilo; Nilssen, Torstein Kastberg; Proske, Frank Norbert & Zhang, Tusheng (2013). A variational approach to the construction and Malliavin differentiability of strong solutions of SDE's. Mathematische Annalen. ISSN 0025-5831. 357(2), p. 761–799. doi: 10.1007/s00208-013-0916-3.

View all works in Cristin

Nilssen, Torstein (2022). A rough path Euler equation.
Nilssen, Torstein (2020). Introduction to rough paths .
Nilssen, Torstein (2020). Rough path variational principles for fluid equations.
Nilssen, Torstein (2020). Rough path variational principles for fluid equations.
Nilssen, Torstein (2020). Rough path variational principles for fluid equations.
Nilssen, Torstein (2020). Rough path variational principles for fluid equations.
Nilssen, Torstein Kastberg; Baños, David Ruiz & Proske, Frank Norbert (2016). Strong Existence and higher order differentiability of stochastic flows of fractional Brownian motion driven SDE's with singular drift.
Nilssen, Torstein Kastberg (2016). Rough path transport equation with discontinuous drift.
Nilssen, Torstein Kastberg (2016). Rough path transport equation with discontinuous drift.
Nilssen, Torstein Kastberg (2014). The Malliavin differentiability of one-dimensional Reflected SDE's with discontinuous drift.
Nilssen, Torstein Kastberg (2012). One-dimensional SDE's with Discontinuous, Unbounded Drift and Continuously Differentiable Solutions to the Stochastic Transport Equation. 7Letras. Full text in Research Archive
Nilssen, Torstein Kastberg; Mohammed, Salah-Eldin & Proske, Frank Norbert (2012). Sobolev Diﬀerentiable Stochastic Flows for SDE’s with SingularCoeﬃcients: Applications to the Transport Equation. 7Letras. Full text in Research Archive
Meyer-Brandis, Thilo; Nilssen, Torstein Kastberg; Pamen, Olivier Menoukeu; Proske, Frank Norbert & Zhang, Tusheng (2011). A variational approach to the construction and Malliavin differentiability of strong solutions of SDE's. Universitetet i Oslo. ISSN 0806-2439. Full text in Research Archive

View all works in Cristin

Published Apr. 16, 2024 11:41 AM