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Torstein Kastberg Nilssen

Associate Professor Department of Mathematical Sciences

Email: torstein.nilssen@uia.no
Phone: +47 38 14 13 20
Office: J2044 (Universitetsveien 25, 4630 Kristiansand, Norway)

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Background and experience

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.

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.

Research groups

Differential Equations and Dynamical Systems

Galeati, Lucio; Leahy, James Michael & Nilssen, Torstein (2026). On the well-posedness of (nonlinear) rough continuity equations. Journal of Differential Equations. ISSN 0022-0396. 462. doi: 10.1016/j.jde.2026.114124. Full text in Research Archive
Leahy, James-Michael & Nilssen, Torstein (2025). Scaled quadratic variation for controlled rough paths and parameter estimation of fractional diffusions. Electronic Journal of Probability (EJP). 30, p. 1–29. doi: 10.1214/25-EJP1308. Full text in Research Archive Show summary
We introduce the concept of finite γ-scaled quadratic variation along a sequence of partitions for paths on a given interval. This concept, with historical roots in the study of Gaussian processes by Gladyshev (1961) and Klein & Giné (1975), includes the fractional Brownian motion (fBm) with Hurst index H, which has finite 1 − 2 H-scaled quadratic variation. We show that a path that is controlled by a path with finite γ-scaled quadratic variation in the sense of M. Gubinelli inherits this property, and the corresponding scaled quadratic variation satisfies an Itô-isometry type formula. Moreover, we prove quantitative error bounds that establish a relationship between the convergence rates of the scaled quadratic variation of the controlled path and that of the controlling path. Additionally, we introduce a consistent estimator for the parameter γ based on a single sample path, complete with quantitative error bounds. We apply these results to the parameter estimation for fractional diffusions. Our findings specify convergence rates for the estimation of both the Hurst index and the parameters in the noise vector fields. The paper concludes with numerical experiments that substantiate our theoretical findings.
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. Full text in Research Archive
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. 94(8). doi: 10.1080/17442508.2022.2027948. Full text in Research Archive Show summary
In this article, we will present a new perspective on the variable-order fractional calculus, which allows for differentiation and integration to a variable order. The concept of multifractional calculus has been a scarcely studied topic within the field of functional analysis in the past 20 years. We develop a multifractional differential operator which acts as the inverse of the multifractional integral operator. This is done by solving the Abel integral equation generalized to a multifractional order. With this new multifractional differential operator, we prove a Girsanov's theorem for multifractional Brownian motions of Riemann–Liouville type. As an application, we show how Girsanov's theorem can then be applied to prove the existence of a unique strong solution to stochastic differential equations where the drift coefficient is merely of linear growth, and the driving noise is given by a non-stationary multifractional Brownian motion with a Hurst parameter as a function of time. The Hurst functions we study will take values in a bounded subset of (0,1/2) . The application of multifractional calculus to SDEs is based on a generalization of the works of D. Nualart and Y. Ouknine [Regularization of differential equations by fractional noise, Stoch Process Appl. 102(1) (2002), pp. 103–116].
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). 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.
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. Full text in Research Archive
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
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.
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 Show summary
We introduce a rough perturbation of the Navier–Stokes system and justify its physical relevance from balance of momentum and conservation of circulation in the inviscid limit. We present a framework for a well-posedness analysis of the system. In particular, we define an intrinsic notion of strong solution based on ideas from the rough path theory and study the system in an equivalent vorticity formulation. In two space dimensions, we prove that well-posedness and enstrophy balance holds. Moreover, we derive rough path continuity of the equation, which yields a Wong–Zakai result for Brownian driving paths, and show that for a large class of driving signals, the system generates a continuous random dynamical system. In dimension three, the noise is not enstrophy balanced, and we establish the existence of local in time solutions.
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. 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. 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 Show summary
We investigate existence, uniqueness and regularity for solutions of rough parabolic equations of the form ∂tu−Atu−f = (X˙t(x)·∇+Y˙ t(x))u on [0, T ]×Rd . To do so, we introduce a concept of “differential rough driver”, which comes with a counterpart of the usual controlled paths spaces in rough paths theory, built on the Sobolev spaces Wk,p. We also define a natural notion of geometricity in this context, and show how it relates to a product formula for controlled paths. In the case of transport noise (i.e. when Y = 0), we use this framework to prove an Ito Formula (in the sense of a chain rule) for Nemytskii operations of the ˆ form u → F (u), where F is C2 and vanishes at the origin. Our method is based on energy estimates, and a generalization of the Moser Iteration argument to prove boundedness of a dense class of solutions of parabolic problems as above. In particular, we avoid the use of flow transformations and work directly at the level of the original equation. We also show the corresponding chain rule for F (u) = |u| p with p ≥ 2, but also when Y = 0 and p ≥ 4. As an application of these results, we prove existence and uniqueness of a suitable class of Lp-solutions of parabolic equations with multiplicative noise. Another related development is the homogeneous Dirichlet boundary problem on a smooth domain, for which a weak maximum principle is shown under appropriate assumptions on the coefficients.
Nilssen, Torstein (2020). Rough linear PDEs with discontinuous coefficients – existence of solutions via regularization by fractional Brownian motion. Electronic Journal of Probability (EJP). 25, p. 1–33. doi: 10.1214/20-EJP437. Full text in Research Archive Show summary
We consider two related linear PDE’s perturbed by a fractional Brownian motion. We allow the drift to be discontinuous, in which case the corresponding deterministic equation is ill-posed. However, the noise will be shown to have a regularizing effect on the equations in the sense that we can prove existence of solutions for almost all paths of the fractional Brownian motion.
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.
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 Show summary
We consider the Navier–Stokes system in two and three space dimensions perturbed by transport noise and subject to periodic boundary conditions. The noise arises from perturbing the advecting velocity field by space–time-dependent noise that is smooth in space and rough in time. We study the system within the framework of rough path theory and, in particular, the recently developed theory of unbounded rough drivers. We introduce an intrinsic notion of a weak solution of the Navier–Stokes system, establish suitable a priori estimates and prove existence. In two dimensions, we prove that the solution is unique and stable with respect to the driving noise.
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. Full text in Research Archive
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. Full text in Research Archive
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. Full text in Research Archive
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. Full text in Research Archive
Nilssen, Torstein Kastberg (2015). Rough path transport equation with discontinuous coefficient - regularization by fractional Brownian motion. arXiv.org. ISSN 2331-8422. Full text in Research Archive
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. Full text in Research Archive
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. 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. Full text in Research Archive

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Nilssen, Torstein (2022). A rough path Euler equation. Full text in Research Archive
Nilssen, Torstein (2020). Rough path variational principles for fluid equations. Full text in Research Archive
Nilssen, Torstein (2020). Rough path variational principles for fluid equations. Full text in Research Archive
Nilssen, Torstein (2020). Rough path variational principles for fluid equations. Full text in Research Archive
Nilssen, Torstein (2020). Rough path variational principles for fluid equations. Full text in Research Archive
Nilssen, Torstein (2020). Introduction to rough paths. Full text in Research Archive
Nilssen, Torstein Kastberg (2016). Rough path transport equation with discontinuous drift. Full text in Research Archive
Nilssen, Torstein Kastberg (2016). Rough path transport equation with discontinuous drift. Full text in Research Archive
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. Full text in Research Archive
Nilssen, Torstein Kastberg (2014). The Malliavin differentiability of one-dimensional Reflected SDE's with discontinuous drift. Full text in Research Archive
Nilssen, Torstein Kastberg; Leahy, James & Hofmanova, Martina (2017). Navier Stokes Equation with Rough Transport Noise. Universitetet i Oslo. 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.
Flandoli, Franco; Nilssen, Torstein Kastberg & Proske, Frank Norbert (2013). Malliavin differentiability and strong solutions for a class of SDE in Hilbert spaces. Universitetet i Oslo. Full text in Research Archive Show summary
We consider a class of Hilbert-space valued SDE’s where the drift coefficients are non- Lipschitzian in the sense of Hölder-continuity. Using a novel technique based on Malliavin calculus we show in this paper the existence and uniqueness of a mild solution to such equations. We emphasize that our approach does not rely on the Yamada-Watanabe principle. Moreover our method gives the important additional insight that the obtained solution is Malliavin differentiable - a property which was recently shown to play a crucial role in the study of the geometry of certain optimal causal transference plans, [12].
Nilssen, Torstein Kastberg (2012). One-dimensional SDE's with Discontinuous, Unbounded Drift and Continuously Differentiable Solutions to the Stochastic Transport Equation. Universitetet i Oslo. Full text in Research Archive Show summary
In this paper we develop a method for constructing strong solutions of one-dimensional SDE’s where the drift may be discontinuous and unbounded. The driving noise is the Brownian Motion. In addition to existence and uniqueness of the strong solution, we show that the solution is Sobolev-differentiable in the initial condition and Malliavin differentiable. The method is based on Malliavin calculus using a sim- ilar technique as initiated in [11] and further developed in [10] and [12] where the authors consider bounded coefficients. This method is not based on a pathwise uniqueness argument. We will apply these results to the stochastic transport equation. More specifically, we ob- tain a continuously differentiable solution of the stochastic transport equation when the driving function is a step function.
Nilssen, Torstein Kastberg; Mohammed, Salah-Eldin & Proske, Frank Norbert (2012). Sobolev Diﬀerentiable Stochastic Flows for SDE’s with SingularCoeﬃcients: Applications to the Transport Equation. Universitetet i Oslo. Full text in Research Archive
Pamen, Olivier Menoukeu; Meyer-Brandis, Thilo; Nilssen, Torstein Kastberg; Proske, Frank Norbert & Zhang, Tusheng (2011). A variational approach to the construction and Malliavin differentiability of strong solutions of SDE's. Universitetet i Oslo. Full text in Research Archive

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Published Apr. 16, 2024 11:41 AM