82 citations · 147 across the 21 of their papers we have counts for
27 papers
Higher Order Unbounded Rough Drivers
Torstein Nilssen, Jonas Pedersen Vean
We construct a framework for higher order unbounded rough drivers defined by rough paths over vector fields. This allows us to consider a purely Eulerian perspective for rough tran…
Unbounded rough drivers, rough PDEs and applications
Antoine Hocquet, Martina Hofmanova, Torstein Nilssen
A summary of recent contributions in the field of rough partial differential equations is given. For that purpose we rely on the formalism of ``unbounded rough driver''. We present…
On the well-posedness of (nonlinear) rough continuity equations
Lucio Galeati, James-Michael Leahy, Torstein Nilssen
Motivated by applications to fluid dynamics, we study rough differential equations (RDEs) and rough partial differential equations (RPDEs) with non-Lipschitz drifts. We prove well-…
Malliavin Calculus for rough stochastic differential equations
Fabio Bugini, Michele Coghi, Torstein Nilssen
In this work we show that rough stochastic differential equations (RSDEs), as introduced by Friz, Hocquet, and Lê (2021), are Malliavin differentiable. We use this to prove existen…
Scaled quadratic variation for controlled rough paths and parameter estimation of fractional diffusions
James-Michael Leahy, Torstein Nilssen
We introduce the concept of finite -scaled quadratic variation along a sequence of partitions for paths on a given interval. This concept, with historical roots in the study of…
Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering
Michele Coghi, Torstein Nilssen, Nikolas Nüsken +1
Motivated by the challenge of incorporating data into misspecified and multiscale dynamical models, we study a McKean-Vlasov equation that contains the data stream as a common driv…
Solution properties of the incompressible Euler system with rough path advection
Dan Crisan, Darryl D. Holm, James-Michael Leahy +1
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 incompressi…
Random dynamical system generated by the 3D Navier-Stokes equation with rough transport noise
Jorge Cardona, Martina Hofmanova, Torstein Nilssen +1
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 wa…
Geometric rough paths on infinite dimensional spaces
Erlend Grong, Torstein Nilssen, Alexander Schmeding
Similar to ordinary differential equations, rough paths and rough differential equations can be formulated in a Banach space setting. For , we give criteria for whe…
Variational principles for fluid dynamics on rough paths
Dan Crisan, Darryl D. Holm, James-Michael Leahy +1
In this paper, we introduce a new framework for parametrization schemes (PS) in GFD. Using the theory of controlled rough paths, we derive a class of rough geophysical fluid dynami…
Global well-posedness of the 3D Navier--Stokes equations perturbed by a deterministic vector field
Franco Flandoli, Martina Hofmanová, Dejun Luo +1
We are concerned with the problem of global well-posedness of the 3D Navier--Stokes equations on the torus with unitary viscosity. While a full answer to this question seems to be…
Non-autonomous rough semilinear PDEs and the multiplicative Sewing Lemma
Andris Gerasimovics, Antoine Hocquet, Torstein Nilssen
We investigate existence, uniqueness and regularity for local solutions of rough parabolic equations with subcritical noise of the form $du_t- L_tu_tdt= N(u_t)dt + \sum_{i = 1}^dF_…
Rough nonlocal diffusions
Michele Coghi, Torstein Nilssen
We consider a nonlinear Fokker-Planck equation driven by a deterministic rough path which describes the conditional probability of a McKean-Vlasov diffusion with "common" noise. To…
On a rough perturbation of the Navier-Stokes system and its vorticity formulation
Martina Hofmanova, James-Michael Leahy, Torstein Nilssen
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…
Existence, uniqueness and stability of semi-linear rough partial differential equations
Peter Friz, Torstein Nilssen, Wilhelm Stannat
We prove well-posedness and rough path stability of a class of linear and semi-linear rough PDE's on using the variational approach. This includes well-posedness of…
An Itô Formula for rough partial differential equations and some applications
Antoine Hocquet, Torstein Nilssen
We investigate existence, uniqueness and regularity for solutions of rough parabolic equations of the form on $[0,T]…
Weak solutions of rough path SDE's via Girsanov
Torstein Nilssen
We consider a differential equation driven by a Brownian motion as well as a rough path. We prove a Girsanov-type result for this equation to construct a weak solution in the proba…
Generalized Burgers equation with rough transport noise
Antoine Hocquet, Torstein Nilssen, Wilhelm Stannat
We introduce a new technique for studying well posedness and energy estimates for evolution equations with a rough transport term. The technique is based on finding suitable space-…
On the Navier-Stokes equation perturbed by rough transport noise
Martina Hofmanová, James-Michael Leahy, Torstein Nilssen
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 th…
Girsanov Theorem for Multifractional Brownian Processes
Fabian Harang, Torstein Nilssen, Frank Proske
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, i.e. one differen…
Rough path continuity equations with discontinuous coefficients - regularization by fractional Brownian motion
Torstein Nilssen
We consider the stochastic continuity equation perturbed by a fractional Brownian motion and the drift is allowed to be discontinuous. We show that for almost all paths of the frac…
Strong existence and higher order Fréchet differentiability of stochastic flows of fractional Brownian motion driven SDE's with singular drift
David Baños, Torstein Nilssen, Frank Proske
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 m…
Rough linear PDE's with discontinuous coefficients - existence of solutions via regularization by fractional Brownian motion
Torstein Nilssen
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 il…
Quasi-linear Stochastic Partial Differential Equations with irregular Coefficients - Malliavin regularity of the Solutions
Torstein Nilssen
We study quasi-linear stochastic partial differential equations with discontinuous drift coefficients. Existence and uniqueness of a solution is already known under weaker conditio…