Nicolas Perkowski

An extension of the stochastic sewing lemma and applications to fractional stochastic calculus

Toyomu Matsuda, Nicolas Perkowski

We give an extension of LÃª's stochastic sewing lemma [Electron. J. Probab. 25: 1 - 55, 2020]. The stochastic sewing lemma proves convergence in $L_{m}$ of Riemann type sums $\sum _{…

Pathwise stochastic integrals for model free finance

Nicolas Perkowski, David J. PrÃ¶mel

We present two different approaches to stochastic integration in frictionless model free financial mathematics. The first one is in the spirit of ItÃ´'s integral and based on a cer…

math.PR2024

Weak well-posedness of energy solutions to singular SDEs with supercritical distributional drift

Lukas GrÃ¤fner, Nicolas Perkowski

We study stochastic differential equations with additive noise and distributional drift on $T^{d}$ or $R^{d}$ and $d ⩾ 2$ . We work in a scaling-supercritical…

Paracontrolled distributions on Bravais lattices and weak universality of the 2d parabolic Anderson model

JÃ¶rg Martin, Nicolas Perkowski

We develop a discrete version of paracontrolled distributions as a tool for deriving scaling limits of lattice systems, and we provide a formulation of paracontrolled distribution…

A Rough Super-Brownian Motion

Nicolas Perkowski, Tommaso Cornelis Rosati

We study the scaling limit of a branching random walk in static random environment in dimension $d = 1, 2$ and show that it is given by a super-Brownian motion in a white noise potent…

The infinitesimal generator of the stochastic Burgers equation

Massimiliano Gubinelli, Nicolas Perkowski

We develop a martingale approach for a class of singular stochastic PDEs of Burgers type (including fractional and multi-component Burgers equations) by constructing a domain for t…

math.PR2015

KPZ reloaded

Massimiliano Gubinelli, Nicolas Perkowski

We analyze the one-dimensional periodic Kardar-Parisi-Zhang equation in the language of paracontrolled distributions, giving an alternative viewpoint on the seminal results of Hair…

math.PR2025

Coming up from $- \infty$ for KPZ via stochastic control

Nicolas Perkowski, Carlos Villanueva Mariz

We derive a lower bound, independent of the initial condition, for the solution of the KPZ equation on the torus through its representation as the value function of a (conditional)…

One-dimensional game-theoretic differential equations

RafaÅ M. Åochowski, Nicolas Perkowski, David J. PrÃ¶mel

We provide a very brief introduction to typical paths and the corresponding ItÃ´ type integration. Relying on this robust ItÃ´ integration, we prove an existence and uniqueness res…

Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the WASEP

Patricia GonÃ§alves, Nicolas Perkowski, Marielle Simon

We consider the weakly asymmetric simple exclusion process on the discrete space ${1, ..., n - 1}$ , in contact with stochastic reservoirs, both with density $I ¨ \in (0, 1)$ at the ext…

physics.data-an2012

Particle filtering in high-dimensional chaotic systems

Nishanth Lingala, N. Sri Namachchivaya, Nicolas Perkowski +1

We present an efficient particle filtering algorithm for multiscale systems, that is adapted for simple atmospheric dynamics models which are inherently chaotic. Particle filters r…

Surface Dean--Kawasaki equations

John Bell, Ana Djurdjevac, Nicolas Perkowski

We consider stochastic particle dynamics on hypersurfaces represented in Monge gauge parametrization. Starting from the underlying Langevin system, we derive the surface Dean-Kawas…

The Compact Support Property of Rough Super Brownian Motion on $R^{2}$

Ruhong Jin, Nicolas Perkowski

We discuss the compact support property of the rough super-Brownian motion constructed as a scaling limit of a branching random walk in static random environment. The semi-linear e…

The Kardar-Parisi-Zhang equation as scaling limit of weakly asymmetric interacting Brownian motions

Joscha Diehl, Massimiliano Gubinelli, Nicolas Perkowski

We consider a system of infinitely many interacting Brownian motions that models the height of a one-dimensional interface between two bulk phases. We prove that the large scale fl…

math.PR2019

The KPZ Equation on the Real Line

Nicolas Perkowski, Tommaso Cornelis Rosati

We prove existence and uniqueness of distributional solutions to the KPZ equation globally in space and time, with techniques from paracontrolled analysis. Our main tool for extend…

An introduction to singular SPDEs

Massimiliano Gubinelli, Nicolas Perkowski

We review recent results on the analysis of singular stochastic partial differential equations in the language of paracontrolled distributions.

On the role of positivity preservation for high order approximations of the Dean--Kawasaki equation

Ana DamnjanoviÄ, Ana Djurdjevac, Nicolas Perkowski

We study a spectral regularization of the Dean--Kawasaki equation and quantify how the failure of positivity preservation affects its weak approximation of the empirical measure of…

Multidimensional SDE with distributional drift and LÃ©vy noise

Helena Kremp, Nicolas Perkowski

We solve multidimensional SDEs with distributional drift driven by symmetric, $I^\pm$ -stable LÃ©vy processes for $I^\pm \in (1, 2]$ by studying the associated (singular) martingale problem…

math.DS2023

Almost Sure Asymptotic Stability of Parabolic SPDEs with Small Multiplicative Noise

Yiming Meng, N. Sri. Namachchivaya, Nicolas Perkowski

A better understanding of the instability margin will eventually optimize the operational range for safety-critical industries. In this paper, we investigate the almost-sure expone…

math.PR2012

Conditioned Martingales

Nicolas Perkowski, Johannes Ruf

It is well known that upward conditioned Brownian motion is a three-dimensional Bessel process, and that a downward conditioned Bessel process is a Brownian motion. We give a simpl…

math.PR2025

Weak Error of Dean-Kawasaki Equation with Smooth Mean-Field Interactions

Ana Djurdjevac, Xiaohao Ji, Nicolas Perkowski

We consider the weak-error rate of the SPDE approximation by regularized Dean-Kawasaki equation with ItÃ´ noise for particle systems with mean-field interactions both on the drift…

math.PR2014

A Fourier approach to pathwise stochastic integration

Massimiliano Gubinelli, Peter Imkeller, Nicolas Perkowski

We develop a Fourier approach to rough path integration, based on the series decomposition of continuous functions in terms of Schauder functions. Our approach is rather elementary…

Paracontrolled distributions and singular PDEs

Massimiliano Gubinelli, Peter Imkeller, Nicolas Perkowski

We introduce an approach to study certain singular PDEs which is based on techniques from paradifferential calculus and on ideas from the theory of controlled rough paths. We illus…

Fractional Kolmogorov equations with singular paracontrolled terminal conditions

Helena Kremp, Nicolas Perkowski

We consider backward fractional Kolmogorov equations with singular Besov drift of low regularity and singular terminal conditions. To treat drifts beyond the socalled Young regime,…

A simple construction of the dynamical $I^¦_{3}^{4}$ model

Aukosh Jagannath, Nicolas Perkowski

The $I^¦_{3}^{4}$ equation is a singular stochastic PDE with important applications in mathematical physics. Its solution usually requires advanced mathematical theories like regularity…

math.PR2024

Fractional stochastic Landau-Lifshitz Navier-Stokes equations in dimension $d \geq 3$ : Existence and (non-)triviality

Ruhong Jin, Nicolas Perkowski

We investigate fractional stochastic Navier-Stokes equations in $d \geq 3$ , driven by the random force $(- I^)^{\frac{I ^}{¸} 2} I^¾$ which, as we show, corresponds to a fractional version o…

math.PR2019

Additive functionals as rough paths

Jean-Dominique Deuschel, Tal Orenshtein, Nicolas Perkowski

We consider additive functionals of stationary Markov processes and show that under Kipnis-Varadhan type conditions they converge in rough path topology to a Stratonovich Brownian…

Weak error analysis for a nonlinear SPDE approximation of the Dean-Kawasaki equation

Ana Djurdjevac, Helena Kremp, Nicolas Perkowski

We consider a nonlinear SPDE approximation of the Dean-Kawasaki equation for independent particles. Our approximation satisfies the physical constraints of the particle system, i.e…

Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity

Rama Cont, Nicolas Perkowski

We construct a pathwise integration theory, associated with a change of variable formula, for smooth functionals of continuous paths with arbitrary regularity defined in terms of t…

math.PR2013

Dimensional reduction in nonlinear filtering: A homogenization approach

Peter Imkeller, N. Sri Namachchivaya, Nicolas Perkowski +1

We propose a homogenized filter for multiscale signals, which allows us to reduce the dimension of the system. We prove that the nonlinear filter converges to our homogenized filte…

math.PR2012

Large deviations for Hilbert space valued Wiener processes: a sequence space approach

Andreas Andresen, Peter Imkeller, Nicolas Perkowski

Ciesielski's isomorphism between the space of alpha-HÃ¶lder continuous functions and the space of bounded sequences is used to give an alternative proof of the large deviation prin…

Approximation of the Filter Equation for Multiple Timescale, Correlated, Nonlinear Systems

Ryne Beeson, N. Sri Namachchivaya, Nicolas Perkowski

This paper considers the approximation of the continuous time filtering equation for the case of a multiple timescale (slow-intermediate, and fast scales) that may have correlation…

The Hairer--Quastel universality result in equilibrium

Massimiliano Gubinelli, Nicolas Perkowski

We use the notion of energy solutions of the stochastic Burgers equation to give a short proof of the Hairer-Quastel universality result for a class of stationary weakly asymmetric…

Periodic homogenization for singular LÃ©vy SDEs

Helena Kremp, Nicolas Perkowski

We generalize the theory of periodic homogenization for multidimensional SDEs with additive Brownian and stable LÃ©vy noise for $I^\pm \in (1, 2)$ to the setting of singular periodic Be…

C-infinity regularization of ODEs perturbed by noise

Fabian A. Harang, Nicolas Perkowski

We study ODEs with vector fields given by general Schwartz distributions, and we show that if we perturb such an equation by adding an "infinitely regularizing" path, then it has a…

q-fin.MF2017

A superhedging approach to stochastic integration

RafaÅ M. Åochowski, Nicolas Perkowski, David J. PrÃ¶mel

Using Vovk's outer measure, which corresponds to a minimal superhedging price, the existence of quadratic variation is shown for "typical price paths" in the space of cÃ dlÃ g func…

A Rough Functional Breuer-Major Theorem

Henri Elad Altman, Tom Klose, Nicolas Perkowski

We extend the functional Breuer-Major theorem by Nourdin and Nualart (2020) to the space of rough paths. The proof of tightness combines the multiplication formula for iterated Mal…

Quantitative heat kernel estimates for diffusions with distributional drift

Nicolas Perkowski, Willem van Zuijlen

We consider the stochastic differential equation on $R^{d}$ given by $d X_{t} = b (t, X_{t}) d t + d B_{t},$ where $B$ is a Brownian motion and…

math.PR2013

The Existence of Dominating Local Martingale Measures

Peter Imkeller, Nicolas Perkowski

We prove that, for locally bounded processes, absence of arbitrage opportunities of the first kind is equivalent to the existence of a dominating local martingale measure. This is…

q-fin.MF2016

Pathwise super-replication via Vovk's outer measure

Mathias BeiglbÃ¶ck, Alexander M. G. Cox, Martin Huesmann +2

Since Hobson's seminal paper [D. Hobson: Robust hedging of the lookback option. In: Finance Stoch. (1998)] the connection between model-independent pricing and the Skorokhod embedd…

Helena Kremp, Nicolas Perkowski

We introduce a weak solution concept (called "rough weak solutions") for singular SDEs with additive alpha-stable LÃ©vy noise (including the Brownian noise case) and prove its equi…

Probabilistic approach to the stochastic Burgers equation

Massimiliano Gubinelli, Nicolas Perkowski

We review the formulation of the stochastic Burgers equation as a martingale problem. One way of understanding the difficulty in making sense of the equation is to note that it is…

Level crossings of fractional Brownian motion

Purba Das, RafaÅ Åochowski, Toyomu Matsuda +1

Since the classical work of LÃ©vy, it is known that the local time of Brownian motion can be characterized through the limit of level crossings. While subsequent extensions of this…

The Fundamental Solution for the Heat Equation on the half-line with Drift and Dirichlet Boundary Condition

Tertuliano Franco, PatrÃcia GonÃ§alves, Nicolas Perkowski +1

By a probabilistic method we provide an explicit fundamental solution of the Cauchy problem associated to the heat equation on the half-line with constant drift and Dirichlet bound…

Energy solutions of singular SPDEs on Hilbert spaces with applications to domains with boundary conditions

Lukas GrÃ¤fner, Nicolas Perkowski, Shyam Popat

In this paper we extend the theory of energy solutions for singular SPDEs, focusing on equations driven by highly irregular noise with bilinear nonlinearities, including scaling cr…

Optimal Investment Decision Under Switching regimes of Subsidy Support

Carlos Oliveira, Nicolas Perkowski

We address the problem of making a managerial decision when the investment project is subsidized, which results in the resolution of an infinite-horizon optimal stopping problem of…

math.PR2015

Local times for typical price paths and pathwise Tanaka formulas

Nicolas Perkowski, David J. PrÃ¶mel

Following a hedging based approach to model free financial mathematics, we prove that it should be possible to make an arbitrarily large profit by investing in those one-dimensiona…

An invariance principle for the two-dimensional parabolic Anderson model with small potential

Khalil Chouk, Jan Gairing, Nicolas Perkowski

We prove an invariance principle for the two-dimensional lattice parabolic Anderson model with small potential. As applications we deduce a Donsker type convergence result for a di…

Quantitative Convergence of the Filter Solution for Multiple Timescale Nonlinear Systems with Coarse-Grain Correlated Noise

Ryne Beeson, N. Sri Namachchivaya, Nicolas Perkowski

In this paper we prove a rate of convergence for the continuous time filtering solution of a multiple timescale correlated nonlinear system to a lower dimensional filtering equatio…

A Littlewood-Paley description of modelled distributions

JÃ¶rg Martin, Nicolas Perkowski

We exhibit a fundamental link between Hairer's theory of regularity structures and the paracontrolled calculus of Gubinelli, Imkeller and Perkowski. By using paraproducts we provid…

Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential

Wolfgang KÃ¶nig, Nicolas Perkowski, Willem van Zuijlen

We consider the parabolic Anderson model (PAM) $\partial_{t} u = \frac{1}{2} I^ u + I^¾ u$ in $R^{2}$ with a Gaussian (space) white-noise potential $I^¾$ . We prove that the almost-sure…

Renormalization destroys a finite time bifurcation in the $I^¦_{2}^{4}$ equation

Alexandra Blessing, Nicolas Perkowski, Chara Zhu

We study the singular $I^¦_{2}^{4}$ equation at a pitchfork bifurcation of the underlying deterministic dynamics. To this aim, we linearize the SPDE along its stationary solution and sh…