Nicolas Perkowski

The existence of dominating local martingale measures

We prove that for locally bounded processes the absence of arbitrage opportunities of the first kind is equivalent to the existence of a dominating local martingale measure. This is related to and motivated by results from the theory of filtration enlargements.

Dimensional reduction in nonlinear filtering: A homogenization approach.

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 filter with rate

Supermartingales as Radon–Nikodym densities and related measure extensions

\sqrt{\varepsilon}

The Kardar–Parisi–Zhang Equation as Scaling Limit of Weakly Asymmetric Interacting Brownian Motions

. This is achieved by a suitable asymptotic expansion of the dual of the Zakai equation, and by probabilistically representing the correction terms with the help of BDSDEs.

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

Certain countably and finitely additive measures can be associated to a given nonnegative supermartingale. Under weak assumptions on the underlying probability space, existence and (non)uniqueness results for such measures are proven.

A Fourier analytic approach to pathwise stochastic integration

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 fluctuations of the system are well approximated by the solution to the KPZ equation provided the microscopic interaction is weakly asymmetric. The proof is based on the martingale solutions of Gonçalves and Jara (Arch Ration Mech Anal 212(2):597–644, 2014) and the corresponding uniqueness result of Gubinelli and Perkowski (Energy solutions of KPZ are unique, 2015).

Large Deviations for Hilbert-Space-Valued Wiener Processes: A Sequence Space Approach

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 discrete random polymer measure, as well as a universality result for the spectrum of discrete random Schrödinger operators on large boxes with small potentials. Our proof is based on paracontrolled distributions and some basic results for multiple stochastic integrals of discrete martingales.

Probabilistic Approach to the Stochastic Burgers Equation

We develop a Fourier analytic approach to rough path integration, based on the series decomposition of continuous functions in terms of Schauder functions. Our approach is rather elementary, the main ingredient being a simple commutator estimate, and it leads to recursive algorithms for the calculation of pathwise stochastic integrals, both of Ito and of Stratonovich type. We apply it to solve stochastic differential equations in a pathwise manner.

An Introduction to Singular SPDEs

Ciesielski’s isomorphism between the space of α-Holder continuous functions and the space of bounded sequences is used to give an alternative proof of the large deviation principle (LDP) for Wiener processes with values in Hilbert space.

PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES

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 a stochastic PDE with distributional drift, so we first review how to construct finite-dimensional diffusions with distributional drift. We then present the uniqueness result for the stationary martingale problem of (M. Gubinelli and N. Perkowski, Energy solutions of KPZ are unique. 2015, [18]), but we mainly emphasize the heuristic derivation and also we include a (very simple) extension of (M. Gubinelli and N. Perkowski, Energy solutions of KPZ are unique. 2015, [18]) to a non-stationary regime.