Related papers: A Fubini type theorem for rough integration

Let the function $f: \bar{\R}^2_+ \to \C$ be such that $f\in L^1_{\loc} (\bar{\R}^2_+)$. We investigate the convergence behavior of the double integral $$\int^A_0 \int^B_0 f(u,v) du dv \quad {\rm as} \quad A,B \to \infty,\leqno(*)$$ where…

This short note shows a limiting behavior of integrals of some centered antipersistent stationary infinitely divisible moving averages as the compact integration domain in $d\ge 1$ dimensions extends to the whole positive quadrant…

We study controlled differential equations with unbounded drift terms, where the driving paths is $\nu$ - H\"older continuous for $\nu \in (\frac{1}{3},\frac{1}{2})$, so that the rough integral are interpreted in the Gubinelli sense…

We study path integrals in the Trotter-type form for the Schr\"odinger equation, where the Hamiltonian is the Weyl quantization of a real-valued quadratic form perturbed by a potential $V$ in a class encompassing that - considered by…

We continue the approach in Part I \cite{duchong19} to study stationary states of controlled differential equations driven by rough paths, using the framework of random dynamical systems and random attractors. Part II deals with driving…

Rough paths techniques give the ability to define solutions of stochastic differential equations driven by signals $X$ which are not semimartingales and whose $p$-variation is finite only for large values of $p$. In this context, rough…

We formulate indefinite integration with respect to an irregular function as an algebraic problem and provide a criterion for the existence and uniqueness of a solution. This allows us to define a good notion of integral with respect to…

One of the essential questions of the theory of multidimensional integrals concerns the evaluation of integrals taken in given domains. In the simplest case, when integrating over parallelepipeds, evaluation can easily be performed by…

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 Malliavin divergences, due to Furlan and Gubinelli (2019), with…

Rough path analysis can be developed using the concept of controlled paths, and with respect to a topology in which L\'evy's area plays a role. For vectors of irregular paths we investigate the relationship between the property of being…

In this paper, we build the foundation for a theory of controlled rough paths on manifolds. A number of natural candidates for the definition of manifold valued controlled rough paths are developed and shown to be equivalent. The theory of…

The classic stochastic Fubini theorem says that if one stochastically integrates with respect to a semimartingale $S$ an $\eta(dz)$-mixture of $z$-parametrized integrands $\psi^z$, the result is just the $\eta(dz)$-mixture of the individual…

In [1], we proved the existence of solutions to reflected rough differential equations based on an idea of Euler approximation of the solutions which is due to Davie [6]. In this paper, we prove the existence theorem under weaker…

Given an arbitrary Lagrangian function on \RR^d and a choice of classical path, one can try to define Feynman's path integral supported near the classical path as a formal power series parameterized by "Feynman diagrams," although these…

We investigate the regular convergence of the $m$-multiple series $$\sum^\infty_{j_1=0} \sum^\infty_{j_2=0}...\sum^\infty_{j_m=0} \ c_{j_1, j_2,..., j_m}\leqno(*)$$ of complex numbers, where $m\ge 2$ is a fixed integer. We prove Fubini's…

I prove a theorem about iterated integrals for non-product measures in a product space. The first task is to show the existence of a family of measures on the second space, indexed by the points on of the first space (outside a negligible…

Recently, Hairer--Pillai proposed the notion of $\theta$-roughness of a path which leads to a deterministic Norris lemma. In the Gubinelli framework (Hoelder, level 2) of rough paths, they were then able to prove a Hoermander type result…

We study a class of controlled rough differential equations. It is shown that the value function satisfies a HJB type equation; we also establish a form of the Pontryagin maximum principle. Deterministic problems of this type arise in the…

Optimal sample path properties of stochastic processes often involve generalized H\"{o}lder- or variation norms. Following a classical result of Taylor, the exact variation of Brownian motion is measured in terms of $\psi (x) \equiv $…

We build a connection between rough path theory and noncommutative algebra, and interpret the integration of geometric rough paths as an example of a non-abelian Young integration. We identify a class of slowly-varying one-forms, and prove…