arXiv subjects

Markus Riedle

Publications and source records attributed to Markus Riedle.

At least 19 recordsLinked to original sources

arXiv2026

Stochastic integration with respect to a L\'evy basis

We develop a stochastic integration theory for predictable integrands with respect to a L\'evy basis. Our approach is based on decoupling inequalities for tangent sequences and reduces the construction of the stochastic integral essentially to the deterministic integration theory for infinitely divisible random measures developed by Rajput and Rosi\'nski. We characterise the corresponding class of integrable predictable processes in terms of the semimartingale characteristics associated with the driving random measure and show that the resulting space of integrands possesses a natural Musielak-Orlicz type structure equipped with an F-norm. Furthermore, we establish continuity properties of the integral operator and a stochastic version of Lebesgue's dominated convergence theorem.

math.PR↗

arXiv2026

Stochastic evolution equations driven by arbitrary cylindrical L\'evy processes

We establish the first existence and uniqueness result for mild solutions of abstract stochastic evolution equations driven by arbitrary cylindrical L\'evy processes in Hilbert spaces. The coefficients are assumed to satisfy global Lipschitz conditions, and no moment assumptions are imposed on the driving noise. The principal difficulty arises from the fact that cylindrical L\'evy processes exist solely in a generalised sense and typically admit no semimartingale or L\'evy-It\^o decomposition, which precludes the use of classical existence methods. To overcome these obstacles, we develop a pathwise adaptive Euler-Peano approximation scheme based on noise-dependent stopping times and a fixed-point formulation of the mild solution operator. The resulting approach avoids stochastic calculus techniques relying on semimartingale decompositions and provides a robust and flexible framework for treating multiplicative cylindrical L\'evy noise in infinite-dimensional systems.

math.PR↗

arXiv2025

Regularity of multiplicative processes on infinite-dimensional Lie groups

This article studies regularity properties of multiplicative stochastic processes on infinite-dimensional Lie groups. We investigate conditions under which these processes admit c\`adl\`ag modifications and derive bounds on their local behavior. Our approach builds on the local equivalence of Banach-Lie groups and Banach spaces via the exponential and logarithm, allowing us to transfer analytic estimates and structural results.

math.PR↗

arXiv2024

Regularisation of cylindrical L\'evy processes in Besov spaces

In this work, we quantify the irregularity of a given cylindrical L\'evy process $L$ in $L^2({\mathbb R}^d)$ by determining the range of weighted Besov spaces $B$ in which $L$ has a regularised version $Y$, that is a stochastic process $Y$ in the classical sense with values in $B$. Our approach is based on characterising L\'evy measures on Besov spaces. As a by-product, we determine those Besov spaces $B$ for which the embedding of $L^2({\mathbb R}^d)$ into $B$ is $0$-Radonifying and $p$-Radonifying for $p>1$.

math.PR↗

arXiv2024

L\'evy measures on Banach spaces

We establish an explicit characterisation of L\'evy measures on both $L^p$-spaces and UMD Banach spaces. In the case of $L^p$-spaces, L\'evy measures are characterised by an integrability condition, which directly generalises the known description of L\'evy measures on sequence spaces. The latter has been the only known description of L\'evy measures on infinite dimensional Banach spaces that are not Hilbert. L\'evy measures on UMD Banach spaces are characterised by the finiteness of the expectation of a random {\gamma}-radonifying norm. Although this description is more abstract, it reduces to simple integrability conditions in the case of $L^p$-spaces.

math.FA↗

arXiv2024

Stochastic integration with respect to cylindrical L\'evy processes in Hilbert spaces

In this work, we present a comprehensive theory of stochastic integration with respect to arbitrary cylindrical L\'evy processes in Hilbert spaces. Since cylindrical L\'evy processes do not enjoy a semi-martingale decomposition, our approach relies on an alternative approach to stochastic integration by decoupled tangent sequences. The space of deterministic integrands is identified as a modular space described in terms of the characteristics of the cylindrical L\'evy process. The space of random integrands is described as the space of predictable processes whose trajectories are in the space of deterministic integrands almost surely. The derived space of random integrands is verified as the largest space of potential integrands, based on a classical definition of stochastic integrability. We apply the introduced theory of stochastic integration to establish a dominated convergence theorem.

math.PR↗

arXiv2024

SPDEs driven by standard symmetric $\alpha$-stable cylindrical L\'evy processes: existence, Lyapunov functionals and It\^{o} formula

We investigate several aspects of solutions to stochastic evolution equations in Hilbert spaces driven by a standard symmetric $\alpha$-stable cylindrical noise. Similarly to cylindrical Brownian motion or Gaussian white noise, standard symmetric $\alpha$-stable noise exists only in a generalised sense in Hilbert spaces. The main results of this work are the existence of a mild solution, long-term regularity of the solutions via Lyapunov functional approach, and an It\^{o} formula for mild solutions to evolution equations under consideration. The main tools for establishing these results are Yosida approximations and an It\^{o} formula for Hilbert space-valued semi-martingales where the martingale part is represented as an integral driven by cylindrical $\alpha$-stable noise. While these tools are standard in stochastic analysis, due to the cylindrical nature of our noise, their application requires completely novel arguments and techniques.

math.PR↗

arXiv2022

Stochastic integration with respect to canonical $α$-stable cylindrical Lévy processes

In this work, we introduce a theory of stochastic integration with respect to symmetric $α$-stable cylindrical Lévy processes. Since $α$-stable cylindrical Lévy processes do not enjoy a semi-martingale decomposition, our approach is based on a decoupling inequality for the tangent sequence of the Radonified increments. This approach enables us to characterise the largest space of predictable Hilbert-Schmidt operator-valued processes which are integrable with respect to an $α$-stable cylindrical Lévy process as the collection of all predictable processes with paths in the Bochner space $L^α$. We demonstrate the power and robustness of the developed theory by establishing a dominated convergence result allowing the interchange of the stochastic integral and limit.

math.PR↗

arXiv2021

Stochastic evolution equations driven by cylindrical stable noise

We prove existence and uniqueness of a mild solution of a stochastic evolution equation driven by a standard $α$-stable cylindrical Lévy process defined on a Hilbert space for $α\in (1,2)$. The coefficients are assumed to map between certain domains of fractional powers of the generator present in the equation. The solution is constructed as a weak limit of the Picard iteration using tightness arguments. Existence of strong solution is obtained by a general version of the Yamada--Watanabe theorem.

math.PR↗

arXiv2019

Variational solutions of stochastic partial differential equations with cylindrical Lévy noise

In this article, the existence of a unique solution in the variational approach of the stochastic evolution equation $$\dX(t) = F(X(t)) \dt + G(X(t)) \dL(t)$$ driven by a cylindrical Lévy process $L$ is established. The coefficients $F$ and $G$ are assumed to satisfy the usual monotonicity and coercivity conditions. The noise is modelled by a cylindrical Lévy processes which is assumed to belong to a certain subclass of cylindrical Lévy processes and may not have finite moments.

math.PR↗

arXiv2019

Stochastic integration with respect to cylindrical Lévy processes by p-summing operators

We introduce a stochastic integral with respect to cylindrical Lévy processes with finite $p$-th weak moment for $p\in [1,2]$. The space of integrands consists of $p$-summing operators between Banach spaces of martingale type $p$. We apply the developed integration theory to establish the existence of a solution for a stochastic evolution equation driven by a cylindrical Lévy process.

math.PR↗

arXiv2019

Modelling Levy space-time white noises

Based on the theory of independently scattered random measures, we introduce a natural generalisation of Gaussian space-time white noise to a Levy-type setting, which we call Levy-valued random measures. We determine the subclass of cylindrical Levy processes which correspond to Levy-valued random measures, and describe the elements of this subclass uniquely by their characteristic function. We embed the Levy-valued random measure, or the corresponding cylindrical Levy process, in the space of general and tempered distributions. For the latter case, we show that this embedding is possible if and only if a certain integrability condition is satisfied. Similar to existing definitions, we introduce Levy-valued additive sheets, and show that integrating a Levy-valued random measure in space defines a Levy-valued additive sheet. This relation is manifested by the result, that a Levy-valued random measure can be viewed as the weak derivative of a Levy-valued additive sheet in the space of distributions.

math.PR↗

arXiv2019

Invariant measure for the stochastic Cauchy problem driven by a cylindrical Lévy process

In this work, we present sufficient conditions for the existence of a stationary solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Lévy process, and show that these conditions are also necessary if the semigroup is stable, in which case the invariant measure is unique. For typical situations such as the heat equation, we significantly simplify these conditions without assuming any further restrictions on the driving cylindrical Lévy process and demonstrate their application in some examples.

math.PR↗

arXiv2018

Stable cylindrical Lévy processes and the stochastic Cauchy problem

In this work, we consider the stochastic Cauchy problem driven by the canonical $α$-stable cylindrical Lévy process. This noise naturally generalises the cylindrical Brownian motion or space-time Gaussian white noise. We derive a sufficient and necessary condition for the existence of the weak and mild solution of the stochastic Cauchy problem and establish the temporal irregularity of the solution.

math.PR↗

arXiv2018

The stochastic Cauchy problem driven by a cylindrical Levy process

In this work, we derive sufficient and necessary conditions for the existence of a weak and mild solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Levy process. Our approach requires to establish a stochastic Fubini result for stochastic integrals with respect to cylindrical Levy processes. This approach enables us to conclude that the solution process has almost surely scalarly square integrable paths. Further properties of the solution such as the Markov property and stochastic continuity are derived.

math.PR↗

arXiv2016

Large deviations for stochastic heat equations with memory driven by Levy-type noise

For a heat equation with memory driven by a Lévy-type noise we establish the existence of a unique solution. The main part of the article focuses on the Freidlin-Wentzell large deviation principle of the solutions of heat equation with memory driven by a Lévy-type noise. For this purpose, we exploit the recently introduced weak convergence approach.

math.PR↗

arXiv2016

Stochastic integration with respect to cylindrical Lévy processes

A cylindrical Levy process does not enjoy a cylindrical version of the semi-martingale decomposition which results in the need to develop a completely novel approach to stochastic integration. In this work, we introduce a stochastic integral for random integrands with respect to cylindrical Levy processes in Hilbert spaces. The space of admissible integrands consists of adapted stochastic processes with values in the space of Hilbert-Schmidt operators. Neither the integrands nor the integrator is required to satisfy any moment or boundedness condition. The integral process is characterised as an adapted, Hilbert space valued semi-martingale with cadlag trajectories.

math.PR↗

arXiv2015

Copulas in Hilbert spaces

In this article, the concept of copulas is generalised to infinite dimensional Hilbert spaces. We show one direction of Sklar's theorem and explain that the other direction fails in infinite dimensional Hilbert spaces. We derive a necessary and sufficient condition which allows to state this direction of Sklar's theorem in Hilbert spaces. We consider copulas with densities and specifically construct copulas in a Hilbert space by a family of pairwise copulas with densities.

math.PR↗