arXiv subjects

Martin Grothaus

Publications and source records attributed to Martin Grothaus.

At least 19 recordsLinked to original sources

arXiv2026

Hypocoercivity for Hamiltonian Diffusions with Singular Drift

We establish $L^2$-exponential strong ergodicity (strong mixing) with an explicit rate of convergence for a class of degenerate diffusions with multiplicative noise and with singular drift in both the noisy and noise-free components. This class includes diffusions with an additional inert drift given by the gradient of a singular potential, as well as singular generalized stochastic Hamiltonian systems. Cases in which the diffusion is confined to a proper, bounded or unbounded subset of $\mathbb{R}^{d_1+d_2}$ are included. Concrete examples of admissible potentials are provided. To obtain these results, we use an analytical approach and study the long-time behavior of the strongly continuous contraction semigroup generated by the formal Kolmogorov backward operator. Using the theory of generalized Dirichlet forms, these objects are then identified with the transition semigroup and generator of the unique weak solution to the original stochastic differential equation. The existence and uniqueness of this solution are established under near-minimal conditions.

math.PR↗

arXiv2026

The Ornstein$-$Uhlenbeck process on $\mathscr P_2$ with a volatility operator

We analyze a diffusion ${(\mu_t)}_{t\geq 0}$ on the $2$-Wasserstein space $\mathscr P_2$ over $\mathbb R^d$ for which \begin{equation*} |\mu_t|_2^2-|\mu_0|_2^2-2ct+2\int_0 ^t|\mu_s|_2^2\,d s,\qquad t\geq 0, \end{equation*} is a martingale, where the constant $c\in(0,\infty)$ equals the trace of a volatility operator on a Hilbert space and $|\mu_t|_2:=(\int_{\mathbb R^d}x^T x\mu_t(d x ))^{1/2}$. The invariant measure of ${(\mu_t)}_{t\geq 0}$ is a Gaussian on $\mathscr P_2$, as introduced by P. Ren and F.-Y. Wang. Moreover, the Dirichlet form and its generator are given explicitly on a dense subspace of $L^2$.

math.PR↗

arXiv2026

On Skorokhod Problems for Reflected and Singular Stochastic Heat Equations

We prove a Skorokhod decomposition for the Markov processes $X^a$ and $X$ associated to the gradient Dirichlet forms with respect to the measures $\rho^a\mu^{\beta}$ and $\rho\mu^{\beta}$, respectively. Here, $\mu^{\beta}$ is the law of the standard Brownian bridge $\beta$, while $\rho^a$ and $\rho$ denote densities which are given by $\rho^a(z) := \mathbf{1}_{[0,\infty)}(\bar{z}_a)$ and $\rho(z) := \int_0^1 \mathbf{1}_{[0,\infty)}(\bar{z}_x) \, dx$, respectively, for all $z\in L^2(0,1)$ which have a (unique) continuous representative $\bar{z}$ which vanishes at zero and one. To this end, we derive infinite-dimensional integration by parts formulas (IbPFs) w.r.t. $\rho^a\mu^{\beta}$ and $\rho\mu^{\beta}$, which contain Hida distributions alongside the usual drift terms. We represent these Hida distributions by integration w.r.t. vector measures of bounded variation. The vector measures in question are constructed via an approximation argument, making use of a generalization of Prokhorov's theorem for vector measures. We further prove that, almost surely, the sample paths of $X^a$ and $X$ take values in the equivalence class of continuous functions vanishing at zero and one for all and $dt$-almost all times, respectively. The main motivation for studying $\rho^a\mu^{\beta}$ and $\rho\mu^{\beta}$ lies in the fact that the distributional terms in their IbPFs are simplifications of the distributional term in the IbPF w.r.t. the law of the reflected Brownian bridge on the unit interval $\mu^{|\beta|}$. Representing the latter by integration w.r.t. a vector measure of bounded variation is still an open problem.

math.PR↗

arXiv2026

Hypocoercive Langevin dynamics on the Lie group $\mathrm{SE}(2)$

We consider a Langevin-type diffusion on the planar motion group $\mathrm{SE}(2)$, describing the coupled evolution of position and orientation with degenerate noise acting only in the rotational direction. Although hypocoercivity for related models on $\mathbb{R}^2 \times \mathbb{S}^1$ is well understood, our purpose is to present an intrinsic formulation on the Lie group $\mathrm{SE}(2)$, and to highlight the underlying geometric mechanism. By expressing the generator in terms of invariant vector fields and using the natural projection onto the kernel of the symmetric part, we show how an effective macroscopic diffusion on $\mathbb{R}^2$ emerges through averaging over the compact rotation subgroup.

math.PR↗

arXiv2026

Characterization of the (fractional) Malliavin-Watanabe-Sobolev spaces $\mathcal{D}^{\alpha,2}$ via the Bargmann-Segal norm

Motivated by an open question going back to P.Malliavin and P.-A.Meyer (and closely related to the foundational work of S.Watanabe) on whether Malliavin-Watanabe-Sobolev regularity admits a characterization in terms of a holomorphic Laplace image similar as for Hida distributions, we establish a characterization of the spaces $\mathcal{D}^{\alpha,2}$ for all $\alpha\in\mathbb{R}$ via the Bargmann-Segal norm of the $S$-transform. More precisely, we express $\mathcal{D}^{\alpha,2}$-regularity, $\alpha > 0$, of $F\in L^{2}(\mu)$, as well as dual regularity of distributions, in terms of integrability, differentiability and growth properties of the function \[ (0,1) \ni \lambda \longmapsto \int_{\mathcal{S}'_{\mathbb{C}}} |SF(\lambda u)|^{2}\,d\nu(u) \] involving integer-order derivatives in $\lambda$ for $\alpha\in\mathbb{N}$ and Riemann-Liouville fractional derivatives/integrals for non-integer $\alpha$. Here $\nu$ is the Gaussian Bargmann-Segal measure. This yields practical criteria for both positive and negative (including fractional) orders of Malliavin regularity and thereby bridges Malliavin calculus and Bargmann-Segal techniques from white noise analysis. Applications are worked out for Donsker's delta, self-intersection local times of Gaussian processes, and Gauss kernels.

math.PR↗

arXiv2024

The essential m-dissipativity for degenerate infinite dimensional stochastic Hamiltonian systems and applications

We consider a degenerate infinite dimensional stochastic Hamiltonian system with multiplicative noise and establish the essential m-dissipativity on $L^2(\mu^{\Phi})$ of the corresponding Kolmogorov (backwards) operator. Here, $\Phi$ is the potential and $\mu^{\Phi}$ the invariant measure with density $e^{-\Phi}$ with respect to an infinite dimensional non-degenerate Gaussian measure. The main difficulty, besides the non-sectorality of the Kolmogorov operator, is the coverage of a large class of potentials. We include potentials that have neither a bounded nor a Lipschitz continuous gradient. The essential m-dissipativity is the starting point to establish the hypocoercivity of the strongly continuous contraction semigroup $(T_t)_{t\geq 0}$ generated by the Kolmogorov operator. By using the refined abstract Hilbert space hypocoercivity method of Grothaus and Stilgenbauer, originally introduced by Dolbeault, Mouhot and Schmeiser, we construct a $\mu^{\Phi}$-invariant Hunt process with weakly continuous paths and infinite lifetime, whose transition semigroup is associated with $(T_t)_{t\geq 0}$. This process provides a stochastically and analytically weak solution to the degenerate infinite dimensional stochastic Hamiltonian system with multiplicative noise. The hypocoercivity of $(T_t)_{t\geq 0}$ and the identification of $(T_t)_{t\geq 0}$ with the transition semigroup of the process leads to the exponential ergodicity. Finally, we apply our results to degenerate second order in time stochastic reaction-diffusion equations with multiplicative noise. A discussion of the class of applicable potentials and coefficients governing these equations completes our analysis.

math.PR↗

arXiv2024

Construction of distorted Brownian motion with permeable sticky behaviour on sets with Lebesgue measure zero

The starting point is a gradient Dirichlet form with respect to $\varrho\lambda^d$ on the space $L^2({\mathbb{R}}^d, \varrho\mu)$. Here $\lambda^d$ is the Lebesgue measure on ${\mathbb R}^d$, $\varrho$ a strictly positive density and $\mu$ puts weight on a set $A\subset {\mathbb R}^d$ with Lebesgue measure zero. We show that the Dirichlet form admits an associated stochastic process $X$. We derive an explicit representation of the corresponding generator if $A$ is a Lipschitz boundary. This representation together with the Fukushima decomposition identifies $X$ as a distorted Brownian motion with drift given by the logarithmic derivative of $\varrho$ in ${\mathbb R}^d \setminus A$. Furthermore, we prove $X$ to be irreducible and recurrent. Finally, via ergodicity we prove positive s\'ejour time of $X$ on $A$. Hence we obtain a stochastic process $X$ with permeable sticky behaviour on $A$.

math.PR↗

arXiv2024

Mosco convergence of gradient forms with non-convex potentials II

This article provides a scaling limit for a family of skew interacting Brownian motions in the context of mesoscopic interface models. Let $d\in\mathbb N$, $y_1,\dots,y_M\in\mathbb R$ and $f\in C_b(\mathbb R)$ be fixed. For each $N\in\mathbb N$ we consider a $k_N$-dimensional, skew reflecting distorted Brownian motion $(X^{N,i}_t)_{i=1,\dots,k_N}$, $t\geq 0$, and investigate the scaling limits for $N\to\infty$. The drift includes skew reflections at height levels $\tilde y_j:=N^{1-\frac{d}{2}}y_j$ with intensities $β_j/N^d$ for $j=1,\dots,M$. The corresponding SDE is given by \begin{equation} d X^{N,i}_t=-\big(A_N X^{N}_t\big)_id t-\frac{1}{2}N^{-\tfrac{d}{2}-1}\,f\big(N^{\frac{d}{2}-1}X^{N,i}_t\big)d t \\+\sum_{j=1}^M\tfrac{1-e^{-β_j/N^d}}{1+e^{-β_j/N^d}}d l_t^{N,i, \tilde y_j} +d B_t^{N,i}, \end{equation} where ${(B_t^{N,i})}_{t\geq 0}$, $i=1,\dots, k_N$, are independent Brownian motions and $ l_t^{N,i, \tilde y_j}$ denotes the local time of ${(X^{N,i}_t)}_{t\geq 0}$ at $\tilde y_j$. We prove the weak convergence of the equilibrium laws of \begin{equation*} u_t^N=Λ_N\circ X^{N}_{N^2t},\quad t\geq 0, \end{equation*} for $N\to\infty$, choosing suitable injective, linear maps $Λ_N:\mathbb R^{k_N}\to \{h\,|\,h:D\to\mathbb R\}$. The scaling limit is a distorted Ornstein-Uhlenbeck process whose state space is the Hilbert space $H=L^2(D, dz)$. We characterize a class of height maps, such that the scaling limit of the dynamic is not influenced by the particular choice of ${(Λ_N)}_{N\in\mathbb N}$ within that class.

math.PR↗

arXiv2024

Stochastic Currents of Fractional Brownian Motion: Existence and Regularity

By using white noise analysis, we study the integral kernel $\xi(x)$, $x\in\mathbb{R}^{d}$, of stochastic currents corresponding to fractional Brownian motion with Hurst parameter $H\in(0,1)$. For $x\in\mathbb{R}^{d}\backslash\{0\}$ and $d\ge1$ we show that the kernel $\xi(x)$ is well-defined as a Hida distribution for all $H\in(0,1)$. For $x=0$ and $d=1$, $\xi(0)$ is a Hida distribution for all $H\in(0,1)$. For $d\ge2$, then $\xi(0)$ is a Hida distribution only for $H\in(0,1/d)$. For $d=1$, $x \neq 0$, and $H \in (0,1)$, we show that $\xi(x) \in \mathcal{G}'$, the space of regular generalized functions. Elements of the space $\mathcal{G}'$ and elements from the negative Sobolev--Watanabe distribution spaces share the property that partial sums of their chaos decomposition are square integrable functions. More precisely, we show that $\xi(x) \in \mathcal{G}_{-s} \subset \mathcal{G}'$ for $x \neq 0$, $H \in (0,1)$, and all $s > 0$.

math.PR↗

arXiv2024

Mosco convergence of gradient forms with non-convex interaction potential

This article provides a new approach to address Mosco convergence of gradient-type Dirichlet forms, $\mathcal E^N$ on $L^2(E,μ_N)$ for $N\in\mathbb N$, in the framework of converging Hilbert spaces by K.~Kuwae and T.~Shioya. The basic assumption is weak measure convergence of the family ${(μ_N)}_{N}$ on the state space $E$ - either a separable Hilbert space or a locally convex topological vector space. Apart from that, the conditions on ${(μ_N)}_{N}$ try to impose as little restrictions as possible. The problem has fully been solved if the family ${(μ_N)}_{N}$ contain only log-concave measures, due to L.~Ambrosio, G.~Savaré and L.~Zambotti, 2009. However for a large class of convergence problems the assumption of log-concavity fails. The article suggests a way to overcome this hindrance, as it presents a new approach. Combining the theory of Dirichlet forms with methods from numerical analysis we find abstract criteria for Mosco convergence of standard gradient forms with varying reference measures. These include cases in which the measures are not log-concave. To demonstrate the accessibility of our abstract theory we discuss a first application, generalizing an approximation result by S.~K.~Bounebache and L.~Zambotti, 2014.

math.PR↗

arXiv2023

Hypocoercivity for infinite-dimensional non-linear degenerate stochastic differential equations with multiplicative noise

We analyze infinite-dimensional non-linear degenerate stochastic differential equations with multiplicative noise. First, essential m-dissipativity of their associated Kolmogorov backward generators on $L^2(μ^Φ)$ defined on smooth finitely based functions is established. Here $Φ$ is an appropriate potential and $μ^Φ$ is the invariant measure with density $e^{-Φ}$ w.r.t. an infinite-dimensional non-degenerate Gaussian measure. Second, we use resolvent methods to construct corresponding right processes with infinite lifetime, solving the martingale problem for the Kolmogorov backward generators. They provide weak solutions, with weakly continuous paths, to the non-linear degenerate stochastic partial differential equations. Moreover, we identify the transition semigroup of such a process with the strongly continuous contraction semigroup $(T_t)_{t\geq 0}$ generated by the corresponding Kolmogorov backwards generator. Afterwards, we apply a refinement of the abstract Hilbert space hypocoercivity method, developed by Dolbeaut, Mouhot and Schmeiser, to derive hypocoercivity of $(T_t)_{t\geq 0}$. I.e. we take domain issues into account and use the formulation in the Kolmogorov backwards setting worked out by Grothaus and Stilgenbauer. The method enables us to explicitly compute the constants determining the exponential convergence rate to equilibrium of $(T_t)_{t\geq 0}$. The identification between $(T_t)_{t\geq 0}$ and the transition semigroup of the process yields exponential ergodicity of the latter. Finally, we apply our results to second order in time stochastic reaction-diffusion and Cahn-Hilliard type equations with multiplicative noise. More generally, we analyze corresponding couplings of infinite-dimensional deterministic with stochastic differential equations.

math.PR↗

arXiv2023

Hypocoercivity for non-linear infinite-dimensional degenerate stochastic differential equations

The aim of this article is to construct solutions to second order in time stochastic partial differential equations and to show hypocoercivity of the corresponding transition semigroups. More generally, we analyze non-linear infinite-dimensional degenerate stochastic differential equations in terms of their infinitesimal generators. In the first part of this article we use resolvent methods developed by Beznea, Boboc and Röckner to construct diffusion processes with infinite lifetime and explicit invariant measures. The processes provide weak solutions to infinite-dimensional Langevin dynamics. The second part deals with a general abstract Hilbert space hypocoercivity method, developed by Dolbeaut, Mouhot and Schmeiser. In order to treat stochastic (partial) differential equations, Grothaus and Stilgenbauer translated these concepts to the Kolmogorov backwards setting taking domain issues into account. To apply these concepts in the context of infinite-dimensional Langevin dynamics we use an essential m-dissipativity result for infinite-dimensional Ornstein-Uhlenbeck operators, perturbed by the gradient of a potential. We allow unbounded diffusion operators as coefficients and apply corresponding regularity estimates. Furthermore, essential m-dissipativity of a non-sectorial Kolmogorov backward operator associated to the dynamic is needed. Poincaré inequalities for measures with densities w.r.t. infinite-dimensional non-degenerate Gaussian measures are studied. Deriving a stochastic representation of the semigroup generated by the Kolmogorov backward operator as the transition semigroup of a diffusion process enables us to show an $L^2$-exponential ergodicity result for the weak solution. Finally, we apply our results to explicit infinite-dimensional degenerate diffusion equations.

math.PR↗

arXiv2023

Singular Degenerate SDEs: Well-Posedness and Exponential Ergodicity

The well-posedness and exponential ergodicity are proved for stochastic Hamiltonian systems containing a singular drift term which is locally integrable in the component with noise. As an application, the well-posedness and uniform exponential ergodicity are derived for a class of singular degenerated McKean-Vlasov SDEs.

math.PR↗

arXiv2022

Essential m-dissipativity and hypocoercivity of Langevin dynamics with multiplicative noise

We provide a complete elaboration of the $L^2$-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behaviour of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in the corresponding Fokker-Planck framework, and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. We extend these results to weakly differentiable diffusion coefficient matrices, introducing multiplicative noise for the corresponding stochastic differential equation. The rate of convergence is explicitly computed depending on the choice of these coefficients and the potential giving the outer force. In order to obtain a solution to the abstract Cauchy problem, we first prove essential self-adjointness of non-degenerate elliptic Dirichlet operators on Hilbert spaces, using prior elliptic regularity results and techniques from Bogachev, Krylov and Röckner. We apply operator perturbation theory to obtain essential m-dissipativity of the Kolmogorov operator, extending the m-dissipativity results from Conrad and Grothaus. We emphasize that the chosen Kolmogorov approach is natural, as the theory of generalized Dirichlet forms implies a stochastic representation of the Langevin semigroup as the transition kernel of a diffusion process which provides a martingale solution to the Langevin equation with multiplicative noise. Moreover, we show that even a weak solution is obtained this way.

math.FA↗

arXiv2021

Stochastic analysis for vector-valued generalized grey Brownian motion

In this article, we show that the standard vector-valued generalization of a generalized grey Brownian motion (ggBm) has independent components if and only if it is a fractional Brownian motion. In order to extend ggBm with independent components, we introduce a vector-valued generalized grey Brownian motion (vggBm). The characteristic function of the corresponding measure is introduced as the product of the characteristic functions of the one-dimensional case. We show that for this measure, the Appell system and a calculus of generalized functions or distributions are accessible. We characterize these distributions with suitable transformations and give a d-dimensional Donsker's delta function as an example for such distributions. From there, we show the existence of local times and self-intersection local times of vggBm as distributions under some constraints, and compute their corresponding generalized expectations. At the end, we solve a system of linear SDEs driven by a vggBm noise in d dimensions.

math.PR↗

arXiv2021

Dirichlet form analysis of the Jacobi process

We construct and analyze the Jacobi process - in mathematical biology referred to as Wright-Fisher diffusion - using a Dirichlet form. The corresponding Dirichlet space takes the form of a Sobolev space with different weights for the function itself and its derivative. Depending on the parameters we characterize the boundary behavior of the functions in the Dirichlet space, show density results, derive Sobolev embeddings and verify functional inequalities of Hardy type. Since the generator is a hypergeometric differential operator, many of the proofs can be carried out by explicit calculations involving hypergeometric functions. We deduce corresponding properties for the associated semigroup and Markov process and show that the latter is up to minor technical modifications a solution to the Jacobi SDE.

math.PR↗

arXiv2021

Convergence Rate for Degenerate Partial and Stochastic Differential Equations via weak Poincaré Inequalities

We employ weak hypocoercivity methods to study the long-term behavior of operator semigroups generated by degenerate Kolmogorov operators with variable second-order coefficients, which solve the associated abstract Cauchy problem. We prove essential m-dissipativity of the operator, which extends previous results and is key to the rigorous analysis required. We give estimates for the $L^2$-convergence rate by using weak Poincaré inequalities. As an application, we obtain estimates for the (sub-)exponential convergence rate of solutions to the corresponding degenerate Fokker-Planck equations and of weak solutions to the corresponding degenerate stochastic differential equation with multiplicative noise.

math.PR↗

arXiv2021

An improved characterisation of regular generalised functions of white noise and an application to singular SPDEs

A characterisation of the spaces $\mathcal{G}_K$ and $\mathcal{G}_K'$ introduced in Grothaus et al. (Methods Funct Anal Topol 3(2):46-64, 1997) and Potthoff and Timpel (Potential Anal 4(6):637-654, 1995) is given. A first characterisation of these spaces provided in Grothaus et al. (Methods Funct Anal Topol 3(2):46-64, 1997) uses the concepts of holomorphy on infinite dimensional spaces. We, instead, give a characterisation in terms of U-functionals, i.e., classic holomorphic function on the one dimensional field of complex numbers. We apply our new characterisation to derive new results concerning a stochastic transport equation and the stochastic heat equation with multiplicative noise.

math.PR↗