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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.

The starting point is a gradient Dirichlet form with respect to $\varrho\lambda^d$ on $L^2({\mathbb{R}}^d, \varrho\mu)$. 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 $A$.

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 $\beta_j/N^d$ 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^{-\beta_j/N^d}}{1+e^{-\beta_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=\Lambda_N\circ X^{N}_{N^2t},\quad t\geq 0, \end{equation*} for $N\to\infty$, choosing suitable injective, linear maps $\Lambda_N:\mathbb R^{k_N}\to \{h\,|\,h:D\to\mathbb R\}$. 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 ${(\Lambda_N)}_{N\in\mathbb N}$ within that class.

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 $H\in(0,1/2]$. 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 $H\in(0,1/d)$. To cover the case $H\in[1/d,1)$ we have to truncate the delta function so that $\xi^{(N)}(0)$ is a Hida distribution whenever $2N(H-1)+Hd>1$., Comment: 17 pages

We analyze infinite-dimensional non-linear degenerate stochastic differential equations with multiplicative noise. First, essential m-dissipativity of their associated Kolmogorov backward generators on $L^2(\mu^{\Phi})$ defined on smooth finitely based functions is established. Here $\Phi$ is an appropriate potential and $\mu^{\Phi}$ is the invariant measure with density $e^{-\Phi}$ 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.

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., Comment: 26 pages

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.

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., Comment: 34 pages

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 Poincar\'e 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.

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 R\"ockner 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. Poincar\'e 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.

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.

In this paper we study stochastic currents of Brownian motion $\xi(x)$, $x\in\mathbb{R}^{d}$, by using white noise analysis. For $x\in\mathbb{R}^{d}\backslash\{0\}$ and for $x=0\in\mathbb{R}$ we prove that the stochastic current $\xi(x)$ is a Hida distribution. Moreover for $x=0\in\mathbb{R}^{d}$ with $d>1$ we show that the stochastic current is not a Hida distribution., Comment: 10 pages

This article provides a new approach to address Mosco convergence of gradient-type Dirichlet forms, $\mathcal E^N$ on $L^2(E,\mu_N)$ 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 ${(\mu_N)}_{N}$ on the state space $E$ - either a separable Hilbert space or a locally convex topological vector space. Apart from that, the conditions on ${(\mu_N)}_{N}$ try to impose as little restrictions as possible. The problem has fully been solved if the family ${(\mu_N)}_{N}$ contain only log-concave measures, due to L.~Ambrosio, G.~Savar\'e 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., Comment: 56 pages, 8 figures

First essential m-dissipativity of an infinite-dimensional Ornstein-Uhlenbeck operator $N$, perturbed by the gradient of a potential, on a domain $\mathcal{F}C_b^{\infty}$ of finitely based, smooth and bounded functions, is shown. Our considerations allow unbounded diffusion operators as coefficients. We derive corresponding second order regularity estimates for solutions $f$ of the Kolmogorov equation $\alpha f-Nf=g$, $\alpha \in (0,\infty)$, generalizing some results of Da Prato and Lunardi. Second we prove essential m-dissipativity for generators $(L_{\Phi},\mathcal{F}C_b^{\infty})$ of infinite-dimensional non-linear degenerate diffusion processes. We emphasize that the essential m-dissipativity of $(L_{\Phi},\mathcal{F}C_b^{\infty})$ is useful to apply general resolvent methods developed by Beznea, Boboc and R\"ockner, in order to construct martingale/weak solutions to infinite-dimensional non-linear degenerate diffusion equations. Furthermore, the essential m-dissipativity of $(L_{\Phi},\mathcal{F}C_b^{\infty})$ and $(N,\mathcal{F}C_b^{\infty})$, as well as the regularity estimates are essential to apply the general abstract Hilbert space hypocoercivity method from Dolbeault, Mouhot, Schmeiser and Grothaus, Stilgenbauer, respectively, to the corresponding diffusions.

We construct Schwinger functions as the superposition of Schwinger functions which correspond to those of free fields with sharp masses $ m $. We prove that all axioms of Osterwalder and Schrader are satisfied. This construction works independently of the space-time dimension $ d $. The Schwinger functions under consideration are the moments of a non-Gaussian measure on the space of tempered distributions $ S'(\mathbb R^d) $., Comment: The proof of the cluster property (E4) for the superposition is wrong. Yoh Tanimoto and Wojciech Dybalski gave a counterexample s.t. the statement of cluster property can't hold in its full generality

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 R\"ockner. 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., Comment: Replaced with accepted version, added DOI reference to published version, 30 pages

In this article we investigate hypocoercivity of Langevin-type dynamics in nonlinear smooth geometries. The main result stating exponential decay to an equilibrium state with explicitly computable rate of convergence is rooted in an appealing Hilbert space strategy by Dolbeault, Mouhot and Schmeiser. This strategy was extended in [GS14] to Kolmogorov backward evolution equations in contrast to the dual Fokker-Planck framework. We use this mathematically complete elaboration to investigate wide ranging classes of Langevin-type SDEs in an abstract manifold setting, i.e. (at least) the position variables obey certain smooth side conditions. Such equations occur e.g. as fibre lay-down processes in industrial applications. We contribute the Lagrangian-type formulation of such geometric Langevin dynamics in terms of (semi-)sprays and point to the necessity of fibre bundle measure spaces to specify the model Hilbert space., Comment: 46 pages, 0 figures, 2 comm. diagrams

First weak solutions of generalized stochastic Hamiltonian systems (gsHs) are constructed via essential m-dissipativity of their generators on a suitable core. For a scaled gsHs we prove convergence of the corresponding semigroups and tightness of the weak solutions. This yields convergence in law of the scaled gsHs to a distorted Brownian motion. In particular, the results confirm the convergence of the Langevin dynamics in the overdamped regime to the overdamped Langevin equation. The proofs work for a large class of (singular) interaction potentials including, e.g., potentials of Lennard--Jones type.

We construct Skorokhod decompositions for diffusions with singular drift and reflecting boundary behavior on open subsets of $\mathbb R^d$ with $C^2$-smooth boundary except for a sufficiently small set. This decomposition holds almost surely under the path measures of the process for every starting point from an explicitly known set. This set is characterized by the boundary smoothness and the singularities of the drift term. We apply modern methods of Dirichlet form theory and $\mathcal L^p$-strong Feller processes. These tools have been approved as useful for the pointwise analysis of stochastic processes with singular drift and various boundary conditions. Furthermore, we apply Sobolev space theorems and elliptic regularity results to prove regularity properties of potentials related to surface measures. These are important ingredients for the pointwise construction of the boundary local time of the diffusions under consideration. As an application we construct stochastic dynamics for particle systems with hydrodynamic and pair interaction. Our approach allows highly singular potentials like Lennard-Jones potentials and position-dependent diffusion coefficients and thus the treatment of physically reasonable models., Comment: Accepted for publication in "Stochastics", electronic pre-print available

For a contraction $C_0$-semigroup on a separable Hilbert space, the decay rate is estimated by using the weak Poincar\'e inequalities for the symmetric and anti-symmetric part of the generator. As applications, non-exponential convergence rate is characterized for a class of degenerate diffusion processes, so that the study of hypocoercivity is extended. Concrete examples are presented.

We prove an infinite dimensional integration by parts formula on the law of the modulus of the Brownian bridge $BB=(BB_t)_{0 \leq t \leq 1}$ from $0$ to $0$ in use of methods from white noise analysis and Dirichlet form theory. Additionally to the usual drift term, this formula contains a distribution which is constructed in the space of Hida distributions by means of a Wick product with Donsker's delta (which correlates with the local time of $|BB|$ at zero). This additional distribution corresponds to the reflection at zero caused by the modulus.

We provide a complete elaboration of the $L^2$-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics via studying the longtime behavior of the strongly continuous contraction semigroup solving the associated Kolmogorov (backward) equation as an abstract Cauchy problem. This hypocoercivity result is proven in previous works before by Dolbeault, Mouhot and Schmeiser in the corresponding dual Fokker-Planck framework, but without including domain issues of the appearing operators. In our elaboration, we include the domain issues and additionally compute the rate of convergence in dependence of the damping coefficient. Important statements for the complete elaboration are the m-dissipativity results for the Langevin operator established by Conrad and the first named author of this article as well as the essential selfadjointness results for generalized Schr\"{o}dinger operators by Wielens or Bogachev, Krylov and R\"{o}ckner. We emphasize that the chosen Kolmogorov approach is natural. Indeed, techniques from the theory of (generalized) Dirichlet forms imply a stochastic representation of the Langevin semigroup as the transition kernel of diffusion process which provides a martingale solution to the Langevin equation. Hence an interesting connection between the theory of hypocoercivity and the theory of (generalized) Dirichlet forms is established besides., Comment: Published in Methods of Functional Analysis and Topology (MFAT), available at http://mfat.imath.kiev.ua/article/?id=847

In this article we develop a new abstract strategy for proving ergodicity with explicit computable rate of convergence for diffusions associated with a degenerate Kolmogorov operator L. A crucial point is that the evolution operator L may have singular and nonsmooth coefficients. This allows the application of the method e.g. to degenerate and singular particle systems arising in Mathematical Physics. As far as we know in such singular cases the relaxation to equilibrium can't be discussed with the help of existing approaches using hypoellipticity, hypocoercivity or stochastic Lyapunov type techniques. The method is formulated in an L2-Hilbert space setting and is based on an interplay between Functional Analysis and Stochastics. Moreover, it implies an ergodicity rate which can be related to L2-exponential convergence of the semigroup. Furthermore, the ergodicity method shows up an interesting analogy with existing hypocoercivity approaches. In the first application we discuss ergodicity of the N-particle degenerate Langevin dynamics with singular potentials. The dual to this equation is also called the kinetic Fokker-Planck equation with an external confining potential. In the second example we apply the method to the so-called (degenerate) spherical velocity Langevin equation which is also known as the fiber lay-down process arising in industrial mathematics., Comment: Older preprint version of the paper (from 2014). The final publication is available at Springer via http://dx.doi.org/10.1007/s00020-015-2254-1 It was published under the name "A hypocoercivity related ergodicity method for singularly distorted non-symmetric diffusions"; see Integral Equations Oper. Theory 83, No. 3, Article ID 2254, 331-379 (2015)

Mittag-Leffler analysis is an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which generalizes the powerful theory of Gaussian analysis and in particular white noise analysis. In this paper we further develop the Mittag-Leffler analysis by characterizing the convergent sequences in the distribution space. Moreover we provide an approximation of Donsker's delta by square integrable functions. Then we apply the structures and techniques from Mittag-Leffler analysis in order to show that a Green's function to the time-fractional heat equation can be constructed using generalized grey Brownian motion (ggBm) by extending the fractional Feynman-Kac formula from Schneider. Moreover we analyse ggBm, show its differentiability in a distributional sense and the existence of corresponding local times., Comment: 45 pages

We construct a fundamental solution to the Schr\"odinger equation for a class of potentials of polynomial type by a complex scaling approach as in [Doss1980]. The solution is given as the generalized expectation of a white noise distribution. Moreover, we obtain an explicit formula as the expectation of a function of Brownian motion. This allows to show its differentiability in the classical sense. The admissible potentials may grow super-quadratically, thus by a result from [Yajima1996] the solution does not belong to the self-adjoint extension of the Hamiltonian., Comment: 32 pages

We give a Dirichlet form approach for the construction of distorted Brownian motion in a bounded domain $\Omega$ of $\mathbb{R}^d$, $d \geq 1$, with boundary $\Gamma$, where the behavior at the boundary is sticky. The construction covers the case of a static boundary behavior as well as the case of a diffusion on the hypersurface $\Gamma$ (for $d \geq 2)$. More precisely, we consider the state space $\overline{\Omega}=\Omega \stackrel{.}{\cup} \Gamma$, the process is a diffusion process inside $\Omega$, the occupation time of the process on the boundary $\Gamma$ is positive and the process may diffuse on $\Gamma$ as long as it sticks on the boundary. The problem is formulated in an $L^2$-setting and the construction is formulated under weak assumptions on the coefficients and $\Gamma$. In order to analyze the process we assume a $C^2$-boundary and some weak differentiability conditions. In this case, we deduce that the process is also a solution to a given SDE for quasi every starting point in $\overline{\Omega}$ with respect to the underyling Dirichlet form. Under the addtional condition that $\{ \varrho =0 \}$ is of capacity zero, we prove ergodicity of the constructed process and consequently, we verify that the boundary behavior is indeed sticky. Moreover, we show ($\mathcal{L}^p$-)strong Feller properties which allow to characterize the constructed process even for every starting point in $\overline{\Omega} \backslash \{ \varrho=0\}$.

Using Girsanov transformations we construct from sticky reflected Brownian motion on $[0,\infty)$ a conservative diffusion on $E:=[0,\infty)^n$, $n \in \mathbb{N}$, and prove that its transition semigroup possesses the strong Feller property for a specified general class of drift functions. By identifying the Dirichlet form of the constructed process, we characterize it as sticky reflected distorted Brownian motion. In particular, the relations of the underlying analytic Dirichlet form methods to the probabilistic methods of random time changes and Girsanov transformations are presented. Our studies are motivated by its applications to the dynamical wetting model with $\delta$-pinning and repulsion.

We give a Dirichlet form approach for the construction of a distorted Brownian motion in $E:=[0,\infty)^n$, $n\in\mathbb{N}$, where the behavior on the boundary is determined by the competing effects of reflection from and pinning at the boundary (sticky boundary behavior). In providing a Skorokhod decomposition of the constructed process we are able to justify that the stochastic process is solving the underlying stochastic differential equation weakly for quasi every starting point with respect to the associated Dirichlet form. That the boundary behavior of the constructed process indeed is sticky, we obtain by proving ergodicity of the constructed process. Therefore, we are able to show that the occupation time on specified parts of the boundary is positive. In particular, our considerations enable us to construct a dynamical wetting model (also known as Ginzburg--Landau dynamics) on a bounded set $D_{\scriptscriptstyle{N}}\subset \mathbb{Z}^d$ under mild assumptions on the underlying pair interaction potential in all dimensions $d\in\mathbb{N}$. In dimension $d=2$ this model describes the motion of an interface resulting from wetting of a solid surface by a fluid., Comment: arXiv admin note: substantial text overlap with arXiv:1203.2078

We construct an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which we call Mittag-Leffler measures. It turns out that the well-known Wick ordered polynomials in Gaussian analysis cannot be generalized to this non-Gaussian case. Instead of using Wick ordered polynomials we prove that a system of biorthogonal polynomials, called Appell system, is applicable to the Mittag-Leffler measures. Therefore we are able to introduce a test function and a distribution space. As an application we construct Donsker's delta in a non-Gaussian setting as a weak integral in the distribution space.

This note is devoted to representation of some evolution semigroups. The semigroups are generated by pseudo-differential operators, which are obtained by different (parametrized by a number $\tau$) procedures of quantization from a certain class of functions (or symbols) defined on the phase space. This class contains functions which are second order polynomials with respect to the momentum variable and also some other functions. The considered semigroups are represented as limits of $n$-fold iterated integrals when $n$ tends to infinity (such representations are called Feynman formulae). Some of these representations are constructed with the help of another pseudo-differential operators, obtained by the same procedure of quantization (such representations are called Hamiltonian Feynman formulae). Some representations are based on integral operators with elementary kernels (these ones are called Lagrangian Feynman formulae and are suitable for computations). A family of phase space Feynman pseudomeasures corresponding to different procedures of quantization is introduced. The considered evolution semigroups are represented also as phase space Feynman path integrals with respect to these Feynman pseudomeasures. The obtained Lagrangian Feynman formulae allow to calculate these phase space Feynman path integrals and to connect them with some functional integrals with respect to probability measures.

The concepts of Hamiltonian Feynman integrals in white noise analysis are used to realize as the first velocity dependent potential the Hamiltonian Feynman integrand for a charged particle in a constant magnetic field in coordinate space as a Hida distribution. For this purpose the velocity dependent potential gives rise to a generalized Gauss kernel. Besides the propagators also the generating functionals are obtained., Comment: arXiv admin note: substantial text overlap with arXiv:1203.0089, arXiv:1012.1125

Using the Feynman path integral representation of quantum mechanics it is possible to derive a model of an electron in a random system containing dense and weakly-coupled scatterers, see [Proc. Phys. Soc. 83, 495-496 (1964)]. The main goal of this paper is to give a mathematically rigorous realization of the corresponding Feynman integrand in dimension one based on the theory of white noise analysis. We refine and apply a Wick formula for the product of a square-integrable function with Donsker's delta functions and use a method of complex scaling. As an essential part of the proof we also establish the existence of the exponential of the self-intersection local times of a one-dimensional Brownian bridge. As result we obtain a neat formula for the propagator with identical start and end point. Thus, we obtain a well-defined mathematical object which is used to calculate the density of states, see e.g. [Proc. Phys. Soc. 83, 495-496 (1964)]., Comment: final version

In this article we extend the modern, powerful and simple abstract Hilbert space strategy for proving hypocoercivity that has been developed originally by Dolbeault, Mouhot and Schmeiser. As well-known, hypocoercivity methods imply an exponential decay to equilibrium with explicit computable rate of convergence. Our extension is now made for studying the long-time behavior of some strongly continuous semigroup generated by a (degenerate) Kolmogorov backward operator L. Additionally, we introduce several domain issues into the framework. Necessary conditions for proving hypocoercivity need then only to be verified on some fixed operator core of L. Furthermore, the setting is also suitable for covering existence and construction problems as required in many applications. The methods are applicable to various, different, Kolmogorov backward evolution problems. As a main part, we apply the extended framework to the (degenerate) spherical velocity Langevin equation. The latter can be seen as some kind of an analogue to the classical Langevin equation in case spherical velocities are required. This model is of important industrial relevance and describes the fiber lay-down in the production process of nonwovens. For the construction of the strongly continuous contraction semigroup we make use of modern hypoellipticity tools and pertubation theory.

In this article we develop geometric versions of the classical Langevin equation on regular submanifolds in euclidean space in an easy, natural way and combine them with a bunch of applications. The equations are formulated as Stratonovich stochastic differential equations on manifolds. The first version of the geometric Langevin equation has already been detected before by Leli\`evre, Rousset and Stoltz with a different derivation. We propose an additional extension of the models, the geometric Langevin equations with velocity of constant absolute value. The latters are seemingly new and provide a galaxy of new, beautiful and powerful mathematical models. Up to the authors best knowledge there are not many mathematical papers available dealing with geometric Langevin processes. We connect the first version of the geometric Langevin equation via proving that its generator coincides with the generalized Langevin operator proposed by Soloveitchik, Jorgensen and Kolokoltsov. All our studies are strongly motivated by industrial applications in modeling the fiber lay-down dynamics in the production process of nonwovens. We light up the geometry occuring in these models and show up the connection with the spherical velocity version of the geometric Langevin process. Moreover, as a main point, we construct new smooth industrial relevant three-dimensional fiber lay-down models involving the spherical Langevin process. Finally, relations to a class of self-propelled interacting particle systems with roosting force are presented and further applications of the geometric Langevin equations are given.

We study a stochastic equation modeling the lay-down of fibers in the production process of nonwovens. The equation can be formulated as some manifold-valued Stratonovich stochastic differential equation. Especially, we study the long time behaviour of the stochastic process. Demanding mathematical difficulties arising due to the degeneracity of the lay-down equation and its associated generator. We prove strong mixing properties by making use of the hypoellipticity of the generator and a new version of Doob's theorem derived recently by Gerlach and Nittka. Moreover, we show convergence to equilibrium exponentially fast with explicitly computable rate of convergence. This analytic approach uses powerful modern Hilbert space methods from the theory of hypocoercivity developed by Dolbeault, Mouhot and Schmeiser. Summarizing, we give interesting mathematical applications of geometric stochastic analysis to real world problems.

In this paper we explore a numerical scheme for a nonlinear fourth order system of partial differential algebraic equations that describes the dynamics of slender inextensible elastica as they arise in the technical textile industry. Applying a semi-discretization in time, the resulting sequence of nonlinear elliptic systems with the algebraic constraint for the local length preservation is reformulated as constrained optimization problems in a Hilbert space setting that admit a solution at each time level. Stability and convergence of the scheme are proved. The numerical realization is based on a finite element discretization in space. The simulation results confirm the analytically predicted properties of the scheme., Comment: Abstract and introduction are partially rewritten. The numerical study in Section 4 is completely rewritten

We give a Dirichlet form approach for the construction of a distorted Brownian motion in $E := [0;\infty)^n$, $n\in\mathbb{N}$, where the behavior on the boundary is determined by the competing effects of reflection from and pinning at the boundary. The problem is formulated in an $L^2$-setting with underlying measure $\mu=\varrho m$. Here $\varrho$ is a positive density, integrable with respect to the measure $m$ and fulfilling the Hamza condition. The measure $m$ is such that the boundary of $E$ is not of $m$-measure zero. A reference measure of this type is needed in order to give meaning to the so-called Wentzell boundary condition which is in literature typical for modeling such kind of boundary behavior. In providing a Skorokhod decomposition of the constructed process we are able to justify that the stochastic process is solving the underlying stochastic differential equation weakly in the sense of N. Ikeda and Sh. Watanabe for quasi every starting point. At the boundary the constructed process indeed is governed by the competing effects of reflection and pinning. In order to obtain the Skorokhod decomposition we need $\varrho$ to be continuously differentiable on $E$, which is equivalent to continuity of the logarithmic derivative of $\varrho$. Furthermore, we assume that the logarithmic derivative of $\varrho$ is square integrable with respect to $\mu$. We do not need that the logarithmic derivative of $\varrho$ is Lipschitz continuous. In particular, our considerations enable us to construct a dynamical wetting model (also known as Ginzburg-Landau dynamics) on a bounded set $D_N\subset\mathbb{Z}^d$ under mild assumptions on the underlying pair interaction potentials in all dimensions $d\in\mathbb{N}$. In dimension $d=2$ this model describes the motion of an interface resulting from wetting of a solid surface by a fluid.