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Marie Bormann

Publications and source records attributed to Marie Bormann.

6 recordsLinked to original sources
arXiv2026

A Dynkin condition for manifolds with boundary

We propose a Dynkin-type condition for smooth Riemannian manifolds with boundary. We show that this condition implies bi-Lipschitz equivalence with a Bakry-\'Emery weighted Riemannian manifold obtained via a time change. As a consequence, we obtain various results, including a local doubling property as well as lower bounds on the Neumann spectral gap and logarithmic Sobolev constant. The local doubling property also yields a new precompactness theorem for manifolds with boundary.

math.DG↗
arXiv2025

A gradient flow that is none: Heat flow with Wentzell boundary condition

We establish a representation of the heat flow with Wentzell boundary conditions on smooth domains as gradient descent dynamics for the entropy in a suitably extended Otto manifold of probability measures with additional boundary parts. Yet it is shown that for weak boundary diffusion, the associated Fokker-Planck dynamics cannot be recovered from any entropy-driven metric JKO-Wasserstein scheme, at least if the underlying point metric satisfies certain natural regularity assumptions. This discrepancy is illustrated in competing large-deviation heuristics in the Sanov and Schilder regimes.

math.AP↗
arXiv2024

Functional Inequalities for doubly weighted Brownian Motion with Sticky-Reflecting Boundary Diffusion

We give upper bounds for the Poincar\'e and Logarithmic Sobolev constants for doubly weighted Brownian motion on manifolds with sticky reflecting boundary diffusion under curvature assumptions on the manifold and its boundary. We therefor use an interpolation approach based on energy interactions between the boundary and the interior of the manifold and the weighted Reilly formula. Along the way we also obtain a lower bound on the first nontrivial doubly weighted Steklov eigenvalue and an upper bound on the norm of the doubly weighted boundary trace operator on Sobolev functions. We also consider the case of weighted Brownian motion with pure sticky reflection.

math.PR↗
arXiv2023

Functional Inequalities for Brownian Motion on Riemannian Manifolds with Sticky-Reflecting Boundary Diffusion

We prove geometric upper bounds for the Poincar\'e and Logarithmic Sobolev constants for Brownian motion on manifolds with sticky reflecting boundary diffusion i.e. extended Wentzell-type boundary condition under general curvature assumptions on the manifold and its boundary. The method is based on an interpolation involving energy interactions between the boundary and the interior of the manifold. As side results we obtain explicit geometric bounds on the first nontrivial Steklov eigenvalue, for the norm of the boundary trace operator on Sobolev functions, and on the boundary trace logarithmic Sobolev constant. The case of Brownian motion with pure sticky reflection is also treated.

math.PR↗
arXiv2022

Subordination principle and Feynman-Kac formulae for generalized time-fractional evolution equations

We consider generalized time-fractional evolution equations of the form $$u(t)=u_0+\int_0^tk(t,s)Lu(s)ds$$ with a fairly general memory kernel $k$ and an operator $L$ being the generator of a strongly continuous semigroup. In particular, $L$ may be the generator $L_0$ of a Markov process $\xi$ on some state space $Q$, or $L:=L_0+b\nabla+V$ for a suitable potential $V$ and drift $b$, or $L$ generating subordinate semigroups or Schr\"{o}dinger type groups. This class of evolution equations includes in particular time- and space- fractional heat and Schr\"odinger type equations. We show that a subordination principle holds for such evolution equations and obtain Feynman-Kac formulae for solutions of these equations with the use of different stochastic processes, such as subordinate Markov processes and randomly scaled Gaussian processes. In particular, we obtain some Feynman-Kac formulae with generalized grey Brownian motion and other related self-similar processes with stationary increments.

math.PR↗