Theory of Markov Semigroups
and Schrödinger Operators

Abstract:

We investigate the $L^p$-boundedness of the Hodge projection in the setting of manifolds with ends. We examine its relationship to the Riesz transform and the space of bounded harmonic functions. In particular, we explore how the $L^p$-boundedness of the Hodge projection is connected to the structure of $L^2$ harmonic one-forms and, subsequently, to the space of bounded harmonic functions.

Based on joint work with Julian Bailey, Himani Sharma, Marcin Preisner, Dangyang He, Andrew Hassell and ...

Abstract:

$L^p$-cohomology is a notion which has the advantage of providing a continuous family of numerical invariants in order to study the large scale geometry of suitable metric spaces. Classical intuitions from algebraic topology are still relevant, even though the ultimate goal is to study groups and metric spaces modulo the equivalence relation given by quasi-isometry: by and large, bi-Lipschitz equivalence relaxed by allowing also additive constants.

Among interesting such spaces are non-compact Lie groups and their associated geometries (for instance: symmetric spaces, but other homogeneous manifolds are considered). The questions are quite different according to whether one studies semisimple or solvable groups. We will discuss the two situations, mostly under the viewpoint of critical exponent and quasi-isometric rigidity phenomena.

The first part will be introductory.

This is a joint work with Marc Bourdon.

Abstract:

In the pioneering work of [1] the solution to the linear Fokker–Planck–Kolmogorov equation, which is a parabolic partial differential equation for a time-dependent probability density on $\mathbb R^d$, is shown to be a gradient flow in the metric space of probability measures equipped with the 2-Wasserstein distance. This result has sparked wide interest to complement and explore the theory of Optimal Transport from various perspectives. In this talk we develop and discuss new methods for stochastic analysis on the 2-Wasserstein space $\mathcal P_2(\mathbb R^d)$. The starting point is a probability measure $\Lambda$ together with a continuous, $\Lambda$-symmetric Markov semigroup $(T_t)_{t\geq 0}$ on $\mathcal P_2(\mathbb R^d)$ whose energy form is of gradient-type, meaning \[ \lim_{t\downarrow 0}\frac{1}{t}\int [u(\mu)-T_tu(\mu)]v(\mu)\textnormal d\Lambda(\mu)=\int\langle \nabla u(\mu),\nabla v(\mu)\rangle_\mu\textnormal d\Lambda(\mu). \] In the above equation, $u,v:\mathcal P_2(\mathbb R^d)\to\mathbb R$ are bounded continuously differentiable in a sense made precise in the talk and $\langle\,\cdot,\cdot\rangle_\mu$ denotes the scalar product of $L^2(\mathbb R^d\to\mathbb R^d,\mu)$. The gradients $\nabla u(\mu),\nabla v(\mu)\in L^2(\mathbb R^d\to\mathbb R^d,\mu)$ at a point $\mu\in\mathcal P_2(\mathbb R^d)$ are consistent with Otto’s structure [1,2].

Our first result is the existence of a diffusion process on $\mathcal P_2(\mathbb R^d)$ with transition semigroup ${(T_t)}_{t\geq 0}$ and invariant measure $\Lambda$. Then, we discuss applications choosing $\Lambda$ as the image measure of a (restricted) Gaussian on $H:=L^2(\mathbb R^d\to\mathbb R^d,\lambda)$ under the map \[ H\ni \phi\mapsto\lambda\circ\phi^{-1}\in\mathcal P_2(\mathbb R^d), \] where $\lambda$ is fixed as an absolutely continuous element of $\mathcal P_2(\mathbb R^d)$. In particular, stochastic perturbations of gradient flows for relevant energy functionals are constructed. Our techniques are mainly from the theory of Dirichlet forms. The talk is based on joint works with P. Ren, M. Röckner and F.-Y. Wang ([3,4]) and a work in progress with M. Grothaus.

References:

• [1] Jordan, R., Kinderlehrer, D., Otto, F., The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29, 1–17 (1998).
• [2] Otto, F. The Geometry of Dissipative Evolution Equations: The Porous Medium Equation. Commun. Partial Differ. Equ. 26, 101–174 (2000).
• [3] Ren, P., Wang, F., Wittmann, S., Diffusion Processes on $p$-Wasserstein Space over Banach Space, Preprint, arXiv:2402.15130 (2024).
• [4] Ren, P., Röckner, M., Wang, F., Wittmann, S., Stochastic intrinsic gradient flows on the Wasserstein space, Preprint, arXiv:2506.12755 (2025).

Abstract:

The Sobolev–Bregman integral forms are an $L^p$ version of quadratic forms defining the fractional Sobolev spaces $H^s$, that emerged in a natural way from potential theory in $L^p$. We will give an example showing that their natural domain is not linear. Despite that, we prove the existence of minimizers for the exterior value problem, and using a special choice of curves we establish an Euler–Lagrange equation for the minimizers. We also prove a Green-type formula and investigate the domain of the polarized form.

Based on joint work with Krzysztof Bogdan, Katarzyna Pietruska-Pałuba and Christian Rose.

Abstract:

The aim of this talk is to present recent results on the asymptotic behaviour of Crump–Mode–Jagers (CMJ) branching processes, with a focus on second-order behaviour and Gaussian fluctuations. I will first discuss joint work with M. Meiners and A. Iksanov on the single-type case, where we extend and unify several existing approaches to the study of fluctuation in specific CMJ processes. I will then outline how these methods can be generalised to the multi-type setting, highlighting the main technical challenges and possible directions for further research.

Abstract:

We study the large-time asymptotic behavior of solutions to the heat equation, i.e., caloric functions, on certain geometric settings. On Euclidean space, it is known that the solution to the heat equation with $L^1$ initial data $f$, behaves asymptotically as the mass $M=\int f$ times the fundamental solution $h_t$ (the heat kernel) in the $L^1$ sense. More generally, \[ \|h_t\|_{L^{p}(\mathbb{R}^n)}^{-1}\,\|f\ast h_t-\, M \, h_{t}\|_{L^{p}(\mathbb{R}^n)}\,\longrightarrow\,0, \quad \text{as } t \rightarrow +\infty, \quad 1\leq p \leq \infty.\]

On the other hand, Gaussian upper and lower bounds of the heat kernel are known to fail on hyperbolic space, where the volume growth is known to be non-doubling. The discrete space analog of hyperbolic space is a homogeneous tree, and the corresponding heat kernel is —now for discrete time— the transition density of a good random walk (aperiodic, isotropic, irreducible). It is thus a natural task to examine the asymptotic behavior of caloric functions in these settings. It turns out that the situation is drastically different: there is no constant $M$ that works for all $p$.

Instead, for each $p \in [1, \infty]$, we introduce a notion of a $p$-mass function and prove that caloric functions with compactly supported initial data, asymptotically decouple as the product of this mass function and the heat kernel. The spatial concentration of the heat kernels in $p$-norm plays an important role in these settings, clarifying the interplay between volume growth and heat diffusion. The results extend to symmetric spaces of non-compact type and even to exotic buildings beyond the Bruhat–Tits framework.

Based on joint works with J.-Ph. Anker, A. Grigor'yan, B. Trojan, H.-W. Zhang.

Abstract:

We study noncommutative Poisson-type limit distributions associated with bm-independence, a notion of independence arising from mixtures of Boolean and monotone independences and naturally linked to positive symmetric cones. We construct discrete bm-Fock spaces with creation, annihilation, and conservation operators, and prove Poisson limit theorems for the corresponding operator sums. The limit distributions are described using labeled noncrossing partitions and depend on geometric properties of the underlying cones, in particular their volume characteristic. (This is joint work with J. Wysoczański.)

Abstract:

We present sharp two-sided estimates of the fundamental solution to the fractional Kolmogorov equation in $\mathbb R \times \mathbb R$ using Fourier methods. Additionally, we provide an explicit form of the fundamental solution in case of the square root of the Laplacian.

Abstract:

A Banach space $X$ satisfies Martingale Type $p$ if there exists $C=C_p$ such that any filtered probability space $(\Omega,\mathcal F,(\mathcal F_n),\mathbb P)$ gives rise to the upper $\ell^p$ estimates,\[\tag{1}\lVert f\rVert_{L^p(\Omega,X)}^p\leq C^p\biggl(\lVert\mathbb Ef\rVert_X^p+\sum_{n=1}^\infty\lVert\mathbf\Delta_nf\rVert_{L^p(\Omega,X)}^p\biggr)\]where $\mathbf\Delta_n f=\mathbb E(f\mid\mathcal F_n)-\mathbb E(f\mid\mathcal F_{n-1}$, and where $L^p(\Omega,X)=L^p(\Omega,\mathcal F,\mathbb P,X)$.

Let $\mathcal I_n$ be the collection of pairwise disjoint dyadic intervals of measure $2^{-n}$ contained in $[0,1)$ and $\mathcal I^n$ be the $\sigma$-algebra generated by $\mathcal{I}_n$. The Theorem of Pisier (see [1,2]) asserts that $X$ satisfies Martingale Type $p$ if there exists $C=C_p$ such that\[\lVert f\rVert_{L^p([0,1),X)}^p\leq C^p\biggl(\lVert\mathbb Ef\rVert_X^p+\sum_{n=1}^\infty\lVert\mathbb E(f\mid\mathcal I^n)-\mathbb E(f\mid\mathcal I^{n-1})\rVert_{L^p([0,1),X)}^p\biggr)\]for $f\in L^p([0,1),X),$ where $L^p([0,1),X)=L^p([0,1),\mathcal I,dt,X)$ and $\mathcal I=\sigma(\bigcup_{n=1}^\infty\mathcal I^n)$. In other words, by [1,2], the dyadic filtration $([0,1),\mathcal I,(\mathcal I^n),dt)$ determines already the martingale type of a Banach space $X$.

Assuming that $\mathcal F_n$ is a sequence of increasing purely atomic sub-$\sigma$-algebras of $\mathcal F$ and $\mathcal F=\sigma(\bigcup_{n=1}^\infty\mathcal F_n)$, we will identify precisely all filtered probability spaces $(\Omega,\mathcal F,(\mathcal F_n),\mathbb P)$, that are able to determine the martingale type of Banach space $X$. We associate explicit intrinsic conditions on the filtration $(\mathcal F_n)$ which determine that the upper $\ell^p$ estimates (1) imply the martingale type $p$.

Theorem. For each fixed $(\Omega,\mathcal F,(\mathcal F_n),\mathbb P)$ the following dichotomy holds true: Either, there exists $C>0$ such that for any Banach space $X$ and any $f\in L^p(\Omega,\mathcal F,\mathbb P,X)$\[\tag{2}\lVert f\rVert_{L^p(\Omega,X)}^p\leq C^p\biggl(\lVert\mathbb Ef\rVert_X^p+\sum_{n=1}^\infty\lVert\mathbb E(f\mid\mathcal F_n)-\mathbb E(f\mid\mathcal F_{n-1})\rVert_{L^p(\Omega,X)}^p\biggr),\]or the filtered probability space $(\Omega,\mathcal F,(\mathcal F_n),\mathbb P)$ and upper $\ell^p$ estimates (2) already determine that the Banach space $X$ is of martingale type $p$.

Talk is based on a joint work with Paul F.X. Müller.

References:

• [1] Gilles Pisier. Martingales with values in uniformly convex spaces. Israel Journal of Mathematics, 20(3–4):326–350, 1975.
• [2] Gilles Pisier. Martingales in Banach Spaces. Cambridge University Press, 2016.

Abstract:

In this talk, we present a quantitative central limit theorem for the simple symmetric exclusion process (SSEP) on a multidimensional discrete torus. Our approach is based on a comparison between the generators of the density fluctuation field of the SSEP and those of a generalized Ornstein–Uhlenbeck process, together with an infinite-dimensional Berry–Esseen bound for the initial particle fluctuations. The analysis relies on regularity estimates for the Ornstein–Uhlenbeck process and precise control of higher-order correlations in the SSEP. As a result, we obtain an optimal rate of convergence.

This is joint work with Benjamin Gess.

Abstract:

Consider a second-order elliptic operator $L$ in the half-plane $\mathbb R \times (0, \infty)$ with coefficients depending only on the second coordinate. The Poisson kernel for $L$ is used in the representation of positive $L$-harmonic functions, that is, solutions of $L u = 0$. In probabilistic terms, the Poisson kernel is the density function of the distribution of the diffusion in $\mathbb R \times (0, \infty)$ with generator $L$ at the hitting time of the boundary. It turns out that the Poisson kernel for $L$ is bell-shaped: its $n$th derivative changes sign $n$ times. In particular, it is unimodal and it has two inflection points (it is concave, then convex, then concave again). This is the main result of my recent paper Poisson kernels on the half-plane are bell-shaped, DOI:10.1112/blms.70303. In my talk I will discuss this result, give a broader perspective, and state a few intriguing, if niche, open problems.

Abstract:

An Orthogonal Polynomial Ensemble (OPE) is a determinantal point process whose correlation functions can be written as determinants of the Christoffel–Darboux kernels. Eigenvalues of Hermitian random matrices with unitary invariant distributions are important examples of OPEs. A popular way of studying point processes is to form a particular random variable, the so-called linear statistic. In the talk I will review some results on limit theorems for linear statistics in microscopic and macroscopic scales. After that I will focus on almost sure convergence for the macroscopic scale. In the study Nevai’s condition known from orthogonal polynomial theory will be useful.

Abstract:

In the first part of my talk I will discuss Stein’s generalised spherical averages. After a brief overview of known results and open problems, I will rephrase Stein’ maximal inequality in terms of derivatives of standard spherical means: if \[ k \ge 0, \qquad d \ge 2 k + 3 , \qquad \frac{d}{d - k - 1} < p < \frac{d - 1}{k} , \] and $\sigma$ is the normalised surface measure on the unit sphere $\mathbb S$, then the maximal operator \[f \mapsto \sup_{r > 0} \, \biggl\lvert r^k (\tfrac{d}{dr})^k \int_{\mathbb S} f(\cdot + r y) \sigma(dy) \biggr\rvert\] is bounded on $L^p$, with a constant that is independent of the dimension $d$.

The second part of my talk will be about $L^p$ estimates for maximal Riesz transforms (of an arbitrary order). This result was originally proved by Mateu, Orobitg, Peréz and Verdera with a constant depending on the dimension, and more recently Kucharski, Wróbel and Zienkiewicz gave a dimension-free bound. I will discuss a shorter and more direct proof of this maximal inequality.

Joint work with Maciej Kucharski and Błażej Wróbel.

Abstract:

In this talk I will present a methodology for finding the Brown measure for operators of the form $x+iy$, where $x$ and $y$ are free and $y$ has the free Poisson (Marchenko–Pastur) distribution.

Based on joint work with: F. Lehner (Graz), A. Nica (Waterloo), P. Zhong (Houston).

Abstract:

We study a variant of the uncentred Hardy–Littlewood maximal operator on hyperbolic plane in which balls are replaced by suitable half balls. Perhaps surprisingly, such modified maximal operator has better boundedness properties than the classical one. In particular, it satisfies an $L\log L$ endpoint estimate and it is bounded on $L^p$ for every $p$ in $(1,\infty]$. The results extend to all rank one non-compact symmetric spaces and even Damek–Ricci spaces.

Abstract:

We present a unified operator-theoretic framework for studying positive solutions of semilinear equations in which the linear part is generated by a nonnegative strongly continuous semigroup acting on $L^p$-space with the exponent greater than or equal to one. Our analysis relies on two structural assumptions—compactness of the resolvent and irreducibility of the underlying semigroup—which allow us to connect nonlinear phenomena with spectral information. For a broad class of nonlinearities satisfying one-sided growth conditions, we introduce two spectral quantities that capture the asymptotic profile of the nonlinearity in the small (near the origin) and in the large (as the amplitude tends to infinity).

The central theorem provides an existence criterion for positive solutions based on comparing these two spectral thresholds: an appropriate crossing of the near-zero and at-infinity spectral levels ensures the existence of a positive solution. We then develop parametric versions of the results (with logistic-type equations as a guiding example), obtaining sharp “if and only if” existence theorems and uniqueness within the cone of positive solutions under a natural monotonicity condition for the quotient $f(u)/u$. These results can be viewed as an extension of the Brezis–Oswald theorem to an abstract semigroup framework.

Although our approach is primarily operator-theoretic, key steps use probabilistic tools: the Feynman–Kac representation for perturbed semigroups, the Itô–Meyer formula, and the notion of supermedian functions from probabilistic potential theory. Importantly, we do not require any smoothing properties of the semigroup (such as ultracontractivity or the Feller property). This makes the framework applicable—within a single scheme—to a wide range of local and nonlocal operators, including settings with low regularity.