Special Session 181: Dirichlet Forms and Related Topics

Feller property and convergence for semi-groups of time-changed processes

Ali BenAmor University of Sousse, high school for transport and logistics Tunisia Co-Author(s):    Kazuhiro Kuwae

Abstract: We give a substitute to the Feller property for semigroups of time-changed processes; under some conditions this leads to sufficient (new) conditions for the semigroups to be Feller. Moreover, given a standard process and a sequence of measures converging vaguely to a final measure, under some assumptions, we establish convergence of the sequence of the semigroups and the resolvents of the corresponding time changed-processes. .

Regularization of the superposition principle: Potential theory meets Fokker-Planck equations

Iulian Cimpean University of Bucharest, Faculty of Mathematics and Computer Science, and Romanian Academy Romania Co-Author(s):    Lucian Beznea, Michael Roeckner

Abstract: For a solution to a nonlinear Fokker-Planck equation (FPE) the powerful superposition principle renders a probability measure on path space with one dimensional time marginals equal to this solution, and additionally solving the martingale problem for the Kolmogorov operator given by the FPE. The superposition principle thus reveals that such parabolic PDEs have a probabilistic counter part. The aim of this talk is to go a substantial further step and, by exploiting the superposition principle, Dirichlet forms, and potential theoretic tools, construct a full fledged Markov process, i.e. a family of path space measures for a large set of space time starting points connected by the Markov property, associated to the (linearized) FPE in the above way. Under very general (merely measurability) conditions on the coefficients of the FPE this is achieved in this paper in such a way that the resulting process is a right process. Furthermore, we introduce a Choquet capacity for such FPEs using the corresponding right process. A main application here is the FPE given by the generalized porous media equation and its corresponding McKean-Vlasov SDE.

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

Martin Grothaus RPTU University Kaiserslautern-Landau Germany Co-Author(s):    Torben Fattler, Nathalie Steil

Abstract: The starting point is a gradient Dirichlet form with respect to $\lambda^d_\varrho$ on the space $L^2({\mathbb R}^d, \mu_\varrho)$. 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 show positive s\`{e}jour time of $X$ on $A$. Hence we obtain a stochastic process $X$ with permeable sticky behaviour on $A$.

Beurling--Deny formula for Sobolev--Bregman forms

Micha{\l} Gutowski Wroc{\l}aw University of Science and Technology Poland Co-Author(s):    Mateusz Kwa{\`s}nicki

Abstract: For an arbitrary regular Dirichlet form $\mathcal{E}$ and the associated symmetric Markovian semigroup $T_t$, we consider the corresponding \emph{Sobolev--Bregman form} $ \mathcal{E}_p [u] := \lim_{t\to0^+} \tfrac{1}{t} \langle u-T_tu,u|u|^{p-2} \rangle $, where $p\in(1,\infty)$. The Sobolev--Bregman form describes the rate of decrease of the $L^p$ norm of $T_t u$ with respect to time: we have $ \mathcal{E}_p [u] = -\tfrac{1}{p} \tfrac{d}{dt} \| T_t u \|_p^p \big\vert_{t = 0} $. When $p=2$, $\mathcal{E}_p$ coincides with the original Dirichlet form $\mathcal{E}$. Therefore, the Sobolev--Bregman form can be treated as a $L^p$-extension of the Dirichlet form. The celebrated Beurling--Deny formula provides a decomposition of a regular Dirichlet form $\mathcal{E}$ into the strongly local term $\mathcal{E}^c$, the purely nonlocal part given in terms of the \emph{jumping kernel} $J$, and the killing term described by the \emph{killing measure} $k$. It is well known that there is a one-to-one correspondence between the class of regular Dirichlet forms $\mathcal{E}$ and the class of symmetric Hunt processes (strong Markovian, quasi-left continuous with c\`adl\`ag paths). The three parts of the Beurling--Deny decomposition describe the local, jumping, and killing behaviour of the process. The aim of the talk will be to provide a similar decomposition of the corresponding Sobolev--Bregman form. The talk is based on joint work with Mateusz Kwa{\`s}nicki (Wroc{\l }aw University of Science and Technology).

Decay of resolvent kernels and Schr\odinger eigenstates for L\`evy operators

Kamil Kaleta Wroclaw University of Science and Technology Poland Co-Author(s):

Abstract: \begin{abstract} I will present results obtained jointly with R. Schilling (Dresden) and P. Sztonyk (Wroclaw) on the spatial decay of resolvent kernels for a broad class of non-local L\`evy operators, as well as on eigenfunctions of the associated Schr\odinger operators [B]. Our results lead to general theorems concerning L\`evy measures with exponential and subexponential decay at infinity. In particular, we identify sharp qualitative transitions in these decay properties, thereby extending classical results of Carmona, Masters, and Simon [A] for the fractional Laplacian (subexponential decay) and relativistic operators (exponential decay). Moreover, these transitions in eigenfunction decay admit a natural energetic interpretation, which will also be discussed during the talk. References: [A] R. Carmona, W. C. Masters, and B. Simon, Relativistic Schr\odinger operators: asymptotic behavior of the eigenfunctions, J. Funct. Anal. 91(1) 117--142 (1990) [B] K. Kaleta, R.L. Schilling, P. Sztonyk, Decay of resolvent kernels and Schr\odinger eigenstates for L\`evy operators, Math. Ann. 394(4), 88 (2026) \end{abstract}

On quasi-ergodic limits for symmetric Markov processes

Daehong Kim Kumamoto University Japan Co-Author(s):    Yiming Zhou

Abstract: We characterize quasi-ergodic limiting measures for additive functionals of a symmetric Markov process $X$. We consider two types of additive functionals beyond the classical occupation-time setting: continuous additive functionals associated with Revuz measures and purely discontinuous additive functionals. Our results do not require the finiteness of the total mass of the underlying measure nor (intrinsic) ultracontractivity of the semigroups, assumptions commonly imposed in the existing literature. As a consequence, we establish Chacon-Ornstein type ratio ergodic limits for additive functionals of $X$.

Non-local Feynman-Kac perturbations for jump diffusions on metric measure spaces

Kyung-Youn Kim National Chung-Hsing University/Applied Mathematics Taiwan Co-Author(s):    Zhen-Qing Chen; Lidan Wang

Abstract: We establish the stability of two-sided heat kernel estimates for a class of (possibly non-symmetric) jump diffusions on metric measure spaces under non-local Feynman-Kac perturbations. Moreover, we obtain parabolic Harnack inequality and Holder regularity for parabolic functions of the non-local Schrodinger operators of the jump diffusions.

Intrinsic geometry of multivariate L\`evy measures and weak solutions to L\`evy driven SDEs

Oleksii Kulyk Wroclaw Univercity of Science and Technology Poland Co-Author(s):    Tadeusz Kulczycki, Michal Ryznar

Abstract: We introduce the notion of the concentration function for multi-dimensional L\`evy measures and related notions of level sets, concentration sets, and concentration norms. We will present a systematic study of the concentration and heat kernel properties of multivariate L\`evy processes based on these geometry notions. This study is originated in our previous research of SDEs driven by spatially heterogeneous L\`evy noises. We will present a general approach for getting weak existence/uniqueness and heat kernel estimates for solutions of L\`evy driven SDEs with the assumptions on the coefficients given in the terms of the intrinsic geometry of the driving noise, which unifies and extends considerably the previously available results.

A remark on subharmonicity for symmetric Dirichlet forms

Kazuhiro Kuwae Fukuoka University/Department of Applied Mathematics Japan Co-Author(s):    Rong Lei and Ludovico Marini

Abstract: We remove the local boundedness for $\mathscr{E}_{\alpha}$-subharmonicity in the framework of (not necessarily strongly local) regular symmetric Dirichlet form $(\mathscr{E},D(\mathscr{E}))$ with $\alpha\geq0$ and establish the stochastic characterization for $\mathscr{E}$-subharmonic functions without assuming the local boundedness.

Emergence of an unsaturated Darcy's law and a Richards-type equation via homogenization of the Stokes-Cahn-Hilliard system

Jun Masamune Kyoto University Japan Co-Author(s):

Abstract: The Centipede Game leads to a counterintuitive Nash equilibrium that destroys wealth. We study this game and investigate versions of the game that lead to a model of formation and burst of financial bubbles, where participants are first incentivize to feed the trend, then run for the exit. We analyze these mechanisms and examine how the model fits behavioral economics interpretations and the observed dynamics of financial bubbles.

On the Markov property of subsequential limits for a sequence of Markov chains

Kouhei Matsuura University of Tsukuba Japan Co-Author(s):

Abstract: In this talk, we consider a sequence of Markov chains defined on point sets in a Euclidean domain and discuss sufficient conditions for its subsequential limits to possess the Markov property. Establishing the Markov property is important, as it implies that the limit generates a symmetric Dirichlet form under appropriate additional conditions including the symmetry of each chain, which in turn allows for characterizing the limit through concepts such as the Silverstein extension. We also discuss conditions under which the subsequential limit becomes a reflected Brownian motion on the domain, as well as a framework applicable to more general Markov processes.

Topological properties of the Revuz correspondence

Takumu Ooi Tokyo University of Science Japan Co-Author(s):    Kaneharu Tsuchida and Toshihiro Uemura

Abstract: We investigate topological properties of the Revuz correspondence. We provide necessary and sufficient conditions for the convergence of Revuz measures of finite energy integrals. In this setting, we show that the Revuz map, restricted to the class of finite energy integrals equipped with the norm induced by a Dirichlet form, is a homeomorphism when the space of positive continuous additive functionals is endowed with the topology induced by $L^2(P_{m+\kappa+\nu_0})$-norm together with the local uniform topology, where $m$ is the underlying measure, $\kappa$ is the killing measure and $\nu_0$ is an energy functional of a Hunt process. We also present a characterization of the Revuz correspondence in terms of killing of the process and further consider continuity properties of the Revuz correspondence for broader classes.

Subharmonic functions and the BMS conjecture for regular Dirichlet forms

Marcel Schmidt Friedrich-Schiller-University Jena Germany Co-Author(s):

Abstract: In this talk we discuss several notions of subharmonic functions in the context of regular Dirichlet forms. A particular emphasis is put on their regularity. We extend recent results by G\uneysu, Pigola, Stollmann and Veronelli `24 to a class of possibly non-local Dirichlet forms and put their attempt at subharmonic functions via so-called locally shift defective functions into perspective. The Bravermann, Milatovic, Shubin conjecture (BMS conjecture, recently confirmed by Pigola, Veronelli `21) asks, whether on a complete Riemannian manifold, for $f \in L^2(M)$ the distributional inequality $\Delta f \leq f$ implies $f \geq 0$. We use our regularity theory for subharmonic functions to verify the validity of this conjecture for a large class of `distributional operators` derived from regular Dirichlet forms.

Dirichlet problem for Lane--Emden type equations with several sublinear terms

Adisak Seesanea Sirindhorn International Institute of Technology, Thammasat University Thailand Co-Author(s):

Abstract: We present the existence, uniqueness, and sharp bilateral pointwise estimates for positive bounded solutions to the Lane--Emden type problem \[ \begin{cases} \mathcal{L} u = \sum\limits_{i=1}^{m}\sigma_{i} u^{q_{i}}+\sigma_0, \quad u\geq0 & \text{in} \; \Omega, \ \liminf \limits_{x \rightarrow y} u(x) = f(y), & y \in \partial^\infty\Omega, \end{cases} \] where $0 < q_{i} < 1$. Here $\mathcal{L} u = - \operatorname{div}(\mathbb A \nabla u)$ is a uniformly elliptic operator with bounded coefficients, $\sigma_{i}$ is a nonnegative locally finite Borel measure on an $\mathbb A$-regular domain $\Omega \subset \mathbb R^n$ which possesses a positive Green function associated with $\mathcal{L}$, and $f$ is a nonnegative continuous function on the boundary $\partial^\infty\Omega$. An analogous result for positive continuous solutions to the problem is also illustrated. Our method can be adapted to address related sublinear problems with zero boundary conditions involving the fractional Laplace operator $(-\Delta)^{\alpha}$ for $0< \alpha < n/2$, in place of $\mathcal{L}$, in $\mathbb{R}^n$ as well. This is a joint work with Kentaro Hirata (Hiroshima) and Toe Toe Shwe (SIIT).

Strongly continuous fields of operators over varying Hilbert spaces

Peter R.M Stollmann TU Chemnitz Germany Co-Author(s):    Ali BenAmor, Batu G\\uneysu, Thomas Kalmes

Abstract: After introducing a natural notion of continuous fields of locally convex spaces, we establish a new theory of strongly continuous families of possibly unbounded self-adjoint operators over varying Hilbert spaces. This setting allows to treat operator families defined on bundles of Hilbert spaces that are not locally trivial (such as e.g. the tangent bundle of Wasserstein space), without referring to identification operators at all.

Homogenization of Non-Symmetric Pure Jump Processes on a Bounded Domain

Toshihiro Uemura Department of Mathematics, Kansai University Japan Co-Author(s):

Abstract: We consider the periodic homogenization of non-symmetric pure jump processes ($\alpha$-stable-like processes with $1

Dirichlet Forms with Jump Kernels Decaying at the Boundary

Zoran Vondracek University of Zagreb and SOIS-FT Croatia Co-Author(s):    S.Cho, P.Kim, R.Song

Abstract: In this talk, I will give an overview of recent results concerning Dirichlet forms and the associated singular nonlocal operators whose kernels vanish at the boundary of a smooth subset of Euclidean space. I will describe the motivation for this line of research, explain the methods - combining probabilistic and analytic techniques - and present the main results, highlighting how they differ from the classical ones.