arXiv subjects

Lukas Gräfner

Publications and source records attributed to Lukas Gräfner.

5 recordsLinked to original sources

arXiv2026

Energy solutions of singular SPDEs on Hilbert spaces with applications to domains with boundary conditions

In this paper we extend the theory of energy solutions for singular SPDEs, focusing on equations driven by highly irregular noise with bilinear nonlinearities, including scaling critical examples. By introducing Gelfand triples and leveraging infinite-dimensional analysis in Hilbert spaces together with an integration by parts formula under the invariant measure, we largely eliminate the need for Fourier series and chaos expansions. This approach broadens the applicability of energy solutions to a wider class of SPDEs, offering a unified treatment of various domains and boundary conditions. Our examples are motivated by recent work on scaling limits of interacting particle systems.

math.PR↗

arXiv2026

Equilibrium fluctuations for a multi-species particle system with long jumps

In the present paper, we study the equilibrium fluctuations of a particle system in infinite volume with two conserved quantities and long-range dependence. More specifically, the model of interest is the so-called ABC model, in which three types of particles (A, B and C) exchange their locations between $x\in\mathbb{Z}$ and $x+z\in\mathbb{Z}$ at a rate that depends on the type of particles involved and is proportional to $|z|^{-γ-1}$ for $γ>0$. After rigorously identifying the normal modes associated to the conserved quantities (the density of particles of types $A$ and $B$, say), we prove that their fluctuations converge to independent fractional stochastic partial differential equations (SPDEs), which are either Gaussian or the Stochastic Burgers equation, and whose nature is determined by the microscopic range of dependence and the strength of the asymmetry.

math.PR↗

arXiv2025

Construction of the 1d Self-repelling Brownian Polymer

We consider the self-repelling Brownian polymer, introduced in [APP83], which is formally defined as the solution of a singular SDE. The singularity comes from the drift term, which is given by the negative gradient of the local time. We construct a solution of the equation in d = 1, give a dynamic characterisation of its law, and show that it is the limiting distribution for a natural family of approximations. In addition, we show that the solution is superdiffusive. As part of the construction, we consider a singular SPDE which is solved by the (recentered) local time. Using the method of energy solutions, we show that this SPDE is well-posed, and prove that this property can be transferred to the original process. Our results hold for a larger class of drift terms, in which the gradient of the local time is a special case.

math.PR↗

arXiv2025

Energy solutions to SDEs with supercritical distributional drift: An extension and weak convergence rates

In this work we consider the SDE \begin{equation} \text{d} X_t = b (t, X_t) \text{d} t + \sqrt{2} \text{d} B_t, \label{mainSDE} \end{equation} in dimension $d \geqslant 2$, where $B$ is a Brownian motion and $b : \mathbb{R}_+ \rightarrow \mathcal{S}' (\mathbb{R}^d , \mathbb{R}^d)$ is distributional, scaling super-critical and satisfies $\nabla \cdot b \equiv 0$. We partially extend the super-critical weak well-posedness result for energy solutions from [GP24] by allowing a mixture of the regularity regimes treated therein: Outside of neighbourhoods of a small (and compared to [GP24] ''time-dependent'') local singularity set $K \subset \mathbb{R}_+ \times \mathbb{R}^d$, $b$ is assumed to be in a certain supercritical $L^q_T H^{s, p}$-type class that allows a direct link between the PDE and the energy solution from a-priori estimates up to the stopping time of visiting $K$. To establish this correspondence, and thus uniqueness, globally in time we then show that $K$ is actually never visited which requires us to impose a relation between the dimension of $K$ and the Hölder regularity of $X$. In the second part of this work we derive weak convergence rates for approximations of the above equation in the case of time-independent drift, in particular with local singularities as above.

math.PR↗

arXiv2024

Weak well-posedness of energy solutions to singular SDEs with supercritical distributional drift

We study stochastic differential equations with additive noise and distributional drift on $\mathbb{T}^d$ or $\mathbb{R}^d$ and $d \geqslant 2$. We work in a scaling-supercritical regime using energy solutions and recent ideas for generators of singular stochastic partial differential equations. We mainly focus on divergence-free drift, but allow for scaling-critical non-divergence free perturbations. In the time-dependent divergence-free case we roughly speaking prove weak well-posedness of energy solutions with initial law $μ\ll \text{Leb}$ for drift $b \in L^p_T B^{-γ}_{p, 1}$ with $p \in (2, \infty]$ and $p \geqslant \frac{2}{1 -γ}$. For time-independent $b$ we show weak well-posedness of energy solutions with initial law $μ\ll \text{Leb}$ under certain structural assumptions on $b$ which allow local singularities such that $b \notin B^{-1}_{2 d/(d-2), 2}$, meaning that for any $p > 2$ in sufficiently high dimension there exists $b \notin B^{-1}_{p, 2}$ such that weak well-posedness holds for energy solutions with drift $b$.

math.PR↗