Titles & Abstracts

• Robert C. Dalang, (École Polytechnique Fédérale de Lausanne, Switzerland)

Sharp upper bounds on hitting probabilities for the solution to the stochastic heat equation on the line

For Gaussian random fields with values in $\R^d$, sharp upper and lower bounds on the probability of hitting a fixed set have been available for many years. These apply in particular to the solutions of systems of linear SPDEs. For non-Gaussian random fields, the available bounds are less sharp. For systems of stochastic heat equations, a sharp lower bound was obtained in [1]. Here, we obtain the corresponding sharp upper bound. This is based on joint work with Fei Pu and David Nualart.

[1] R.C. Dalang and F. Pu, Optimal lower bounds on hitting probabilities for stochastic heat equations in spatial dimension k≥1.  Electron. J. Probab. 25 (2020), Paper No. 40, 31 pp.

• Beom-Seok Han, (Sungshin Women's University, Republic of Korea)

Well-posedness and space-time regularity for stochastic reaction-diffusion-advection equations with variable-order nonlocal operators and spatially homogeneous colored noise

In this talk, we discuss the well-posedness and H\"older regularity of strong solutions to stochastic reaction-diffusion-advection equations driven by spatially homogeneous colored noise and nonlinear multiplicative noise. The equations involve variable-order nonlocal operators defined via Bernstein functions. While various types of SPDEs with variable-order nonlocal operators and SPDEs with nonlinear terms have been studied individually, we present a unified result that recovers and generalizes many of the existing results. We introduce a condition on the noise - strongly reinforced Dalang's condition, which provides a refined criterion for establishing well-posedness and solution regularity. This talk is based on joint work with Jae-Hwan Choi and Daehan Park.

• Joscha Henheik, (Institute of Science and Technology, Austria)

Law of fractional logarithm for random matrices

In this talk, we study the top eigenvalues of the Wigner minor process (a sequence of appropriately scaled upper left corners of a doubly infinite symmetric array of i.i.d. random variables). While LLN (almost sure convergence to 2) and CLT (Tracy-Widom fluctuations) type results for the top eigenvalue sequence are well known, we now make a step further and establish the analogue of the Hartman-Wintner law of iterated logarithm. This result was initially coined the law of fractional logarithm (LFL) by E. Paquette and O. Zeitouni, who resolved the special case of GUE matrices. Our work verifies their 10-year-old conjecture, proving LFL in full generality for both symmetry classes. Based on recent joint works with Z. Bao, G. Cipolloni, L. Erdös, and O. Kolupaiev.

• Yaozhong Hu, (University of Alberta, Canada)

Hyperbolic Anderson equations with general time independent  Gaussian noise: Stratonovich regime

In this talk, I will present a joint work with Xia Chen about two results on the hyperbolic Anderson equation generated by a time-independent Gaussian noise. First one is that we prove that Dalang's condition is necessary and sufficient for existence  of the solution. Second, we establish the precise long time  and high moment asymptotics for the solution under the usual homogeneity assumption of the covariance of the Gaussian noise. An interesting tool is the relation between the solutions of stochastic heat and wave equations.

• Kunwoo Kim, (University of Science and Technology, South Korea)

Long-Time Behavior and an Invariance Principle for Stochastic Reaction-Diffusion Equations with a Multiplicative Random Source 

In this talk, we discuss the long-time behavior of nonlinear stochastic heat equations on an interval with periodic boundary conditions. We begin with a review of recent results on stochastic heat equations, in particular, the parabolic Anderson model (PAM) in which Gu and Komorowski identified the almost-sure Lyapunov exponent and derived a central limit theorem as time tends to infinity. We then present our recent work establishing an invariance principle for a broad class of stochastic reaction-diffusion equations. This principle provides a notion of universality for the PAM, showing that these equations have the same long-time behavior as the one for the parabolic Anderson model. This is joint work with Davar Khoshnevisan and Carl Mueller.

• Honghu Liu, (Virginia Tech, USA)

Transitions in stochastic non-equilibrium systems: Efficient reduction and analysis

It is still a central challenge in physics to describe non-equilibrium systems driven by randomness, such as a randomly growing interface, or fluids subject to random fluctuations that account e.g. for local stresses and heat fluxes in the fluid which are not related to the velocity and temperature gradients. For deterministic systems with infinitely many degrees of freedom, normal form and center manifold theory have shown a prodigious efficiency to often completely characterize how the onset of linear instability translates into the emergence of nonlinear patterns, associated with genuine physical regimes. However, in the presence of random fluctuations, the underlying reduction principle to the center manifold is seriously challenged due to large excursions caused by the noise, and the approach needs to be revisited. In this talk, we present a framework to address some of these difficulties by exploiting the approximation theory of stochastic invariant manifolds, on the one hand, and energy estimates measuring the defect of parameterization of the high-modes, on the other. To operate for fluid problems subject to stochastic stirring forces, these error estimates are derived under assumptions regarding dissipation effects brought by the high-modes in order to suitably counterbalance the loss of regularity due to the nonlinear terms. As a result, the approach enables us to predict, from reduced equations of the stochastic fluid problem, the occurrence in large probability of a stochastic analogue to the pitchfork bifurcation, as long as the noise's intensity and the eigenvalue's magnitude of the mildly unstable mode scale accordingly. Application to a stochastic Rayleigh-B\'enard model will also be presented.

• Hiroshi Matano, (Meiji University, Japan)

Front propagation through a two-dimensional cylinder with saw-toothed boundaries

In this talk, I will discuss a curvature-dependent motion of plane curves in a two-dimensional infinite cylinder with bumpy boundaries. The bumps are arrayed periodically. The two ends of the curve slide freely along the both boundaries of the domain while keeping the constant contact angle of $\pi/2$.

The question is whether the curve continues to travel to infinity (propagation) or remains in a bounded area (blocking). The same problem was studied in my earlier work under the assumption that the maximum opening angle and closing angle of the boundaries are less than $\pi/4$ (2006 and 2013, joint work with K.-I. Nakamura and B. Lou). Under this condition, one can show that no singularity develops and the solution remains classical all the time. On the other hand, if the opening and closing angles are larger than $\pi/4$, the middle part of the curve may hit the boundary bumps and split into pieces, thereby creating singularities. In this talk, I will discuss the long-time behavior of curves that propagates while creating singularities. Among other things, we give necessary and sufficient conditions for blocking and propagation, and show that a well-defined propagation speed exists that is independent of the choice of initial data. 

Next we discuss the homogenization limit as the boundary bumps become finer and finer while keeping the same shape, and derive an explicit formula of the limit propagation speed. Quite intriguingly, our formula shows that the propagation speed increases if the distance between adjacent bumps becomes smaller; in other words, densely arrayed obstacles are more advantageous for propagation than scarcely arrayed obstacles, which may sound somewhat paradoxical. We call this phenomenon "Obstacle-aided propagation". This is joint work with Ryunosuke Mori.

• Cyrill Muratov, (University of Pisa, Italy)

A micromagnetic theory of skyrmion lifetime in ultrathin ferromagnetic films

We use the continuum micromagnetic framework to derive the formulas for compact skyrmion lifetime due to thermal noise in ultrathin ferromagnetic films with relatively weak interfacial Dzyaloshinskii–Moriya interaction. In the absence of a saddle point connecting the skyrmion solution to the ferromagnetic state, we interpret the skyrmion collapse even tas “capture by an absorber” at microscale. This yields an explicit Arrhenius collapse rate with both the barrier height and the prefactor as functions of all the material parameters, as well as the dynamical paths to collapse.

• Lluís Quer-Sardanyons, (Universitat Autònoma Barcelona, Spain)

Convergence in law for stochastic PDEs

We consider the stochastic wave and heat equations in $\mathbb{R}^d$ with $d\in \{1,2,3\}$ and $d\geq 1$, respectively, and perturbed by an additive Gaussian noise which is white in time and has a homogeneous spatial correlation with spectral measure $\mu_n$. We allow the Fourier transform of $\mu_n$ to be a genuine distribution. Let $u^n$ be the solution of our equations with spectral measure $\mu_n$. We provide sufficient conditions on the sequence $\{\mu_n\}_n$ to ensure that $u^n$ converges in law, in the space of continuous functions, to the solution of our equations with spectral measure $\mu$, where $\mu_n\to\mu$ in some sense. We apply our main result to various types of noises, such as fractional correlations with $H\in (0,1)$. The talk is based on joint work with Maria Jolis (Barcelona) and Salvador Ortiz-Latorre (Oslo). 

• Tommaso Rosati, (University of Warwick, UK)

Universality in perturbative expansions of the 4D Anderson Hamiltonian

We prove a functional central limit theorem for perturbative expansions of the (critical) 4D Anderson Hamiltonian. The proof is based on BPHZ-type estimates. Joint work with Simon Gabriel. 

• Hao Shen, (University of Wisconsin-Madison, USA)

Dynamical proof of correlation decay for lattice Yang-Mills

Lattice Yang-Mills or lattice gauge theory are natural lattice models where the field takes values in a matrix group. One important question is the exponential decay of correlations (mass gap). The Langevin dynamics, or so called stochastic quantization, can be exploited to obtain results in these directions, in a large coupling regime. If time permitted, I will also discuss more general models, such as lattice Yang-Mills coupled with Higgs fields. Based on joint work with Rongchan Zhu and Xiangchan Zhu.

• Israel Michael Sigal, (University of Toronto, Canada)

On propagation of electrical pulses in neurons

Alongside the Nobel prize winning Hodgkin-Huxley system (HHS), the FitzHugh-Nagumo (FHN) one is at the foundation of quantitative neuroscience, giving a qualitatively, and often quantitatively, faithful description of the propagation of electrical impulses (pulses) in neurons. Though pulses propagate on a surface of neural axons which are cylindrical surfaces of a complicated geometry, in computations and theoretical work, the latter are modelled by the zero thickness infinite straight line. In this talk I will describe the recent mathematical results on propagation of pulses in a more realistic model of neural axons as cylindrical surfaces of variable radii. The talk is based on the recent joint work with Afroditi Talidou and Almut Burchard and with Georgia Karali and Kostas Tzirakis.

• Scott Andrew Smith, (Chinese Academy of Sciences, China)

An application of Talagrand’s inequality to the Linear Sigma Model

The Linear Sigma Model is the N-component and O(N)-invariant generalization of the well-known $\Phi^{4}_{2}$ model.  In the present work, we show that on $\T^{2}$ at large N, each marginal distribution is close to a massive Gaussian free field, quantified in the 2-Wasserstein distance. The proof is a simple application of classical tools in Euclidean field theory combined with the Feyel/\"Ust\"unel variant of Talagrand's inequality.  In contrast to prior work using stochastic quantization, our proof avoids perturbative assumptions on the mass or the coupling constant.  Based on joint work with Mat\'ias Delgadino.

• Alexandra Stavrianidi, (Stanford University, USA)

Long-time behavior of systems of Fisher-KPP type 

Reaction-diffusion equations and branching processes appear naturally in fields such as physics, biology, epidemiology and other disciplines to model a wide range of phenomena such as chemical reactions, combustion, disease propagation. The Fisher-KPP equation is a thoroughly studied model with interesting long-time behavior that captures how the stable steady state invades the unstable one. Moreover, by virtue of a beautiful connection with Branching Brownian Motion, we can study how Branching Brownian Motion invades its environment with analytical tools provided by the Fisher-KPP equation. In this talk, I will present results on the long-time behavior of systems of Fisher-KPP type and establish their connection with multitype Branching Brownian Motion.

• Johannes Zimmer, (Technical University of Munich, Germany)

Fluctuating hydrodynamics for equations of Vlasov-Fokker-Planck type

We consider systems of interacting particles which are described by a second order Langevin equation, in particular particles experiencing inertia. We introduce an associated stochastic PDE of fluctuating hydrodynamics type ("Dean-Kawasaki-type equation"), which can be interpreted as stochastic version of a Vlasov-Fokker-Planck equation. We show a dichotomy previously known for purely diffusive systems holds here as well: Solutions exist only for suitable atomic initial data, but not for smooth initial data. The class of systems covered includes several models of active matter. Joint work with Fenna Müller and Max von Renesse.