Special Session 9: Stochastics and randomness in physical models

Moments estimates and sharp interface limit for the stochastic Cahn-Hilliard equation

Dimitra Antonopoulou National and Kapodistrian University of Athens Greece Co-Author(s):

Abstract: The phase separation process in a binary alloy is parameterized by a small positive parameter $\varepsilon$ representing the order of the width of the evolving layers. Stochastic versions for the equations modelling the evolution of the concentration of the phases arise due to thermal fluctuations, external mass supply or impurities in the alloy. We consider the $\varepsilon$-dependent stochastic Cahn-Hilliard equation with multiplicative and sufficiently regular in space noise with strength of order $O(\varepsilon^\gamma)$, $\gamma > 0$ in dimensions $d = 1,2,3$. We prove $p$-moments estimates in $H^1$, $H^2$ and $L^\infty$ norms. When the initial data are layered by using the energy ($H^1$) estimate we prove that, as $\varepsilon$ tends to $0$, the solution $u\rightarrow \pm 1$ in the $L^2$ norm with probability tending to $1$. This implies the complete separation of the two phases on the sharp interface limit even in the presence of noise.

Absence of anomalous dissipation in passive scalars advected by random autonomous drifts

Marco Bagnara Imperial College London England Co-Author(s):    Daniel W. Boutros, Camillo De Lellis, Svitlana Mayboroda

Abstract: We prove the absence of anomalous dissipation for passive scalars driven by some random autonomous divergence-free vector fields in $\mathbb T^d$. In dimension $d=2$ we just need continuity almost surely and a mild nondegeneracy condition on the randomness. In dimension $d\geq 3$ we assume a special geometric structure and almost sure H\older regularity with exponent bigger than $\frac{1}{8}$. No regularity is assumed on the passive scalar except for boundedness in the initial data. The proof relies on dimension-theoretic arguments, as opposed to commutator estimates. A consequence of these results is that the same assumptions prevent (almost surely) many other expected properties of turbulent flows, such as anomalous regularization, the Yaglom-Obukhov-Corrsin law, and Richardson diffusion.

Continuous data assimilation for 2d stochastic Navier-Stokes

Hakima Bessaih Florida Internaational University USA Co-Author(s):    B. Ferrario, O. Landoulsi, M. Zanella

Abstract: We extend the Azouani-Olson-Titi (AOT) data assimilation framework to the two-dimensional stochastic Navier-Stokes equations with additive and multiplicative noise. We establish conditions on the nudging parameter and observation scale ensuring convergence of the assimilated solution to the true flow. Convergence rates are obtained for both the multiplicative and additive noises These results provide a stochastic extension of AOT and highlight the role of noise in synchronization of fluid flows

Pathwise uniqueness for stochastic PDEs with singular Holder continuous drift

Davide Augusto Bignamini Universita degli studi dell`Insubria Italy Co-Author(s):    D. Addona, C. Orrieri, L. Scarpa

Abstract: In this talk, we will discuss pathwise uniqueness for mild solutions to stochastic PDEs with drift given in differential form. The key example that we want to study is the following SPDE evolving in $H=L^2([0,1]^d)$ with $d\in \{1,2,3\}$ \[ dX(t) + A^\gamma X(t) \, dt=B(X(t))\, d t +A^{-\rho}\, d W(t)\,, \qquad X(0)=x\,, \] where $\{W(t)\}_{t\geq 0}$ is an $H$-cylindrical Wiener process, $-A$ is a suitable realization of the Laplacian in $H$, $B:D(A^\mu)\rightarrow D(A^{-\nu})$ is locally $\theta$-Holder continuous with $\theta\in (0,1)$, $\gamma>0$ and $\mu, \nu, \rho\geq 0$ are given constants. Under suitable assumptions on $\gamma>0$ and $\mu, \nu, \rho\geq 0$, we will show that the pathwise uniqueness holds in $D(A^\mu)$. The singularity of the drift perturbation $B$ allows to achieve novel pathwise uniqueness results for several classes of examples, ranging from fluid-dynamics to phase-separation models.

The stochastic Cahn--Hilliard equation with singular potential, degenerate mobility and transport noise

Andrea Di Primio Scuola Normale Superiore Italy Co-Author(s):

Abstract: In this talk, I will introduce the stochastic Cahn--Hilliard equation with singular potential, variable mobility and driven by transport noise. In particular, the mobility is allowed to be degenerate. After illustrating some generalities concerning the model, results on the well-posedness of the system are addressed, in terms of existence and uniqueness of martingale and/or probabilistically-strong solutions. This is joint work with Andrea Papini (Chalmers University of Technology), Luca Scarpa (Politecnico di Milano) and Margherita Zanella (Politecnico di Milano).

The nonlinear Schr\odinger equation with multiplicative noise and arbitrary power of the nonlinearity

Benedetta Ferrario University of Pavia Italy Co-Author(s):    Zdzislaw Brzezniak, Mario Maurelli, Margherita Zanella

Abstract: We consider the stochastic nonlinear Schr\odinger equation with the polynomial nonlinearity \[ {\rm d} u(t,x)+\left[ \mathrm{i} \Delta u(t,x)+\mathrm{i} \alpha |u(t,x)|^{2\sigma} u(t,x) \right] \,{\rm d}t = \phi(u(t,x)) \,{\rm d} W(t) \] Classical results of global existence are obtained for power $\sigma$ not too large, depending on the spatial dimension $d$ and the parameter $\alpha$ ($\alpha>0$ is the focusing case and $\alpha\frac d2$. The effect of the noise is to prevent blow-up in finite time, differently from the deterministic setting. Moreover, we prove the existence of an invariant measure and its uniqueness under more restrictive assumptions on the noise term. As an example, one can consider a one dimensional real Wiener process $W$ and diffusion $\phi(u)=[a(1+\|u\|_{L^\infty(\mathbb T^d)})^\sigma+\mathrm{i} b(1+\|u\|_{L^\infty(\mathbb T^d)})^\sigma]u$ for real values $a,b$ with $a$ large enough. The choice $s>\frac d2$ provides the helpful estimate $\|u\|_{L^\infty(\mathbb T^d)} \le C \|u\|_{H^s(\mathbb T^d)}$, because of the continuous embedding $H^s(\mathbb T^d) \subset L^\infty(\mathbb T^d)$. Therefore the local existence result is a trivial fact. Our proof of global existence relies on a tightness method based on the choice of a suitable Lyapunov function. In particular, the global existence holds in both focusing and defocusing cases.

Fluctuations for mean field limits of singular interacting particle systems driven by fBm

Lucio Galeati University of L`Aquila Italy Co-Author(s):    Avi Mayorcas, Johanna Weinberger

Abstract: We consider a system of $N$ particles, subject to a mean-field type pairwise interaction kernel $K$, each driven by an independent fractional Brownian motion (idiosyncratic noises). Previous works established that, for a large class of non-Lipschitz, possibly singular kernels, the associated McKean-Vlasov equation is well-posed, and the empirical measure converges to its law as $N\to\infty$, with rate of order $N^{-1/2}$ in suitable negative Sobolev norms. In this talk I will present results concerning the Gaussian fluctuations underlying this mean field convergence, validating the optimality of this rate; they are valid for both first order interactions and for kinetic systems. The proofs are based on the use of Girsanov transform and the method of U-statistics first introduced by Sznitman. Based on joint work with Avi Mayorcas (Bath) and Johanna Weinberger (Technion).

On new results in the well-posedness of stochastic reaction diffusion models

Fabian Germ Delft University of Technology Netherlands Co-Author(s):    Mark Veraar, Antonio Agresti

Abstract: In this talk I will present some new results regarding the existence, uniqueness and regularity of solutions to stochastic reaction-diffusion models. The solution theory will be established in general Banach spaces. Using refined estimates, we obtain well-posedness for a wider class of systems, allowing for instance non-linearities with higher growth. Additional properties like blow-up criteria and instantaneous regularization of solutions will also be presented. If time permits, extensions to models with non-trace class noise will also be shown. While the results will be formulated in general Banach spaces, they will be motivated by concrete examples from Biology and related disciplines. This is joint work with Antonio Agresti and Mark Veraar.

Small Noise Asymptotics for Nonlinear Reaction-Diffusion Equations

giuseppina guatteri Politecnico di Milano Italy Co-Author(s):    Sandra Cerrai, Gianmario Tessitore

Abstract: This talk is based on joint work with Sandra Cerrai and Gianmario Tessitore. We present well-posedness and small-noise asymptotic results for a class of stochastic reaction-diffusion equations on bounded domains, driven by nonlinear and temporally nonlocal multiplicative noise. The diffusion coefficient depends on the solution through a conditional expectation at the final time, while the reaction term is only assumed to be continuous and quasi-dissipative, with no growth bounds or local Lipschitz assumptions. To handle these difficulties, we avoid relying on the theory of infinite-dimensional quasilinear PDEs and instead work directly at the SPDE level, combining a mild formulation with a Salins-type solution map and contraction arguments. This yields existence and uniqueness for sufficiently small noise, as well as a large deviation principle for the laws of the solution trajectories.

Motion planing with fractals - a rough path approach

Erika Hausenblas Technical University of Leoben Austria Co-Author(s):

Abstract: A control system is called non-holonomic when it cannot move instantaneously in all directions of its state space: at each configuration, only a lower-dimensional set of admissible velocities is available. A canonical example is the unicycle, which can roll forward and backwards and rotate, but cannot slide sideways, so lateral motion is not directly actuated. A similar constraint arises in the classical parallel-parking manoeuvre. From a dynamical viewpoint, the missing directions can be generated through Lie brackets of the control vector fields. Traditionally, motion along Lie bracket directions is achieved using carefully designed switching or oscillatory controls, such as bang-bang strategies or sinusoidal averaging. In this work, we took another approach. We introduce explicit Koch-type fractal controls that generate targeted motions along Lie-bracket directions using short-time inputs. To establish well-posedness for control systems driven by such fractal signals, we apply tools from rough path theory. As model problems, we illustrate our approach through bracket-induced lateral motion for the unicycle and a higher-order steering mechanisms in coupled trailer-type systems.

Anomalous Regularization and Dissipation for 2D Euler Equations with Rough Kraichnan Noise

Eliseo Luongo Bielefeld University Germany Co-Author(s):    Lucio Galeati, Umberto Pappalettera

Abstract: In the 1960s, Robert Kraichnan [Kraichnan 1968] proposed a synthetic model for passive scalar turbulence, consisting of a scalar advected by a random Gaussian velocity field that is white in time and H\{o}lder continuous in space. Despite its simplicity, this linear SPDE exhibits key features of realistic turbulent flows, such as anomalous dissipation. Renewed interest in this model followed [Coghi, Maurelli 2023], where it was proved that the same transport-type noise restores well-posedness in regimes where the deterministic 2D Euler equations admit non-unique weak solutions. In this talk, we further develop this line of research by investigating additional properties of the solutions constructed in [Coghi, Maurelli 2023]. In particular, we present new results on anomalous fractional Sobolev regularity and anomalous dissipation of the mean enstrophy for solutions to the 2D Euler equations with rough Kraichnan noise. Time permitting, we will also discuss implications for the well-posedness theory of more singular nonlinear advection models, such as the Surface Quasi-Geostrophic and Incompressible Porous Media equations. This talk is based on ongoing joint work with L. Galeati and U. Pappalettera.

Bismut-Elworthy type formulae for BSDEs with degenerate noise and related Kolmogorov equations

Federica Masiero Milano-Bicocca University Italy Co-Author(s):    Davide Addona

Abstract: In this talk we present how to derive Bismut-Elworthy formula under assumptions weaker than non degeneracy of the noise. We can apply our results to stochastic (damped) wave equations, whose regularizing properties of the transition semigroup are discussed. We present a nonlinear version of the Bismut formula for BSDEs, and we discuss applications to the solution of semilinear Kolmogorov equations.

On the stochastic H^1-critical Non Linear Sch\odinger equation

Annie A MILLET University Paris 1 Pantheon Sorbonne France Co-Author(s):    Svetlana Roudenko

Abstract: We prove local well posedness for the focusing/defocusing stochastic NLS equation in dimension 3 to 5 for an $H^1$-critical nonlinearity and an additive/multiplicative stochastic perturbation. In the case of a focuslng nonlinearity, we give criteria on the initial condition to obtain either some quantitative information about the maximal existence time, or blow-up with positive probability.

Anomalous dissipation and regularization in isotropic Gaussian turbulence

Umberto Pappalettera University of Basel Switzerland Co-Author(s):    Theodore D. Drivas, Lucio Galeati

Abstract: In this work we rigorously establish a number of properties of turbulent solutions to the stochastic transport and the stochastic continuity equations constructed by Le Jan and Raimond in [Ann. Probab. 30(2): 826-873, 2002].

Finite Speed of Propagation and Waiting Time Phenomena for Stochastic Porous-Media Equations with Nonlinear Conservative Noise

Joshua Utley Friedrich Alexander University Erlangen Nuremberg Germany Co-Author(s):    Guenther Gruen, Max Sauerbrey

Abstract: We are concerned with stochastic porous media equations with nonlinear conservative noise and show that for super-linear and sub-critical noise parameters, kinetic solutions have finite speed of propagation. In particular, we propose a novel iteration technique which allows us to obtain a Stampacchia-type inequality involving one single integral quantity, despite the possibly different scaling behaviors of the porous media and the noise term. This allows us to apply directly the stochastic filtering argument developed in [SIAM J. Math. Anal. 47 (2015), 825--854]. Using related ideas, we identify flatness conditions on initial data which guarantee locally the occurrence of a waiting time phenomenon, i.e., the onset of forward propagation of the solution`s support is locally delayed. The condition for the latter matches the one for $B=0$ up to a logarithmic correction if $n=(m+1)/2$, but it requires more and more flatness of the initial data as $n\to 1$. This is in line with the expected behavior of $u$: For $n=1$ an instantaneous forward motion is possible due to the effects of stochastic transport, no matter how flat the initial profile is.

New methods in Large deviations for SPDE

Mark Veraar TU Delft Netherlands Co-Author(s):    Antonio Agresti and Esmee Theewis

Abstract: In this talk, I will give an overview of new results on large deviations for stochastic evolution equations. Even in the variational setting, this leads to new results, as we assume neither monotonicity nor a compact embedding in the Gelfand triple. In particular, our framework applies to 2D Navier-Stokes equations on (un)bounded domains, 3D Primitive Equations, and reaction-diffusion equations. While the arguments are still based on the weak convergence approach, the main new ingredients come from the theory of critical spaces and maximal regularity techniques. Based on joint work with Antonio Agresti and Esmee Theewis