Special Session 75: Stochastic Evolution Systems Across Scales: Theory and Applications

$L^2$-Wasserstein ergodicity of modified Euler schemes for SDEs with high diffusivity and applications

Jianhai Bao Tianjin University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we are concerned with a modified Euler scheme for the SDE under consideration, where the drift is of super-linear growth and dissipative merely outside a closed ball. By adopting the synchronous coupling, along with the construction of an equivalent metric, the $L^2$-Wasserstein ergodicity of the modified Euler scheme is addressed provided that the diffusivity is large enough. In particular, as a by-product, the $L^2$-Wasserstein ergodicity of the projected Euler scheme and the tamed Euler algorithm is treated under much more explicit conditions imposed on drifts. The theory derived on the $L^2$-Wasserstein ergodicity has numerous applications on various aspects. In addition to applications on Poincar\`{e} inequalities, concentration inequalities for empirical averages, and bounds concerning KL-divergence, in this paper we present another two potential applications. One concerns the $L^2$-Wasserstein error bound between the exact invariant probability measure and the numerical counterpart corresponding to the projected Euler scheme and the tamed Euler recursion, respectively. It is worthy to emphasize that the associated convergence rate is improved greatly in contrast to the existing literature. Another is devoted to the strong law of large numbers of additive functionals related to the modified Euler algorithm, where the observable functions involved are allowed to be of polynomial growth.

Well-posedness of stochastic Degasperis-Procesi equation

Nikolai V Chemetov University of Sao Paulo Brazil Co-Author(s):    Fernanda Cipriano

Abstract: Well-posedness of stochastic Degasperis-Procesi equation \bigskip Nikolai V. Chemetov (DCM - FFCLRP, University of Sao Paulo, Brazil) \bigskip This talk is concerned with the existence of a solution to the stochastic Degasperis-Procesi equation on R with an infinite dimensional multiplicative noise and integrable initial data. Writing the equation as a system composed of a stochastic nonlinear conservation law and an elliptic equation [1], we are able to develop a method based on the conjugation of kinetic theory [2] with stochastic compactness arguments. More precisely, we apply the stochastic Jakubowski-Skorokhod representation theorem to show the existence of a weak kinetic martingale solution [3]. We also demonstrate the uniqueness result [4]. This is a joint work with Fernanda Cipriano (Universidade Nova de Lisboa, Portugal). \medskip Bibliography: 1. L.K. Arruda, N.V. Chemetov, F. Cipriano, Solvability of the Stochastic Degasperis-Procesi Equation. J. Dynamics and Differential Equations, 35(1) (2023), 523-542. 2. N.V. Chemetov, W Neves, The generalized Buckley--Leverett system: solvability. Archive for Rational Mechanics and Analysis, 208 (1) (2013), 1-24. 3. N.V. Chemetov, F. Cipriano, Weak solution for stochastic Degasperis-Procesi equation. J. Differential Equations, Vol. 382 (15) (2024), 1-49. 4. N.V. Chemetov, F. Cipriano, The uniqueness result for the weak solution for stochastic Degasperis-Procesi equation. To be submitted.

Invariant measures for a class of stochastic third grade fluid equations in 2D and 3D bounded domains

Fernanda F. Cipriano NOVA University of Lisbon Portugal Co-Author(s):    Yassine Tahraoui, Fernanda Cipriano

Abstract: This work aims to investigate the well-posedness and the existence of ergodic invariant measures for a class of third grade fluid equations in bounded domain of $\mathbb{R}^d,d=2,3,$ in the presence of a multiplicative noise. First, we show the existence of a martingale solution by coupling a stochastic compactness and monotonicity arguments. Then, we prove a stabilty result, which gives the pathwise uniqueness of the solution and therefore the existence of strong probabilistic solution. Secondly, we use the stability result to show that the associated semigroup is Feller and by using Krylov-Bogoliubov Theorem we get the existence of an invariant probability measure. Finally, we show that all the invariant measures are concentrated on a compact subspace of $L^2$, which leads to the existence of an ergodic invariant measure.

The Camassa-Holm equation with transport noise

Helge Holden Norwegian University of Science and Technology Norway Co-Author(s):

Abstract: We will discuss recent work regarding the Camassa--Holm equation with transport noise, more precisely, the equation $u_t+uu_x+P_x+\sigma u_x \circ dW=0$ and $P-P_{xx}=u^2+u_x^2/2$. In particular, we will show existence of a weak, global, dissipative solution of the Cauchy initial-value problem on the torus. This is joint work with L. Galimberti (King`s College), K.H. Karlsen (Oslo), and P.H.C. Pang (NTNU/Oslo).

Compactness of Solutions to Stochastic Kinetic Equations

Kenneth H. Karlsen Department of Mathematics, University of Oslo Norway Co-Author(s):

Abstract: We consider stochastically forced kinetic equations in heterogenous environments and stochastic conservation laws with spatially irregular flux. We present new results on the strong compactness of the velocity averages of solutions under general integrability conditions. The talk draws upon papers authored with M. Erceg, M. Kunzinger, and D. Mitrovic.

Restricted path characteristic function determines the law of stochastic processes

Siran Li Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):    Zijiu Lyu; Hao Ni; Jiajie Tao

Abstract: A central question in rough path theory is characterising the law of stochastic processes. It is established in [I. Chevyrev $\&$ T. Lyons, Characteristic functions of measures on geometric rough paths, \textit{Ann. Probab.} \textbf{44} (2016), 4049--4082] that the path characteristic function (PCF), \emph{i.e.}, the expectation of the unitary development of the path, uniquely determines the law of the unparametrised path. We show that PCF restricted to certain subspaces of sparse matrices is sufficient to achieve this goal. The key to our arguments is an explicit algorithm --- as opposed to the nonconstructive approach in [I. Chevyrev $\&$ T. Lyons, \emph{op. cit.}] --- for determining a generic element $X$ of the tensor algebra $\bigoplus_{n=0}^\infty\left(\mathbb{R}^d\right)^{\otimes n}$ from its moment generating function. Our only assumption is that $X$ has a nonzero radius of convergence, which relaxes the condition of having an infinite radius of convergence in the literature. As applications of the above theoretical findings, we propose the restricted path characteristic function distance (RPCFD), a novel distance function for probability measures on the path space that offers enormous advantages for dimension reduction. Its effectiveness is validated via hypothesis testing on fractional Brownian motions, thus demonstrating the potential of RPCFD in generative modeling for synthetic time series generation.

On the Lagrange multiplier method to the Euler and Navier-Stokes equations

Xiangdong Li AMSS, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Guoping Liu

Abstract: In this talk, we use the Lagrange multiplier method to derive the incompressible Euler and Navier-Stokes equations on a compact Riemannian manifold $M$, in which the pressure is given by a variant of the Lagrange multiplier for the incompressible condition ${\rm{div}}~u=0$. Moreover, we give a new derivation of the incompressible Navier-Stokes equation on a compact Riemannian manifold $M$ via the Bellman dynamic programming principle on the infinite dimensional group of diffeomorphisms $G={\rm Diff}(M)$. In particular, in the inviscid case, we give a new derivation of the incompressible Euler equation on a compact Riemannian manifold $M$. Our method provides an explicit construction of a solution to the incompressible Euler and Navier-Stokes equations via the value function and the Lagrange multiplier of a deterministic or stochastic optimal control problem on $G={\rm Diff}(M)$. Joint work with Guoping Liu (HUST, Wuhan).

Fluctuations of SHE

Xue-Mei Li EPFL Switzerland Co-Author(s):

Abstract: We explore the stochastic heat equation (HE) with space time Gaussian noise exhibiting long-range spatial dependence. These equations produce solutions that admit a stationary field. Our focus is on the fluctuation problem associated with diffusively scaled solutions from their average. We demonstrate that the fluctuations of the appropriately scaled solutions from their mean converge weakly to the solution of a stochastic heat equation with additive noise, where the spatial correlation function is governed by the Riesz potential. This is joint work with L. Gerolla and M. Hairer

2D Smagorinsky-Type Large Eddy Models as Limits of Stochastic PDEs

Dejun Luo Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):

Abstract: We prove that a version of Smagorinsky large eddy model for a 2D fluid in vorticity form is the scaling limit of suitable stochastic models for large scales, where the influence of small turbulent eddies is modeled by a transport-type noise. This talk is based on a joint work with F. Flandoli and E. Luongo.

Existence and uniqueness of weak solutions for the generalized stochastic Navier-Stokes-Voigt equations

Manil T. Mohan Indian Institute of Technology Roorkee India Co-Author(s):    Ankit Kumar and Hermenegildo Borges de Oliveira

Abstract: In this talk, we consider the incompressible generalized Navier-Stokes-Voigt equations in a bounded domain $\mathcal{O}\subset\mathbb{R}^d$, $d\geq 2$, driven by a multiplicative Gaussian noise. The considered momentum equation is given by: $\begin{align*} \mathrm{d}\left(\boldsymbol{u} - \kappa \Delta \boldsymbol{u}\right) = \left[\boldsymbol{f} +\operatorname{div} \left(-\pi\mathbf{I}+\nu|\mathbf{D}(\boldsymbol{u})|^{p-2}\mathbf{D}(\boldsymbol{u})-\boldsymbol{u}\otimes \boldsymbol{u}\right)\right]\mathrm{d} t + \Phi(\boldsymbol{u})\mathrm{d} \mathrm{W}(t). \end{align*}$ In the case of $d=2,3$, $\boldsymbol{u}$ accounts for the velocity field, $\pi$ is the pressure, $\boldsymbol{f}$ is a body force and the final term stay for the stochastic forces. Here, $\kappa$ and $\nu$ are given positive constants that account for the kinematic viscosity and relaxation time, and the power-law index $p$ is another constant (assumed $p>1$) that characterizes the flow. We use the usual notation $\mathbf{I}$ for the unit tensor and $\mathbf{D}(\boldsymbol{u}):=\frac{1}{2}\left(\nabla \boldsymbol{u} + (\nabla \boldsymbol{u})^{\top}\right)$ for the symmetric part of velocity gradient. For $p\in\big(\frac{2d}{d+2},\infty\big)$, we first prove the existence of a martingale solution. Then we show the \emph{pathwise uniqueness of solutions}. We employ the classical Yamada-Watanabe theorem to ensure the existence of a unique probabilistic strong solution. This is a joint work with Dr. Ankit Kumar and Dr. H. B. de Oliveira.

Stochastic extrinsic derivative flows on the space of absolutely continuous measures

Simon Wittmann The Hong Kong Polytechnic University Hong Kong Co-Author(s):    Panpan Ren, Feng-Yu Wang

Abstract: Right Markov processes, whose state space consists of absolutely continuous measures (resp.~probabilities) w.r.t.~a fixed measure $\lambda$ on a Polish space $M$, are the central objects in this talk.These stand in one-to-one correspondence with quasi-regular Dirichlet forms. Our first result states the quasi-regularity for a broad class of Dirichlet forms, whose diffusion part is of extrinsic derivative type and which have a non-local (killing and jumping) part. A natural way to construct closed forms of this class is to take as reference measure the push-forward of a Gaussian on $L^2(M,\lambda)$ under $f\mapsto f^2\lambda$, resp.~$f\mapsto (f^2/\lambda(f^2))\lambda$. Hence, we obtain Gaussian-type Sobolev spaces on absolutely continuous measures and associated Ornstein-Uhlenbeck processes. In case $M=\mathbb R^d$ the entropy functional is identified as a member of such a Sobolev space and we construct its stochastic extrinsic derivative flow driven by the Ornstein-Uhlenbeck process.

On the empty balls of super-Brownian motion and branching random walk

Jie Xiong Southern University of Science and Technology Peoples Rep of China Co-Author(s):    Shuxiong Zhang

Abstract: In this talk, I will explore various limiting behavior of the radius of the largest ball around the origin which is not occupied by a super-Brownian motion and that not by a branching random walk according to the spatial dimension as time tends to infinity.

Metastability of Random Dynamical Systems

Tusheng Zhang University of Science and Technology of China Peoples Rep of China Co-Author(s):

Abstract: In this talk, we first present a criterion on uniform large deviation principles (ULDP) of stochastic differential equations under Lyapunov conditions on the coefficients. In the second part, using the ULDP criterion we preclude the concentration of limiting measures of invariant measures of stochastic dynamical systems on repellers and acyclic saddle chains. Of particular interest, we determine the limiting measures of the invariant measures of the famous stochastic van der Pol equation and van der Pol Duffing equation whose noises are naturally degenerate.