We show that, in one spatial and arbitrary jump dimension, the averaged solution of a Marcustype SPDE with pure jump Lévy transport noise satisfies a dissipative deterministic equation involving a fractional Laplace-type operator. To this end, we identify the correct associated Lévy measure for the driving noise. We consider this a first step in the direction of a non-local version of enhanced dissipation, a phenomenon recently proven to occur for Brownian transport noise and the associated local parabolic PDE by the first author. Moreover, we present numerical simulations, supporting the fact that dissipation occurs for the averaged solution, with a behavior akin to the diffusion due to a fractional Laplacian, but not in a pathwise sense.
The stochastic $p$-Laplace equation with multiplicative transport noise is studied on the torus $\mathbb T^d\, (d\geq 2)$. It is shown that the dissipation is enhanced by transport noise in both the averaged sense and the pathwise sense.
Structure preservation and emergent dissipation in stochastic wave equations with transport noise
School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
In this paper, we establish the global well-posedness of stochastic 3D Leray-$α$ model with general fractional dissipation driven by multiplicative noise. This model is the stochastic 3D Navier-Stokes equation regularized through a smoothing kernel of order $θ_1$ in the nonlinear term and a $θ_2$-fractional Laplacian. In the case of $θ_1 \ge 0, θ_2 > 0$ and $θ_1+θ_2 \geq\frac{5}{4}$, we prove the global existence and uniqueness of strong solutions. The main results cover many existing works in the deterministic cases, and also generalize some known results of stochastic models as our special c…
A stochastic version of the two-layer quasi-geostrophic model (2LQG) with multiplicative transport noise is analysed. This popular intermediate complexity model describes large scale atmosphere and ocean dynamics at the mid-latitudes. The transport noise, which acts on both layers, accounts for the unresolved small scales. After establishing the well-posedness of the perturbed equations, we show that, under a suitable scaling of the noise, the solutions converge to the deterministic 2LQG model with enhanced dissipation. Moreover, these solutions converge to the deterministic stationary ones on…
The Energy-Dissipation Principle provides a variational tool for the analysis of parabolic evolution problems: solutions are characterized as so-called null-minimizers of a global functional on entire trajectories. This variational technique allows for applying the general results of the calculus of variations to the underlying differential problem and has been successfully applied in a variety of deterministic cases, ranging from doubly nonlinear flows to curves of maximal slope in metric spaces. The aim of this note is to extend the Energy-Dissipation Principle to stochastic parabolic evolut…
We formulate explicit bounds to guarantee the exponential dissipation for some non-gradient stochastic differential equations towards their invariant distributions. Our method extends the connection between Gamma calculus and Hessian operators in $L^2$--Wasserstein space. In details, we apply Lyapunov methods in the space of probabilities, where the Lyapunov functional is chosen as the relative Fisher information. We derive the Fisher information induced Gamma calculus to handle non-gradient drift vector fields. We obtain the explicit dissipation bound in terms of $L_1$ distance and formulate …
We derive the fluctuation-dissipation relation and explore its connection with the equipartition theorem and Maxwell-Boltzmann statistics through the use of different stochastic analytical techniques. Our first approach is the theory of backward stochastic differential equations, which arises naturally in this context, and facilitates the understanding of the interplay between these classical results of statistical mechanics. The second approach consists in deriving forward stochastic differential equations for the energy of an electric system according to both Itô and Stratonovich stochastic …
We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the `fluctuation-dissipation theorem', the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic …
An exact relation is derived between scalar dissipation due to molecular diffusivity and the randomness of stochastic Lagrangian trajectories for flows without bounding walls. This "Lagrangian fluctuation-dissipation relation" equates the scalar dissipation for either passive or active scalars to the variance of scalar inputs associated to initial scalar values and internal scalar sources, as those are sampled backward in time by the stochastic Lagrangian trajectories. As an important application, we reconsider the phenomenon of "Lagrangian spontaneous stochasticity" or persistent non-determin…
We consider degenerate diffusion equations of the form $\partial_tp_t = Δf(p_t)$ on a bounded domain and subject to no-flux boundary conditions, for a class of nonlinearities $f$ that includes the porous medium equation. We derive for them a trajectorial analogue of the entropy dissipation identity, which describes the rate of entropy dissipation along every path of the diffusion. Our approach is based on applying stochastic calculus to the underlying probabilistic representations, which in our context are stochastic differential equations with normal reflection on the boundary. This trajector…
We study the mixing and dissipation properties of the advection-diffusion equation with diffusivity $0 < κ\ll 1$ and advection by a class of random velocity fields on $\mathbb T^d$, $d=\{2,3\}$, including solutions of the 2D Navier-Stokes equations forced by sufficiently regular-in-space, non-degenerate white-in-time noise. We prove that the solution almost surely mixes exponentially fast uniformly in the diffusivity $κ$. Namely, that there is a deterministic, exponential rate (independent of $κ$) such that all mean-zero $H^1$ initial data decays exponentially fast in $H^{-1}$ at this rate wit…
