The paper deals with linear passive scalar transport-diffusion equation subject to a velocity field which is white noise in time and is mainly active at small scales in space. The purpose is investigating the enhancement of dissipation and decay given by the small-scale transport. We modify and improve estimates from the previous work by Flandoli et al. (J. Differ. Equations 394, 237–277, 2024), in order to investigate a different regime, namely small molecular diffusion and small noise intensity – corresponding to small turbulent kinetic energy. The noise specification is carefully tuned to Kolmogorov theory of turbulent fluids.
The purpose of this paper is to investigate the problem of autocorrelated moving boundaries for a one-dimensional heat equation, where the moving boundary is a fractional Brownian motion B^H with Hurst parameter H >1/2 and the dependent Volterra-type Dirichlet boundary term ξ is verified to lie in the exponential Orlicz space 𝒮_Ψ^(v)(Ω× (0,T]) with v∈ (0,1] . This problem is interpreted as a single-layer potential involving stochastic heat kernel G( t-s,x-B^H(s)) = 1/√(4π (t-s))exp( -( x-B^H(s)) ^2/4(t-s)) , with 0z ) , z> E [ sup _(s,t)∈𝒬 B^H(t)-B^H(s)/(t-s)^1/2+γ] , where 𝒬={(s,t): 0≤ s ≤ t≤ T } and γ∈ [0,H-1/2) . By potential theoretic arguments, we establish the unique weak solution to the fractional Brownian moving boundary problem in L^p(Ω ; C_(v)((0,T];C_b(-∞ ,B^H(· )))) with p>1 and v∈ (0,1] .
We present an example of a linear partial differential equation whose Cauchy problem becomes well-posed when perturbed by noise. Specifically, we make clear how a suitable multiplicative Stratonovich perturbation of Brownian type renders a weakly hyperbolic operator with double involutive characteristics well-posed in the C^∞-category, while its deterministic counterpart is only well-posed in the Gevrey s classes with 1 ≤ s <2 .
We consider time-dependent singular stochastic partial differential equations on the three-dimensional torus. These equations are only well-posed after one adds renormalization terms. In order to construct a well-defined notion of solution, one should put the equation in a more general setting. In this article, we consider the paradigm of paracontrolled distributions, and get concentration results around a stable deterministic equilibrium for solutions of non-autonomous generalizations of the (Phi 34) model. Specifically, we obtain Gaussian-type tail bounds.
In this work, we deal with the stochastic counterpart of the nonlocal Cahn–Hilliard equation with regular potential in a smooth bounded one-, two- or three-dimensional domain. The problem is endowed with homogeneous Neumann boundary conditions and random initial data. Furthermore, the system is driven by cylindrical noise of multiplicative type. For the resulting system, we are able to show the existence of probabilistically-weak (or martingale) solutions in two and three dimensions, that are unique and probabilistically-strong under suitable assumptions on the stochastic diffusion. Moreover, we investigate the nonlocal-to-local asymptotics toward solutions of the local stochastic Cahn–Hilliard equations, establishing, under regularity conditions, a precise rate of convergence as well.
The work focuses on averaging for a class of multivalued Dirichlet-Neumann problems. Firstly, we establish an averaging principle for general multivalued stochastic differential equations in the weak sense. Subsequently, we present another averaging principle for general forward-backward multivalued stochastic systems. Finally, we apply these results to investigate the averaging behavior of a class of multivalued Dirichlet-Neumann problems.
This paper establishes a probabilistic representation for the solution of the parabolic obstacle problem associated with the normalized p-Laplacian. We introduce a zero-sum stochastic tug-of-war game with noise in a space-time cylinder, where one player has the option to stop the game at any time to collect a payoff given by an obstacle function. We prove that the value functions of this game exist, satisfy a dynamic programming principle, and converge uniformly to the unique viscosity solution of the continuous obstacle problem as the step size ε tends to zero.
We investigate a fully discrete finite element approximation for the stochastic Kuramoto–Sivashinsky equation, combining the standard finite element methods in spatial discretization with the implicit Euler–Maruyama scheme in time. Rigorous error estimates are established for two distinct noise regimes. In the case of bounded multiplicative noise, we prove optimal strong convergence rates in full expectation. The analysis relies crucially on a stochastic Gronwall inequality and an exponential stability estimate for the PDE solution, which together control the interplay between the nonlinear drift and the multiplicative stochastic forcing. For general multiplicative noise, where boundedness no longer holds, we derive sub-optimal convergence rates in probability by introducing a localization technique based on carefully constructed subsets of the sample space. This dual framework demonstrates that the proposed fully discrete scheme achieves strong convergence under bounded noise and probabilistic convergence under general multiplicative noise, thus providing the first comprehensive error analysis for numerical approximations of the stochastic Kuramoto–Sivashinsky equation. Numerical experiments are also provided to demonstrate the efficiency of the numerical method and validate the theoretical results.
We consider fractional stochastic heat equations with space-time Lévy white noise of the form ∂ X/∂ t(t,x)=ℒ_αX(t,x)+σ (X(t,x))Λ̇(t,x). Here, the principal part ℒ_α=-(-Δ )^α /2 is the d-dimensional fractional Laplacian with α∈ (0,2) , the noise term Λ̇(t,x) denotes the space-time Lévy white noise, and the function σ : ℝ→ℝ is Lipschitz continuous. Under suitable assumptions, we obtain bounds for the Lyapunov exponents and the growth indices of exponential type on pth moments of the mild solutions, which are connected with the weakly intermittency properties and the characterizations of the high peaks propagate away from the origin. Unlike the case of the Gaussian noise, the proofs heavily depend on the heavy tail property of heat kernel estimates for the fractional Laplacian. The results complement these in [8, 12] for fractional stochastic heat equations driven by space-time white noise and stochastic heat equations with Lévy noise, respectively.
We introduce an abstract Hilbert space-valued framework of Markovian lifts for stochastic Volterra equations with operator-valued Volterra kernels. Our main results address the existence and characterisation of possibly multiple limit distributions and stationary processes, a law of large numbers including a convergence rate, and the central limit theorem for time averages of the process within the Gaussian domain of attraction. As particular examples, we study Markovian lifts based on Laplace transforms in a weighted Hilbert space of densities and Markovian lifts based on the shift semigroup on the Filipović space. We illustrate our results for the case of fractional stochastic Volterra equations with additive or multiplicative Gaussian noise.
We consider the asymptotic behaviour of the fluctuation process for large stochastic systems of interacting particles driven by both idiosyncratic and common noise with an interaction kernel k ∈ L^2(ℝ^d) ∩ L^∞ (ℝ^d) . Our analysis relies on uniform relative entropy estimates and Kolmogorov’s compactness criterion to establish tightness and convergence of the fluctuation process. In this framework, an extension of the exponential law of large numbers is used to derive the necessary uniform estimates, while a conditional Fubini theorem is employed in the identification of the limit in the presence of common noise. We demonstrate that the fluctuation process converges in distribution to the unique solution of a linear stochastic evolution equation. This work extends previous fluctuation results beyond the classical Lipschitz framework.
We derive the nonlinear stochastic Fokker-Planck equation from stochastic particle systems with individual and environmental noises via relative entropy method, with pathwise quantitative bounds. Moreover, we prove the existence of a unique strong solution to the associated Fokker-Planck equation. Our proof is based on tools from PDE analysis, stochastic analysis, functional inequalities, and also we use the dissipation of entropy which provides some bound on the Fisher information of the particle system. The approach applies to repulsive and attractive kernels.
In this article, we continue the investigations initiated by the first author in Balan [2] related to the study of stochastic partial differential equations (SPDEs) with Lévy colored noise on ℝ_+×ℝ^d . This noise is constructed from a Lévy white noise on ℝ_+×ℝ^d (which is in turn built from a Poisson random measure on ℝ_+×ℝ^d ×ℝ_0 with intensity dtdx ν (dz) ), using the convolution with a suitable spatial kernel κ . We assume that the Lévy measure ν has finite variance. Therefore, the stochastic integral with respect to this noise is constructed similarly to the integral with respect to the spatially-homogeneous Gaussian case considered in Dalang [15]. Using Rosenthal’s inequality, we provide an upper bound for the p-th moment of the stochastic integral with respect to the Lévy colored noise, which allows us to identify sufficient conditions for the solution of an SPDE driven by this noise to have higher order moments. We first analyze this question for the linear SPDE (in which the noise enters in an additive way), considering as examples the stochastic heat and wave equations in any dimension d, for three examples of kernels κ : the heat kernel, the Riesz kernel, and the Bessel kernel. Then, we present a general theory for a non-linear SPDE with Lipschitz coefficients, and perform a detailed analysis in the case of the heat equation (in dimension d≥ 1 ), and wave equation (in dimension d≤ 3 ), for the same kernels κ . We show that the solution of each of these equations has a finite upper Lyapounov exponent of order p≥ 2 , and in some cases, is weakly intermittent (in the sense of Foondun and Khoshnevisan [19]). In the case of the parabolic/hyperbolic Anderson model with Lévy colored noise, we provide the Poisson chaos expansion of the solution and the explicit form of the second-order Lyapounov exponent.
We study the stability and dynamics of solitons in the Korteweg-de Vries (KdV) equation in the presence of noise and deterministic forcing. The noise is space-dependent and statistically translation-invariant. We show that, for small forcing, solitons remain close to the family of traveling waves in a weighted Sobolev norm, with high probability. We study the effective dynamics of the soliton amplitude and position via their variational phase, for which we derive explicit modulation equations. The stability result holds on a time scale where the deterministic forcing induces significant amplitude modulation.
We study a stochastic complex Ginzburg-Landau equation (SCGL) on compact surfaces with magnetic Laplacian and polynomial nonlinearity, forced by a space-time white noise. After renormalizing the equation in a suitable manner, we show that the dynamics is locally well-posed. Moreover, we prove deterministic global well-posedness for the defocusing SCGL in the weakly dispersive regime.
We consider time-dependent singular stochastic partial differential equations on the three-dimensional torus. These equations are only well-posed after one adds renormalization terms. In order to construct a well-defined notion of solution, one should put the equation in a more general setting. In this article, we consider the paradigm of paracontrolled distributions, and get concentration results around a stable deterministic equilibrium for solutions of non-autonomous generalizations of the (Φ _3^4) model. Specifically, we obtain Gaussian-type tail bounds.
We consider a prototypical parabolic SPDE with finite-dimensional multiplicative noise, which, subject to a nonnegative initial datum, has a unique nonnegative solution. Inspired by well-established techniques in the deterministic case, we introduce a finite element discretization of this SPDE that is convergent and which, subject to a nonnegative initial datum and unconditionally with respect to the spatial discretization parameter, preserves nonnegativity of the numerical solution throughout the course of evolution. We perform a mathematical analysis of this method. In addition, in the associated linear setting, we develop a fully discrete scheme that also preserves nonnegativity, and we present numerical experiments that illustrate the advantages of the proposed method over alternative finite element and finite difference methods that were previously considered in the literature, which do not necessarily guarantee nonnegativity of the numerical solution.
