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The purpose of this work is to provide an explicit construction of a strong Feller semigroup on the space of probability measures over the real line that additionally maps bounded measurable functions into Lipschitz continuous functions, with a Lipschitz constant that blows up in an integrable manner in small time. Our construction relies on a rearranged version of the stochastic heat equation on the circle driven by a coloured noise. Formally, this stochastic equation writes as a reflected equation in infinite dimension. Under the action of the rearrangement, the solution is forced to live in a space of quantile functions that is isometric to the space of probability measures on the real line. We prove the equation to be solvable by means of an Euler scheme in which we alternate flat dynamics in the space of random variables on the circle with a rearrangement operation that projects back the random variables onto the subset of quantile functions. A first challenge is to prove that this scheme is tight. A second one is to provide a consistent theory for the limiting reflected equation and in particular to interpret in a relevant manner the reflection term. The last step in our work is to establish the aforementioned Lipschitz property of the semigroup by adapting earlier ideas from the Bismut–Elworthy–Li formula. [ABSTRACT FROM AUTHOR]

This paper is devoted to the continuity of the weak solution for tempered fractional general diffusion equations driven by tempered fractional Brownian motion (TFBM). Based on the Feynman-Kac formula (1.2), by using the Itô isometry for the stochastic integral with respect to TFBM, Parseval's identity and some ingenious calculations, we establish the continuities of the solution with respect to Hurst index H and tempering parameter λ of TFBM. [ABSTRACT FROM AUTHOR]

This study introduces three numerical techniques to obtain the solutions of stochastic evolution equations driven by infinite-dimensional fractional Brownian motion. The fractional Brownian motion, which is defined by a Hurst parameter H ∈ 1 2 , 1 , is a Gaussian process widely applied in modeling various natural and engineered systems. We use the Galerkin method for spatial discretization with three distinct time discretization schemes to approximate the solutions. We specifically consider Q-fractional Brownian motion, where Q represents a trace class operator. This paper seeks to establish theoretical results on the convergence and error bounds of the proposed methods, supplemented by validation through numerical experiments. [ABSTRACT FROM AUTHOR]

We estimate the finite-time Lyapunov exponents for a stochastic partial differential equation driven by a fractional Brownian motion (fbm) with Hurst index H ∈ (0 , 1) close to a bifurcation of pitchfork type. We characterize regions depending on the distance from bifurcation, the Hurst parameter of the fbm and the noise strength where finite-time Lyapunov exponents are positive and thus indicate a change of stability. The results on finite-time Lyapunov exponents are novel also for SDEs perturbed by fractional noise. [ABSTRACT FROM AUTHOR]

The paper establishes the strong convergence rates of a spatio-temporal full discretization of the stochastic wave equation with nonlinear damping in dimension one and two. We discretize the SPDE by applying a spectral Galerkin method in space and a modified implicit exponential Euler scheme in time. The presence of the super-linearly growing damping in the underlying model brings challenges into the error analysis. To address these difficulties, we first achieve upper mean-square error bounds, and then obtain mean-square convergence rates of the considered numerical solution. This is done without requiring the moment bounds of the full approximations. The main result shows that, in dimension one, the scheme admits a convergence rate of order 1 2 in space and order 1 in time. In dimension two, the error analysis is more subtle and can be done at the expense of an order reduction due to an infinitesimal factor. Numerical experiments are performed and confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

We prove the existence of a B-continuous viscosity solution for a class of infinite dimensional semilinear partial differential equations (PDEs) using probabilistic methods. Our approach also yields a stochastic representation formula for the solution in terms of a scalar-valued backward stochastic differential equation. The uniqueness is proved under additional assumptions using a comparison theorem for viscosity solutions. Our results constitute the first nonlinear Feynman–Kac formula using the notion of B-continuous viscosity solutions and thus introduces a framework allowing for generalizations to the case of fully nonlinear PDEs. [ABSTRACT FROM AUTHOR]

Purpose: The main motivation of this paper is to present the Yosida approximation of a semi-linear backward stochastic differential equation in infinite dimension. Under suitable assumption and condition, an L2-convergence rate is established. Design/methodology/approach: The authors establish a result concerning the L2-convergence rate of the solution of backward stochastic differential equation with jumps with respect to the Yosida approximation. Findings: The authors carry out a convergence rate of Yosida approximation to the semi-linear backward stochastic differential equation in infinite dimension. Originality/value: In this paper, the authors present the Yosida approximation of a semi-linear backward stochastic differential equation in infinite dimension. Under suitable assumption and condition, an L2-convergence rate is established. [ABSTRACT FROM AUTHOR]

We provide necessary and sufficient conditions for stochastic invariance of finite dimensional submanifolds for solutions of stochastic partial differential equations (SPDEs) in continuously embedded Hilbert spaces with non-smooth coefficients. Furthermore, we establish a link between invariance of submanifolds for such SPDEs in Hermite Sobolev spaces and invariance of submanifolds for finite dimensional SDEs. This provides a new method for analyzing stochastic invariance of submanifolds for finite dimensional Itô diffusions, which we will use in order to derive new invariance results for finite dimensional SDEs. [ABSTRACT FROM AUTHOR]

We study the long time behavior of the standard linear implicit Euler scheme for the discretization of a class of erdogic parabolic semilinear SPDEs driven by additive space-time white noise. When the nonlinearity is a gradient, the invariant distribution is of Gibbs form, but it cannot be approximated in the total variation sense by the standard Euler scheme. We prove that the numerical scheme gives an approximation in the total variation sense of a modified Gibbs distribution, which is the invariant distribution of a modified SPDE. The modified distribution and the modified equation depend on the time-step size. This original result goes beyond existing results in the literature where the weak error estimates for the approximation of the invariant distribution do not imply convergence in total variation when the time-step size vanishes. The proof of the main result requires regularity properties of associated infinite dimensional Kolmogorov equations. [ABSTRACT FROM AUTHOR]

We study Laplace equation and heat equation with the Gaussian white noise as the boundary/initial condition. Using the stochastic pre-orthogonal adaptive Fourier decomposition on the Bergman space, we give sparse expansions of the solutions of the boundary/initial value problems. The proposed methodology breaks away from the H - H K structure. The key point to the solution is to use the Wiener isometry property of the Gaussian white noise integral and the corresponding Green's function to represent the covariance of the solution. For the heat equation with the Gaussian white noise initial value, we define a class of parabolic Bergman space following E. Stein, prove existence of Bergman kernel, and obtain its expression, and further verify the boundary vanishing property needed for generating a sparse expansion. In order to make the heat flow satisfy a long-time decay property, we induce a subspace of the parabolic Bergman space, which corresponds to a class of homogeneous Sobolev spaces. [ABSTRACT FROM AUTHOR]

We study the continuity set (the set of all continuous points) of pullback random attractors from a parametric space into the space of all compact subsets of the state space with Hausdorff metric. We find a general theorem that the continuity set is an IOD-type (a countable intersection of open dense sets) with the local similarity under appropriate conditions of random dynamical systems, and we further show that any IOD-type set in the parametric space has the continuous cardinality, which affirmatively answers the unsolved question about the cardinality of the continuity set of attractors in the literature. Applying to the random nonautonomous nonlocal parabolic equations on an unbounded domain driven by colored noise, we establish the existence and IOD-type continuity of pullback random attractors in time, sample-translation and noise-size, moreover, we prove that the continuity set of the pullback random attractor on the plane of time and sample-translation is composed of diagonal rays whose number of bars is the continuous cardinality. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with an operator-splitting scheme for linear fractional and fractional degenerate stochastic conservation laws driven by multiplicative Lévy noise. More specifically, using a variant of the classical Kružkov doubling of variables approach, we show that the approximate solutions generated by the splitting scheme converge to the unique stochastic entropy solution of the underlying problems. Finally, the convergence analysis is illustrated by several numerical examples. [ABSTRACT FROM AUTHOR]

In Tao 2016, the author constructs an averaged version of the deterministic three-dimensional Navier–Stokes equations (3D NSE) which experiences blow-up in finite time. In the last decades, various works have studied suitable perturbations of ill-behaved deterministic PDEs in order to prevent or delay such behavior. A promising example is given by a particular choice of stochastic transport noise closely studied in Flandoli et al. 2021. We analyze the model in Tao 2016 in view of these results and discuss the regularization skills of this noise in the context of the averaged 3D NSE. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of Stepanov-like (weighted) pseudo S-asymptotically Bloch type periodic processes in the square mean sense, and establish some basic results on the function space of such processes like completeness, convolution and composition theorems. Under the situation that the functions forcing are Stepanov-like (weighted) pseudo S-asymptotically Bloch type periodic and verify some suitable assumptions, we establish the existence and uniqueness of square-mean (weighted) pseudo S-asymptotically Bloch type periodic mild solutions of some fractional stochastic integrodifferential equations (driven by fractional Brownian motion). Finally, the most important findings are substantiated with the assistance of an illustration. [ABSTRACT FROM AUTHOR]

We investigate the fractional stochastic heat equation, driven by a random noise which admits a covariance measure structure with respect to the time variable and has a spatial covariance given by the Riesz kernel. This class of process includes White-colored noise, fractional colored noise and other related processes. We give a sufficient condition for the existence of the mild solution and we establish some properties of its. Then, we study the self similarity and the path regularity of this solution with respect to time variable on the particular case when the noise behaves as a fractional Brownian motion in time. [ABSTRACT FROM AUTHOR]

This work is concerned with limit behavior of the Navier–Stokes–Voigt equation with degenerate white noise in a 3D thin domain. Although the individual solutions may be chaotic in fluid dynamics, the stationary statistical solutions are essential to capture complex dynamical behaviors in the view of statistic. We therefore prove that the stationary statistical solution of the system converges weakly to the counterpart of the 2D stochastic Euler equation as the viscosity, the elasticity parameter and the thickness of the thin domain tend to zero. [ABSTRACT FROM AUTHOR]

We consider multiple stochastic integrals with respect to càdlàg martingales, which approximate a cylindrical Wiener process. We define a chaos expansion, analogous to the case of multiple Wiener stochastic integrals, for these integrals and use it to show moment bounds. Key tools include an iteration of the Burkholder–Davis–Gundy inequality and a multi-scale decomposition similar to the one developed in Hairer and Quastel (Forum Math Pi 6:e3, 2018). Our method can be combined with the recently developed discretisation framework for regularity structures (Hairer and Matetski in Ann Probab 46(3):1651–1709, 2018, Erhard and Hairer in Ann Inst Henri Poincaré Probab Stat 55(4):2209–2248, 2019) to prove convergence of interacting particle systems to singular stochastic PDEs. A companion article (Grazieschiet al. in The dynamical Ising–Kac model in 3D converges to Φ 3 4 , 2023. arXiv:2303.10242) applies the results of this paper to prove convergence of a rescaled Glauber dynamics for the three-dimensional Ising–Kac model near criticality to the Φ 3 4 dynamics on a torus. [ABSTRACT FROM AUTHOR]

Considering the effects of spatial diffusion and environmental noise, this paper establishes a stochastic reaction–diffusion tuberculosis (TB) model that includes varying immunity period. Compared to the classical TB models, the TB models driven by stochastic partial differential equations are more versatile. Mathematically, the boundness of moment on the solutions is given. Furthermore, the regularity of an invariant measure and the unique geometric ergodicity of model are proved by the generalized coupling method. Some numerical simulations are also displayed, which demonstrate the effectiveness of theories. [ABSTRACT FROM AUTHOR]

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 D ⊂ 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 stability 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 subset of L 2 , which leads to the existence of an ergodic invariant measure. [ABSTRACT FROM AUTHOR]

The stochastic nonlinear Kodama (SNLK) equation forced in the Stratonovich sense by multiplicative noise is considered here. New elliptic, hyperbolic, trigonometric, and rational stochastic solutions are acquired using (G ′/ G)-expansion method and mapping method. Because the SNLK equation is extensively used in is extensively used in nonlinear optics, fluid dynamics, and plasma physics, the obtained solutions may be used to study a broad range of relevant physical phenomena. In order to interpret the effects of multiplicative noise, the dynamic performances of the various obtained solutions are displayed utilizing 3D and 2D graphs. We infer that multiplicative noise impacts and stabilizes the solutions of SNLK equation. [ABSTRACT FROM AUTHOR]

In this article we investigate the blowup behavior of semilinear stochastic partial differential equations of the prototype d u (t , x) = - (- Δ) α / 2 u (t , x) - q (x) u (t , x) + γ u (t , x) + G (u (t , x)) d t + κ u (t , x) d Z t on the space R d , where α ∈ (0 , 2 ] , γ , κ ∈ R , q is a nonnegative locally bounded function such that q (x) → ∞ as | x | → ∞ , G is a locally Lipschitz continuous function, and Z is a real-valued centered Gaussian process with Hölder continuous paths with exponent θ > 1 / 3 . We show that there exists a random maximal interval [ 0 , τ) where the solution exists and is unique. Moreover, we show that the solution explodes in L ∞ -norm on the event { τ < ∞ } . For the case where G(x) is of the form C | x | 1 + β for some constants β , C > 0 , we obtain estimates for the blowup probability, the blowup time, and the blowup rate of the solution. These bounds are given in terms of the exponential functional of Z. [ABSTRACT FROM AUTHOR]

In this paper, we prove that stochastic porous media equations over σ -finite measure spaces (E , B , μ) , driven by time-dependent multiplicative noise, with the Laplacian replaced by a self-adjoint transient Dirichlet operator L and the diffusivity function given by a maximal monotone multi-valued function Ψ of polynomial growth, have a unique solution. This generalizes previous results in that we work on general measurable state spaces, allow non-continuous monotone functions Ψ , for which, no further assumptions (as e.g. coercivity) are needed, but only that their multi-valued extensions are maximal monotone and of at most polynomial growth. Furthermore, an L p (μ) -Itô formula in expectation is proved, which is not only crucial for the proof of our main result, but also of independent interest. The result in particular applies to fast diffusion stochastic porous media equations (in particular self-organized criticality models) and cases where E is a manifold or a fractal, and to non-local operators L, as e.g. L = - f (- Δ) , where f is a Bernstein function. [ABSTRACT FROM AUTHOR]

We establish a framework for the existence and uniqueness of solutions to stochastic nonlinear (possibly multi-valued) diffusion equations driven by multiplicative noise, with the drift operator L being the generator of a transient Dirichlet form on a finite measure space (E , B , μ) and the initial value in F e ∗ , which is the dual space of an extended transient Dirichlet space. L and F e ∗ replace the Laplace operator Δ and H - 1 , respectively, in the classical case. This framework includes stochastic fast diffusion equations, stochastic fractional fast diffusion equations, the Zhang model, and applies to cases with E being a manifold, a fractal, or a graph. In addition, our results apply to operators - f (- L) , where f is a Bernstein function, e.g., f (λ) = λ α or f (λ) = (λ + 1) α - 1 , 0 < α < 1 . [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the stochastic Landau–Lifshitz–Slonczewski (LLS) equation that describes magnetisation of an infinite nanowire evolving under current-driven spin torque. The current brings into the system a multiplicative gradient noise that appears as a transport term in the equation. We prove the existence, uniqueness and regularity of pathwise solutions to the equation. [ABSTRACT FROM AUTHOR]

In this work we establish weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise, in particular, for the hyperbolic Anderson model. For this class of stochastic partial differential equations the weak convergence rates we obtain are indeed twice the known strong rates. To the best of our knowledge, our findings are the first in the scientific literature which provide essentially sharp weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise. Key ideas of our proof are a sophisticated splitting of the error and applications of the recently introduced mild Itô formula. We complement our analytical findings by means of numerical simulations in Python for the decay of the weak approximation error for SPDEs for four different test functions. [ABSTRACT FROM AUTHOR]

We study Laplace equation and heat equation with the Gaussian white noise as the boundary/initial condition. Using the stochastic pre-orthogonal adaptive Fourier decomposition on the Bergman space, we give sparse expansions of the solutions of the boundary/initial value problems. The proposed methodology breaks away from the H - H K structure. The key point to the solution is to use the Wiener isometry property of the Gaussian white noise integral and the corresponding Green's function to represent the covariance of the solution. For the heat equation with the Gaussian white noise initial value, we define a class of parabolic Bergman space following E. Stein, prove existence of Bergman kernel, and obtain its expression, and further verify the boundary vanishing property needed for generating a sparse expansion. In order to make the heat flow satisfy a long-time decay property, we induce a subspace of the parabolic Bergman space, which corresponds to a class of homogeneous Sobolev spaces. [ABSTRACT FROM AUTHOR]

We prove the null controllability of forward and backward linear stochastic parabolic equations with first and zero order coupling terms, and with bounded coefficients. The null controllability results rely on novel Carleman estimates for the backward and forward adjoint equations, established through the application of the duality technique. In doing so, we resolve an open question (see Remark 2.2 in [Tang and Zhang in SIAM J Control Optim 48:2191–2216, 2009]). Moreover, we provide more accurate estimates for the null-control costs. [ABSTRACT FROM AUTHOR]

Loosely speaking, the Navier–Stokes- α model and the Navier–Stokes equations differ by a spatial filtration parametrized by a scale denoted α . Starting from a strong two-dimensional solution to the Navier–Stokes- α model driven by a multiplicative noise, we demonstrate that it generates a strong solution to the stochastic Navier–Stokes equations under the condition α → 0 . The initially introduced probability space and the Wiener process are maintained throughout the investigation, thanks to a local monotonicity property that abolishes the use of Skorokhod’s theorem. High spatial regularity a priori estimates for the fluid velocity vector field are carried out within periodic boundary conditions. [ABSTRACT FROM AUTHOR]

We consider the stochastic Cahn-Hilliard-Brinkman system with regular potential in a bounded domain D⊂ℝ3 driven by a multiplicative noise. Since the noise term and coupling terms create some difficulties in the process of showing the existence of weak solutions, we will first prove the existence of weak solutions by the Faedo-Galerkin methods and stochastic compactness method when the initial data satisfies some "regular" condition. For the case of general initial data, we will establish the existence of weak solutions by taking a sequence of "regular" initial data and proving the convergence in probability as well as some weak convergence of the corresponding solution sequences. [ABSTRACT FROM AUTHOR]

The primary focus of this article is to explore the existence and controllability of a new class of Sobolev-type Hilfer fractional stochastic Volterra-Fredholm integrodifferential systems in Hilbert spaces. Our main findings are grounded in ideas from fractional calculus, stochastic analysis, cosine families, and multivalued maps. To begin, we use Bohnenblust-Karlin's fixed point theorem to examine the existence of mild solutions for the given problem. Furthermore, we introduce a novel set of sufficient conditions for the considered nonlinear Hilfer fractional stochastic differential systems, and assuming that the corresponding linear system is approximately controllable. Finally, an example is provided to illustrate the proposed theory. [ABSTRACT FROM AUTHOR]

In this article, we investigate the precise asymptotics for complete convergence and complete moment convergence for the spatial averages of the solution to the stochastic heat equation over a Euclidean ball as the radius of the ball diverges to infinity. Some general results on precise asymptotics are obtained, which can describe the relations among the boundary function, weighted function, convergence rate, and limit value. [ABSTRACT FROM AUTHOR]

This paper discusses the fractional stochastic neutral evolution equation driven by a fractional Brownian motion. The existence of square-mean S-asymptotically periodic mild solutions for the associated stochastic evolution equation driven by a fractional Brownian motion is established. Some new existence results of mean square S-asymptotically periodic mild solutions for neutral fractional stochastic evolution equation are showed by means of the operator semigroup theory, stochastic analysis and fixed point theorems. [ABSTRACT FROM AUTHOR]

In the present paper we deal with stochastic semilinear partial differential equations L u = γ (u) + σ (u) Ξ ˙ of parabolic type with (t, x)-depending coefficients which may admit a polynomial growth with respect to the space variable. Under suitable assumptions on the coefficients of the parabolic operator L, on the initial data and on the stochastic noise Ξ (more precisely, on the spectral measure M associated with Ξ ) we prove existence of a unique (mild) function-valued solution for the associated Cauchy problem. [ABSTRACT FROM AUTHOR]