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Weak error analysis for a nonlinear SPDE approximation of the Dean–Kawasaki equation Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2024-03-15 Ana Djurdjevac, Helena Kremp, Nicolas Perkowski
We consider a nonlinear SPDE approximation of the Dean–Kawasaki equation for independent particles. Our approximation satisfies the physical constraints of the particle system, i.e. its solution is a probability measure for all times (preservation of positivity and mass conservation). Using a duality argument, we prove that the weak error between particle system and nonlinear SPDE is of the order
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Convergence of stratified MCMC sampling of non-reversible dynamics Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2024-03-01 Gabriel Earle, Jonathan C. Mattingly
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Regularity of random elliptic operators with degenerate coefficients and applications to stochastic homogenization Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2024-02-27 Peter Bella, Michael Kniely
We consider degenerate elliptic equations of second order in divergence form with a symmetric random coefficient field a. Extending the work of Bella et al. (Ann Appl Probab 28(3):1379–1422, 2018), who established the large-scale \(C^{1,\alpha }\) regularity of a-harmonic functions in a degenerate situation, we provide stretched exponential moments for the minimal radius \(r_*\) describing the minimal
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Weak error analysis for the stochastic Allen–Cahn equation Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2024-02-22 Dominic Breit, Andreas Prohl
We prove strong rate resp. weak rate \({{\mathcal {O}}}(\tau )\) for a structure preserving temporal discretization (with \(\tau \) the step size) of the stochastic Allen–Cahn equation with additive resp. multiplicative colored noise in \(d=1,2,3\) dimensions. Direct variational arguments exploit the one-sided Lipschitz property of the cubic nonlinearity in the first setting to settle first order strong
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Higher order homogenization for random non-autonomous parabolic operators Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2024-02-13 Marina Kleptsyna, Andrey Piatnitski, Alexandre Popier
We consider Cauchy problem for a divergence form second order parabolic operator with rapidly oscillating coefficients that are periodic in spatial variables and random stationary ergodic in time. As was proved in Zhikov et al. (Mat Obshch 45:182–236, 1982) and Kleptsyna and Piatnitski (Homogenization and applications to material sciences. GAKUTO Internat Ser Math Sci Appl vol 9, pp 241–255. Gakkōtosho
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Hitting properties of generalized fractional kinetic equation with time-fractional noise Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-12-08 Derui Sheng, Tau Zhou
This paper investigates the hitting properties of a system of generalized fractional kinetic equations driven by Gaussian noise that is fractional in time and either white or colored in space. The considered model encompasses various examples such as the stochastic heat equation and the stochastic biharmonic heat equation. Under relatively general conditions, we derive the mean square modulus of continuity
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Importance sampling for stochastic reaction–diffusion equations in the moderate deviation regime Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-12-08 Ioannis Gasteratos, Michael Salins, Konstantinos Spiliopoulos
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Delayed blow-up and enhanced diffusion by transport noise for systems of reaction–diffusion equations Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-11-28 Antonio Agresti
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Sticky nonlinear SDEs and convergence of McKean–Vlasov equations without confinement Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-11-06 Alain Durmus, Andreas Eberle, Arnaud Guillin, Katharina Schuh
We develop a new approach to study the long time behaviour of solutions to nonlinear stochastic differential equations in the sense of McKean, as well as propagation of chaos for the corresponding mean-field particle system approximations. Our approach is based on a sticky coupling between two solutions to the equation. We show that the distance process between the two copies is dominated by a solution
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Regularity theory for a new class of fractional parabolic stochastic evolution equations Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-10-30 Kristin Kirchner, Joshua Willems
A new class of fractional-order parabolic stochastic evolution equations of the form \((\partial _t + A)^\gamma X(t) = {\dot{W}}^Q(t)\), \(t\in [0,T]\), \(\gamma \in (0,\infty )\), is introduced, where \(-A\) generates a \(C_0\)-semigroup on a separable Hilbert space H and the spatiotemporal driving noise \({\dot{W}}^Q\) is the formal time derivative of an H-valued cylindrical Q-Wiener process. Mild
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Global existence and non-uniqueness for the Cauchy problem associated to 3D Navier–Stokes equations perturbed by transport noise Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-10-30 Umberto Pappalettera
We show global existence and non-uniqueness of probabilistically strong, analytically weak solutions of the three-dimensional Navier–Stokes equations perturbed by Stratonovich transport noise. We can prescribe either: (i) any divergence-free, square integrable intial condition; or (ii) the kinetic energy of solutions up to a stopping time, which can be chosen arbitrarily large with high probability
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Norm inflation for a non-linear heat equation with gaussian initial conditions Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-10-15 Ilya Chevyrev
We consider a non-linear heat equation \(\partial _t u = \Delta u + B(u,Du)+P(u)\) posed on the d-dimensional torus, where P is a polynomial of degree at most 3 and B is a bilinear map that is not a total derivative. We show that, if the initial condition \(u_0\) is taken from a sequence of smooth Gaussian fields with a specified covariance, then u exhibits norm inflation with high probability. A consequence
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Local strong solutions to the stochastic third grade fluid equations with Navier boundary conditions Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-10-09 Yassine Tahraoui, Fernanda Cipriano
This work is devoted to the study of non-Newtonian fluids of grade three on two-dimensional and three-dimensional bounded domains, driven by a nonlinear multiplicative Wiener noise. More precisely, we establish the existence and uniqueness of the local (in time) solution, which corresponds to an addapted stochastic process with sample paths defined up to a certain positive stopping time, with values
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Multilevel Monte Carlo FEM for elliptic PDEs with Besov random tree priors Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-09-30 Christoph Schwab, Andreas Stein
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On partially observed jump diffusions II: the filtering density Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-10-03 Alexander Davie, Fabian Germ, István Gyöngy
A partially observed jump diffusion \(Z=(X_t,Y_t)_{t\in [0,T]}\) given by a stochastic differential equation driven by Wiener processes and Poisson martingale measures is considered when the coefficients of the equation satisfy appropriate Lipschitz and growth conditions. Under general conditions it is shown that the conditional density of the unobserved component \(X_t\) given the observations \((Y_s)_{s\in
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A support theorem for parabolic stochastic PDEs with nondegenerate Hölder diffusion coefficients Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-09-27 Yi Han
In this paper we work with parabolic SPDEs of the form $$\begin{aligned} \partial _t u(t,x)=\partial _x^2 u(t,x)+g(t,x,u)+\sigma (t,x,u)\dot{W}(t,x) \end{aligned}$$ with Neumann boundary conditions, where \(x\in [0,1]\), \(\dot{W}(t,x)\) is the space-time white noise on \((t,x)\in [0,\infty )\times [0,1]\), g is uniformly bounded, and the solution \(u\in \mathbb {R}\) is real valued. The diffusion
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Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-09-29 David Candil, Le Chen, Cheuk Yin Lee
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An $$L_q(L_p)$$ -theory for space-time non-local equations generated by Lévy processes with low intensity of small jumps Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-08-25 Jaehoon Kang, Daehan Park
We investigate an \(L_{q}(L_{p})\)-regularity (\(1
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On the wellposedness of periodic nonlinear Schrödinger equations with white noise dispersion Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-08-08 Gavin Stewart
We study the wellposedness of the periodic nonlinear Schrödinger equation with white noise dispersion and a power nonlinearity given by \(idu = \Delta u \circ dW_t + |u |^{p-1}u\;dt\). We develop Strichartz estimates for this equation, which we then use to prove almost sure global wellposedness of this equation with \(L^2\) initial data for nonlinearities with exponent \(1
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Absolute continuity of the solution to stochastic generalized Burgers–Huxley equation Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-07-31 Ankit Kumar, Manil T. Mohan
The present work deals with the global solvability as well as absolute continuity of the law of the solution to stochastic generalized Burgers–Huxley (SGBH) equation driven by multiplicative space-time white noise in a bounded interval of \({\mathbb {R}}\). We first prove the existence of a unique local mild solution to SGBH equation with the help of a truncation argument and contraction mapping principle
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Asymptotic stability of evolution systems of probability measures of stochastic discrete modified Swift–Hohenberg equations Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-07-25 Fengling Wang, Tomás Caraballo, Yangrong Li, Renhai Wang
This paper is concerned with the asymptotic stability of evolution systems of probability measures for non-autonomous stochastic discrete modified Swift–Hohenberg equations driven by local Lipschitz nonlinear noise. We first show the existence of evolution systems of probability measures of the original equation. Then, using the theoretical results in Wang et al. (Proc Am Math Soc 151:2449–2458, 2023)
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Moderate deviations for fully coupled multiscale weakly interacting particle systems Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-07-10 Z. W. Bezemek, K. Spiliopoulos
We consider a collection of fully coupled weakly interacting diffusion processes moving in a two-scale environment. We study the moderate deviations principle of the empirical distribution of the particles’ positions in the combined limit as the number of particles grow to infinity and the time-scale separation parameter goes to zero simultaneously. We make use of weak convergence methods, which provide
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Existence and uniqueness of maximal solutions to SPDEs with applications to viscous fluid equations Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-07-06 Daniel Goodair, Dan Crisan, Oana Lang
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Central limit theorems for nonlinear stochastic wave equations in dimension three Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-07-01 Masahisa Ebina
In this paper, we consider three-dimensional nonlinear stochastic wave equations driven by the Gaussian noise which is white in time and has some spatial correlations. Using the Malliavin–Stein’s method, we prove the Gaussian fluctuation for the spatial average of the solution under the Wasserstein distance in the cases where the spatial correlation is given by an integrable function and by the Riesz
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Inviscid limit for stochastic second-grade fluid equations Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-06-29 Eliseo Luongo
We consider in a smooth bounded and simply connected two dimensional domain the convergence in the \(L^2\) norm, uniformly in time, of the solution of the stochastic second-grade fluid equations with transport noise and no-slip boundary conditions to the solution of the corresponding Euler equations. We prove, that assuming proper regularity of the initial conditions of the Euler equations and a proper
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Path continuity of Markov processes and locality of Kolmogorov operators Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-06-29 Lucian Beznea, Iulian Cîmpean, Michael Röckner
We prove that if we are given a generator of a right Markov process with càdlàg paths and an open domain G in the state space, on which the generator has the local property expressed in a suitable way on a class \({\mathcal {C}}\) of test functions that is sufficiently rich, then the Markov process has continuous paths when it passes through G. The result holds for any Markov process which is associated
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Statistical solutions for the Navier–Stokes–Fourier system Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-06-21 Eduard Feireisl, Mária Lukáčová-Medvid’ová
We show a general stability result in the framework of strong solutions of the Navier–Stokes–Fourier system describing the motion of a compressible viscous and heat conducting gas. As a corollary, we develop a concept of statistical solution in the class of regular solutions “beyond the blow up time”.
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Hypocoercivity for non-linear infinite-dimensional degenerate stochastic differential equations Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-06-10 Benedikt Eisenhuth, Martin Grothaus
The aim of this article is to construct solutions to second order in time stochastic partial differential equations and to show hypocoercivity of the corresponding transition semigroups. More generally, we analyze non-linear infinite-dimensional degenerate stochastic differential equations in terms of their infinitesimal generators. In the first part of this article we use resolvent methods developed
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Strong Feller semigroups and Markov processes: a counterexample Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-06-07 Lucian Beznea, Iulian Cîmpean, Michael Röckner
The aim of this note is to show, by providing an elementary way to construct counterexamples, that the strong Feller and the joint (space-time) continuity for a semigroup of Markov kernels on a Polish space are not enough to ensure the existence of an associated càdlàg Markov process on the same space. One such simple counterexample is the Brownian semigroup on \({\mathbb {R}}\) restricted to \({\mathbb
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Renormalization of stochastic nonlinear heat and wave equations driven by subordinate cylindrical Brownian noises Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-04-28 Hirotatsu Nagoji
In this paper, we study the stochastic nonlinear heat equations (SNLH) and stochastic nonlinear wave equations (SNLW) on two-dimensional torus \({\mathbb {T}}^2 = ({\mathbb {R}}/2\pi {\mathbb {Z}})^2\) driven by a subordinate cylindrical Brownian noise, which we define by the time-derivative of a cylindrical Brownian motion subordinated to a nondecreasing càdlàg stochastic process. To construct the
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Global well-posedness of the two-dimensional stochastic viscous nonlinear wave equations Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-04-27 Ruoyuan Liu
We study well-posedness of viscous nonlinear wave equations (vNLW) on the two-dimensional torus with a stochastic forcing. In particular, we prove pathwise global well-posedness of the stochastic defocusing vNLW with an additive stochastic forcing \(D^\alpha \xi \), where \(\alpha < \frac{1}{2}\) and \(\xi \) denotes the space–time white noise.
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Non-linear Young equations in the plane and pathwise regularization by noise for the stochastic wave equation Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-04-26 Florian Bechtold, Fabian A. Harang, Nimit Rana
We study pathwise regularization by noise for equations on the plane in the spirit of the framework outlined by Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016). To this end, we extend the notion of non-linear Young equations to a two dimensional domain and prove existence and uniqueness of such equations. This concept is then used in order to prove regularization by noise for stochastic
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Non-uniqueness in law of three-dimensional Navier–Stokes equations diffused via a fractional Laplacian with power less than one half Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-03-31 Kazuo Yamazaki
Non-uniqueness of three-dimensional Euler equations and Navier-Stokes equations forced by random noise, path-wise and more recently even in law, have been proven by various authors. We prove non-uniqueness in law of the three-dimensional Navier–Stokes equations forced by random noise and diffused via a fractional Laplacian that has power between zero and one half. The solution we construct has Hölder
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LDP and CLT for SPDEs with transport noise Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-03-28 Lucio Galeati, Dejun Luo
In this work we consider solutions to stochastic partial differential equations with transport noise, which are known to converge, in a suitable scaling limit, to solution of the corresponding deterministic PDE with an additional viscosity term. Large deviations and Gaussian fluctuations underlying such scaling limit are investigated in two cases of interest: stochastic linear transport equations in
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A branching particle system approximation for solving partially observed stochastic optimal control problems via stochastic maximum principle Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-03-24 Hexiang Wan, Guangchen Wang, Jie Xiong
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Well-posedness for a stochastic Camassa–Holm type equation with higher order nonlinearities Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-03-21 Yingting Miao, Christian Rohde, Hao Tang
This paper aims at studying a generalized Camassa–Holm equation under random perturbation. We establish a local well-posedness result in the sense of Hadamard, i.e., existence, uniqueness and continuous dependence on initial data, as well as blow-up criteria for pathwise solutions in the Sobolev spaces \(H^s\) with \(s>3/2\) for \(x\in \mathbb R\). The analysis on continuous dependence on initial data
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Analysis of fully discrete mixed finite element scheme for stochastic Navier–Stokes equations with multiplicative noise Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-02-18 Hailong Qiu
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Global martingale solutions for stochastic Shigesada–Kawasaki–Teramoto population models Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-02-14 Marcel Braukhoff, Florian Huber, Ansgar Jüngel
The existence of global nonnegative martingale solutions to cross-diffusion systems of Shigesada–Kawasaki–Teramoto type with multiplicative noise is proven. The model describes the stochastic segregation dynamics of an arbitrary number of population species in a bounded domain with no-flux boundary conditions. The diffusion matrix is generally neither symmetric nor positive semidefinite, which excludes
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Well posedness and stochastic derivation of a diffusion-growth-fragmentation equation in a chemostat Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-02-13 Josué Tchouanti
We study the existence and uniqueness of the solution of a non-linear coupled system constituted of a degenerate diffusion-growth-fragmentation equation and a differential equation, resulting from the modeling of bacterial growth in a chemostat. This system is derived, in a large population approximation, from a stochastic individual-based model where each individual is characterized by a non-negative
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Invariant measures and global well-posedness for a fractional Schrödinger equation with Moser-Trudinger type nonlinearity Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-02-06 Jean-Baptiste Casteras, Léonard Monsaingeon
In this paper, we construct invariant measures and global-in-time solutions for a fractional Schrödinger equation with a Moser–Trudinger type nonlinearity $$\begin{aligned} i\partial _t u= (-\Delta )^{\alpha }u+ 2\beta u e^{\beta |u|^2} \qquad \text{ for }\qquad (x,t)\in \ M\times \mathbb {R}\end{aligned}$$(E) on a compact Riemannian manifold M without boundary of dimension \(d\ge 2\). To do so, we
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A maximal $$L_p$$ -regularity theory to initial value problems with time measurable nonlocal operators generated by additive processes Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-01-30 Jae-Hwan Choi, Ildoo Kim
Let \(Z=(Z_t)_{t\ge 0}\) be an additive process with a bounded triplet \((0,0,\varLambda _t)_{t\ge 0}\). Then the infinitesimal generators of Z is given by time dependent nonlocal operators as follows: $$\begin{aligned} {\mathcal {A}}_Z(t)u(t,x)&=\lim _{h \downarrow 0}\frac{{\mathbb {E}}[u(t,x+Z_{t+h}-Z_t)-u(t,x)]}{h} \\&=\int _{{\mathbb {R}}^d}(u(t,x+y)-u(t,x)-y\cdot \nabla _x u(t,x)1_{|y|\le 1})\varLambda
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Statistical analysis of discretely sampled semilinear SPDEs: a power variation approach Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-01-16 Igor Cialenco, Hyun-Jung Kim, Gregor Pasemann
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Degenerate Kolmogorov equations and ergodicity for the stochastic Allen–Cahn equation with logarithmic potential Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-01-11 Luca Scarpa, Margherita Zanella
Well-posedness à la Friedrichs is proved for a class of degenerate Kolmogorov equations associated to stochastic Allen–Cahn equations with logarithmic potential. The thermodynamical consistency of the model requires the potential to be singular and the multiplicative noise coefficient to vanish at the respective potential barriers, making thus the corresponding Kolmogorov equation not uniformly elliptic
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Stationary stochastic Navier–Stokes on the plane at and above criticality Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2023-01-11 G. Cannizzaro, J. Kiedrowski
In the present paper, we study the fractional incompressible Stochastic Navier–Stokes equation on \({\mathbb {R}}^2\), formally defined as $$\begin{aligned} \partial _t v = -\tfrac{1}{2} (-\Delta )^\theta v - \lambda v \cdot \nabla v + \nabla p + \nabla ^{\perp }(-\Delta )^{\frac{\theta -1}{2}} \xi , \qquad \nabla \cdot v = 0 \, , \end{aligned}$$(0.1) where \(\theta \in (0,1]\), \(\xi \) is the space-time
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Gaussian fluctuations of a nonlinear stochastic heat equation in dimension two Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2022-12-23 Ran Tao
We study the Gaussian fluctuations of a nonlinear stochastic heat equation in spatial dimension two. The equation is driven by a Gaussian multiplicative noise. The noise is white in time, smoothed in space at scale \(\varepsilon \), and tuned logarithmically by a factor \(\frac{1}{\sqrt{\log \varepsilon ^{-1}}}\) in its strength. We prove that, after centering and rescaling, the solution random field
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Stochastic evolution equations with Lévy noise in the dual of a nuclear space Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2022-11-22 C. A. Fonseca-Mora
In this article we give sufficient and necessary conditions for the existence of a weak and mild solution to stochastic evolution equations with (general) Lévy noise taking values in the dual of a nuclear space. As part of our approach we develop a theory of stochastic integration with respect to a Lévy process taking values in the dual of a nuclear space. We also derive further properties of the solution
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Trace theorem and non-zero boundary value problem for parabolic equations in weighted Sobolev spaces Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2022-11-18 Doyoon Kim, Kyeong-Hun Kim, Kwan Woo
We present weighted Sobolev spaces \(\widetilde{\mathfrak {H}}_{p, \theta }^{\gamma }(S, T)\) and prove a trace theorem for the spaces. As an application, we discuss non-zero boundary value problems for parabolic equations. The weighted parabolic Sobolev spaces we consider are designed, in particular, for the regularity theory of stochastic partial differential equations on bounded domains.
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The stochastic primitive equations with transport noise and turbulent pressure Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2022-10-27 Antonio Agresti, Matthias Hieber, Amru Hussein, Martin Saal
In this paper we consider the stochastic primitive equation for geophysical flows subject to transport noise and turbulent pressure. Admitting very rough noise terms, the global existence and uniqueness of solutions to this stochastic partial differential equation are proven using stochastic maximal \(L^2\)-regularity, the theory of critical spaces for stochastic evolution equations, and global a priori
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Matching upper and lower moment bounds for a large class of stochastic PDEs driven by general space-time Gaussian noises Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2022-10-21 Yaozhong Hu, Xiong Wang
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A derivative-free Milstein type approximation method for SPDEs covering the non-commutative noise case Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2022-10-04 Claudine von Hallern, Andreas Rößler
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Wellposedness and regularity estimates for stochastic Cahn–Hilliard equation with unbounded noise diffusion Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2022-09-15 Jianbo Cui, Jialin Hong
In this article, we consider the one dimensional stochastic Cahn–Hilliard equation driven by multiplicative space-time white noise with diffusion coefficient of sublinear growth. By introducing the spectral Galerkin method, we obtain the well-posedness of the approximated equation in finite dimension. Then with help of the semigroup theory and the factorization method, the approximation processes are
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Numerical approximation of nonlinear SPDE’s Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2022-09-12 Martin Ondreját, Andreas Prohl, Noel J. Walkington
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Existence of strong solutions for Itô’s stochastic equations via approximations: revisited Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2022-09-10 I. Gyöngy, N. V. Krylov
Given strong uniqueness for an Itô’s stochastic equation, we prove that its solution can be constructed on “any” probability space by using, for example, Euler’s polygonal approximations. Stochastic equations in \({\mathbb {R}}^{d}\) and in domains in \({\mathbb {R}}^{d}\) are considered. This is almost a copy of an old article in which we correct errors in the original proof of Lemma 4.1 found by
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Feynman–Kac formula for perturbations of order $$\le 1$$ ≤ 1 , and noncommutative geometry Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2022-08-29 Sebastian Boldt, Batu Güneysu
Let Q be a differential operator of order \(\le 1\) on a complex metric vector bundle \(\mathscr {E}\rightarrow \mathscr {M}\) with metric connection \(\nabla \) over a possibly noncompact Riemannian manifold \(\mathscr {M}\). Under very mild regularity assumptions on Q that guarantee that \(\nabla ^{\dagger }\nabla /2+Q\) canonically induces a holomorphic semigroup \(\mathrm {e}^{-zH^{\nabla }_{Q}}\)
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Total variation distance between a jump-equation and its Gaussian approximation Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2022-08-24 Vlad Bally, Yifeng Qin
We deal with stochastic differential equations with jumps. In order to obtain an accurate approximation scheme, it is usual to replace the “small jumps” by a Brownian motion. In this paper, we prove that for every fixed time t, the approximate random variable \(X^\varepsilon _t\) converges to the original random variable \(X_t\) in total variation distance and we estimate the error. We also give an
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Correction to: Convergent numerical approximation of the stochastic total variation flow Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2022-08-04 Ĺubomír Baňas, Michael Röckner, André Wilke
Abstract We correct two errors in our paper [4]. First error concerns the definition of the SVI solution, where a boundary term which arises due to the Dirichlet boundary condition, was not included. The second error concerns the discrete estimate [4, Lemma 4.4], which involves the discrete Laplace operator. We provide an alternative proof of the estimate in spatial dimension \(d=1\) by using a mass
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Conditional propagation of chaos in a spatial stochastic epidemic model with common noise Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2022-07-27 Yen V. Vuong, Maxime Hauray, Étienne Pardoux
We study a stochastic spatial epidemic model where the N individuals carry two features: a position and an infection state, interact and move in \({\mathbb {R}}^d\). In this Markovian model, the evolution of infection states are described with the help of the Poisson Point Processes , whereas the displacement of individuals are driven by mean field interactions, a (state dependence) diffusion and also
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Positive recurrence of a solution of an SDE with variable switching intensities Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2022-07-09 Alexander Veretennikov
Positive recurrence of a d-dimensional diffusion with an additive Wiener process, with switching and with one recurrent and one transient regimes and variable switching intensities is established under suitable conditions. The approach is based on embedded Markov chains.
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Local martingale solutions and pathwise uniqueness for the three-dimensional stochastic inviscid primitive equations Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2022-07-04 Ruimeng Hu, Quyuan Lin
We study the stochastic effect on the three-dimensional inviscid primitive equations (PEs, also called the hydrostatic Euler equations). Specifically, we consider a larger class of noises than multiplicative noises, and work in the analytic function space due to the ill-posedness in Sobolev spaces of PEs without horizontal viscosity. Under proper conditions, we prove the local existence of martingale
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Estimates in $$L_{p}$$ L p for solutions of SPDEs with coefficients in Morrey classes Stoch. PDE Anal. Comp. (IF 1.5) Pub Date : 2022-07-01 N. V. Krylov
For solutions of a certain class of SPDEs in divergence form we present some estimates of their \(L_{p}\)-norms and the \(L_{p}\)-norms of their first-order derivatives. The main novelty is that the low-order coefficients are supposed to belong to certain Morrey classes instead of \(L_{p}\)-spaces. Our results are new even if there are no stochastic terms in the equation.