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Abstract: Abstract We prove that some time Euler schemes for the 3D Navier–Stokes equations modified by adding a Brinkman–Forchheimer term and a random perturbation converge in \(L^2(\varOmega )\) . This extends previous results concerning the strong rate of convergence of some time discretization schemes for the 2D Navier Stokes equations. Unlike the 2D case, our proposed 3D model with the Brinkman–Forchheimer term allows for a strong rate of convergence of order almost 1/2, that is independent of the viscosity parameter. PubDate: 2022-05-10

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Abstract: Abstract We introduce the uniqueness, existence, \(L_p\) -regularity, and maximal Hölder regularity of the solution to semilinear stochastic partial differential equation driven by a multiplicative space-time white noise: $$\begin{aligned} u_t = au_{xx} + bu_{x} + cu + {{\bar{b}}} u ^\lambda u_{x} + \sigma (u)\dot{W},\quad (t,x)\in (0,\infty )\times {\mathbb {R}}; \quad u(0,\cdot ) = u_0, \end{aligned}$$ where \(\lambda > 0\) . The function \(\sigma (u)\) is either bounded Lipschitz or super-linear in u. The noise \(\dot{W}\) is a space-time white noise. The coefficients a, b, c depend on \((\omega ,t,x)\) , and \({{\bar{b}}}\) depends on \((\omega ,t)\) . The coefficients \(a,b,c,{\bar{b}}\) are uniformly bounded, and a satisfies ellipticity condition. The random initial data \(u_0 = u_0(\omega ,x)\) is nonnegative. To establish the \(L_p\) -regularity theory, we impose an algebraic condition on \(\lambda \) depending on the nonlinearity of the diffusion coefficient \(\sigma (u)\) . For example, if \(\sigma (u)\) has Lipschitz continuity, linear growth, and boundedness in u, \(\lambda \) is assumed to be less than or equal to 1; \(\lambda \in (0,1]\) . However, if \(\sigma (u) = u ^{1+\lambda _0}\) with \(\lambda _0\in [0,1/2)\) , \(\lambda \) is taken to be less than 1; \(\lambda \in (0,1)\) . Under those conditions, the uniqueness, existence, and regularity of the solution are obtained in stochastic \(L_p\) spaces. Also, we have the maximal Hölder regularity by employing the Hölder embedding theorem. For example, if \(\lambda \in (0,1]\) and \(\sigma (u)\) has Lipschitz continuity, linear growth, and boundedness in u, for \(T<\infty \) and \(\varepsilon >0\) , PubDate: 2022-05-05

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Abstract: Abstract We consider the continuous Anderson Hamiltonian with white noise potential on \((-L/2,L/2)^d\) in dimension \(d\le 3\) , and derive the asymptotic of the smallest eigenvalues when L goes to infinity. We show that these eigenvalues go to \(-\infty \) at speed \((\log L)^{1/(2-d/2)}\) and identify the prefactor in terms of the optimal constant of the Gagliardo–Nirenberg inequality. This result was already known in dimensions 1 and 2, but appears to be new in dimension 3. We present some conjectures on the fluctuations of the eigenvalues and on the asymptotic shape of the corresponding eigenfunctions near their localisation centers. PubDate: 2022-04-28

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Abstract: Abstract Strong convergence rates for fuly discrete numerical approximations of space-time white noise driven SPDEs with superlinearly growing nonlinearities, such as the stochastic Allen–Cahn equation with space-time white noise, are shown. The obtained strong rates of convergence are essentially sharp. PubDate: 2022-04-27

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Abstract: Abstract We investigate the quality of space approximation of a class of stochastic integral equations of convolution type with Gaussian noise. Such equations arise, for example, when considering mild solutions of stochastic fractional order partial differential equations but also when considering mild solutions of classical stochastic partial differential equations. The key requirement for the equations is a smoothing property of the deterministic evolution operator which is typical in parabolic type problems. We show that if one has access to nonsmooth data estimates for the deterministic error operator together with its derivative of a space discretization procedure, then one obtains error estimates in pathwise Hölder norms with rates that can be read off the deterministic error rates. We illustrate the main result by considering a class of stochastic fractional order partial differential equations and space approximations performed by spectral Galerkin methods and finite elements. We also improve an existing result on the stochastic heat equation. PubDate: 2022-04-26

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Abstract: Abstract Let \({{\mathcal {X}}}\) be a real separable Hilbert space. Let C be a linear, bounded, non-negative self-adjoint operator on \({{\mathcal {X}}}\) and let A be the infinitesimal generator of a strongly continuous semigroup in \({{\mathcal {X}}}\) . Let \(\{W(t)\}_{t\ge 0}\) be a \({{\mathcal {X}}}\) -valued cylindrical Wiener process on a filtered (normal) probability space \((\Omega ,{\mathcal {F}},\{{\mathcal {F}}_t\}_{t\ge 0},{\mathbb {P}})\) . Let \(F:{{\text {Dom}}}(F)\subseteq {{\mathcal {X}}}\rightarrow {{\mathcal {X}}}\) be a smooth enough function. We are interested in the generalized mild solution \({\lbrace X(t,x)\rbrace _{t\ge 0}}\) of the semilinear stochastic partial differential equation $$\begin{aligned} {\left\{ \begin{array}{ll} dX(t,x)=\big (AX(t,x)+F(X(t,x))\big )dt+ \sqrt{C}dW(t), &{} t>0;\\ X(0,x)=x\in {{\mathcal {X}}}. \end{array}\right. } \end{aligned}$$ We consider the transition semigroup defined by $$\begin{aligned} P(t)\varphi (x):={{\mathbb {E}}}[\varphi (X(t,x))], \qquad \varphi \in B_b({{\mathcal {X}}}),\ t\ge 0,\ x\in {{\mathcal {X}}}. \end{aligned}$$ If \({{\mathcal {O}}}\) is an open set of \({{\mathcal {X}}}\) , we consider the Dirichlet semigroup defined by $$\begin{aligned} P^{{\mathcal {O}}}(t)\varphi (x):={\mathbb {E}}\left[ \varphi (X(t,x)){\mathbb {I}}_{\{\omega \in \Omega \; :\;\tau _x(\omega )> t\}}\right] ,\quad \varphi \in B_b({\mathcal {O}}),\; x\in {{\mathcal {O}}},\; t>0 \end{aligned}$$ where \(\tau _x\) is the exit time defined by $$\begin{aligned} \tau _x=\inf \{ s> 0\; : \; X(s,x)\in {{\mathcal {O}}}^c \}. \end{aligned}$$ We study the infinitesimal generator of P(t), \(P^{{\mathcal {O}}}(t)\) in \(L^2({{\mathcal {X}}},\nu )\) , \(L^2({{\mathcal {O}}},\nu )\) respectively, where \(\nu \) is the unique invariant measure of P(t). PubDate: 2022-04-23

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Abstract: Abstract We develop further in this work the high order paracontrolled calculus setting to deal with the analytic part of the study of quasilinear singular PDEs. Continuity results for a number of operators are proved for that purpose. Unlike the regularity structures approach of the subject by Gerencser & Hairer and Otto, Sauer, Smith & Weber, or Furlan & Gubinelli’ study of the two dimensional quasilinear parabolic Anderson model equation, we do not use parametrised families of models or paraproducts to set the scene. We use instead infinite dimensional paracontrolled structures that we introduce here. PubDate: 2022-04-16

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Abstract: Abstract Additive noise in Partial Differential equations, in particular those of fluid mechanics, has relatively natural motivations. The aim of this work is showing that suitable multiscale arguments lead rigorously, from a model of fluid with additive noise, to transport type noise. The arguments apply both to small-scale random perturbations of the fluid acting on a large-scale passive scalar and to the action of the former on the large scales of the fluid itself. Our approach consists in studying the (stochastic) characteristics associated to small-scale random perturbations of the fluid, here modelled by stochastic 2D Euler equations with additive noise, and their convergence in the infinite scale separation limit. PubDate: 2022-04-15

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Abstract: Abstract We study the stochastic cubic nonlinear wave equation (SNLW) with an additive noise on the three-dimensional torus \({\mathbb {T}}^3\) . In particular, we prove local well-posedness of the (renormalized) SNLW when the noise is almost a space-time white noise. In recent years, the paracontrolled calculus has played a crucial role in the well-posedness study of singular SNLW on \({\mathbb {T}}^3\) by Gubinelli et al. (Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity, 2018, arXiv:1811.07808 [math.AP]), Oh et al. (Focusing \(\Phi ^4_3\) -model with a Hartree-type nonlinearity, 2020. arXiv:2009.03251 [math.PR]), and Bringmann (Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics, 2020, arXiv:2009.04616 [math.AP]). Our approach, however, does not rely on the paracontrolled calculus. We instead proceed with the second order expansion and study the resulting equation for the residual term, using multilinear dispersive smoothing. PubDate: 2022-04-13

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Abstract: Abstract We consider NLS on \(\mathbb {T}^2\) with multiplicative spatial white noise and nonlinearity between cubic and quartic. We prove global existence, uniqueness and convergence almost surely of solutions to a family of properly regularized and renormalized approximating equations. In particular we extend a previous result by A. Debussche and H. Weber available in the cubic and sub-cubic setting. PubDate: 2022-04-09

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Abstract: Abstract This work introduces and studies a new family of velocity jump Markov processes directly amenable to exact simulation with the following two properties: (i) trajectories converge in law, when a time-step parameter vanishes, towards a given Langevin or Hamiltonian dynamics; (ii) the stationary distribution of the process is always exactly given by the product of a Gaussian (for velocities) by any target log-density. The simulation itself, in addition to the computability of the gradient of the log-density, depends on the knowledge of appropriate explicit upper bounds on lower order derivatives of this log-density. The process does not exhibit any velocity reflections (maximum size of jumps can be controlled) and is suitable for the ’factorization method’. We provide rigorous mathematical proofs of the convergence towards Hamiltonian/Langevin dynamics when the time step vanishes, and of the exponentially fast convergence towards the target distribution when a suitable noise on velocities is present. Numerical implementation is detailed and illustrated. PubDate: 2022-03-30

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Abstract: Abstract We study the asymptotic behavior of stochastic hyperbolic–parabolic equations with slow–fast time scales. Both the strong and weak convergence in the averaging principle are established. Then we study the stochastic fluctuations of the original system around its averaged equation. We show that the normalized difference converges weakly to the solution of a linear stochastic wave equation. An extra diffusion term appears in the limit which is given explicitly in terms of the solution of a Poisson equation. Furthermore, sharp rates for the above convergence are obtained, which are shown not to depend on the regularity of the coefficients in the equation for the fast variable. PubDate: 2022-03-25

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Abstract: Abstract In this article, we develop a martingale approach to localized Bismut-type Hessian formulas for heat semigroups on Riemannian manifolds. Our approach extends the Hessian formulas established by Stroock (An estimate on the Hessian of the heat kernel: 355–371, 1996) and removes in particular the compact manifold restriction. To demonstrate the potential of these formulas, we give as application explicit quantitative local estimates for the Hessian of the heat semigroup, as well as for harmonic functions on regular domains in Riemannian manifolds. PubDate: 2022-03-23

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Abstract: Abstract The BBM equation is a Hamiltonian PDE which revealed to be a very interesting test-model to study the transformation property of Gaussian measures along the flow, after Tzvetkov (Sigma 3:e28-35). In this paper we study the BBM equation with critical dispersion (which is a Benjamin-Ono type model). We prove that the image of the Gaussian measures supported on fractional Sobolev spaces of increasing regularity are absolutely continuous, but we cannot identify the density, for which new ideas are needed. PubDate: 2022-03-20

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Abstract: Abstract To model subsurface flow in uncertain heterogeneous or fractured media an elliptic equation with a discontinuous stochastic diffusion coefficient—also called random field—may be used. In case of a one-dimensional parameter space, Lévy processes allow for jumps and display great flexibility in the distributions used. However, in various situations (e.g. microstructure modeling), a one-dimensional parameter space is not sufficient. Classical extensions of Lévy processes on two parameter dimensions suffer from the fact that they do not allow for spatial discontinuities [see for example Barth and Stein (Stoch Part Differ Equ Anal Comput 6(2):286–334, 2018)]. In this paper a new subordination approach is employed [see also Barth and Merkle (Subordinated gaussian random fields. ArXiv e-prints, arXiv:2012.06353 [math.PR], 2020)] to generate Lévy-type discontinuous random fields on a two-dimensional spatial parameter domain. Existence and uniqueness of a (pathwise) solution to a general elliptic partial differential equation is proved and an approximation theory for the diffusion coefficient and the corresponding solution provided. Further, numerical examples using a Monte Carlo approach on a Finite Element discretization validate our theoretical results. PubDate: 2022-03-18

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Abstract: Abstract We introduce a new class of stochastic partial differential equations (SPDEs) with seed bank modeling the spread of a beneficial allele in a spatial population where individuals may switch between an active and a dormant state. Incorporating dormancy and the resulting seed bank leads to a two-type coupled system of equations with migration between both states. We first discuss existence and uniqueness of seed bank SPDEs and provide an equivalent delay representation that allows a clear interpretation of the age structure in the seed bank component. The delay representation will also be crucial in the proofs. Further, we show that the seed bank SPDEs give rise to an interesting class of “on/off”-moment duals. In particular, in the special case of the F-KPP Equation with seed bank, the moment dual is given by an “on/off-branching Brownian motion”. This system differs from a classical branching Brownian motion in the sense that independently for all individuals, motion and branching may be “switched off” for an exponential amount of time after which they get “switched on” again. On/off branching Brownian motion shows qualitatively different behaviour to classical branching Brownian motion and is an interesting object for study in itself. Here, as an application of our duality, we show that the spread of a beneficial allele, which in the classical F-KPP Equation, started from a Heaviside intial condition, evolves as a pulled traveling wave with speed \(\sqrt{2}\) , is slowed down significantly in the corresponding seed bank F-KPP model. In fact, by computing bounds on the position of the rightmost particle in the dual on/off branching Brownian motion, we obtain an upper bound for the speed of propagation of the beneficial allele given by \(\sqrt{\sqrt{5}-1}\approx 1.111\) under unit switching rates. This shows that seed banks will indeed slow down fitness waves and preserve genetic variability, in line with intuitive reasoning from population genetics and ecology. PubDate: 2022-03-14

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Abstract: Abstract We consider a stochastic partial differential equation (SPDE) model for chemorepulsion, with non-linear sensitivity on the one-dimensional torus. By establishing an a priori estimate independent of the initial data, we show that there exists a pathwise unique, global solution to the SPDE. Furthermore, we show that the associated semi-group is Markov and possesses a unique invariant measure, supported on a Hölder–Besov space of positive regularity, which the solution law converges to exponentially fast. The a priori bound also allows us to establish tail estimates on the \(L^p\) norm of the invariant measure which are heavier than Gaussian. PubDate: 2022-03-13

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Abstract: Abstract We consider singular quasilinear stochastic partial differential equations (SPDEs) studied in Funaki et al. (Ann Inst Henri Poincaré Probab Stat 57:1702-1735, 2021), which are defined in paracontrolled sense. The main aim of the present article is to establish the global-in-time solvability for a particular class of SPDEs with origin in particle systems and, under a certain additional condition on the noise, prove the convergence of the solutions to stationary solutions as \(t\rightarrow \infty \) . We apply the method of energy inequality and Poincaré inequality. It is essential that the Poincaré constant can be taken uniformly in an approximating sequence of the noise. We also use the continuity of the solutions in the enhanced noise, initial values and coefficients of the equation, which we prove in this article for general SPDEs discussed in Funaki et al. (Ann Inst Henri Poincaré Probab Stat 57:1702-1735, 2021) except that in the enhanced noise. Moreover, we apply the initial layer property of improving regularity of the solutions in a short time. PubDate: 2022-03-06

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Abstract: Abstract We show that perturbing ill-posed differential equations with (potentially very) smooth random processes can restore well-posedness—even if the perturbation is (potentially much) more regular than the drift component of the solution. The noise considered is of fractional Brownian type, and the familiar regularity condition \(\alpha >1-1/(2H)\) is recovered for all non-integer \(H>1\) . PubDate: 2022-03-04

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Abstract: Abstract We prove the pathwise uniqueness of solutions of the nonlinear Schrödinger equation with conservative multiplicative noise on compact 3D manifolds. In particular, we generalize the result by Burq, Gérard and Tzvetkov, [7], to the stochastic setting. The proof is based on the deterministic and new stochastic spectrally localized Strichartz estimates and the Littlewood-Paley decomposition. PubDate: 2022-03-02