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Abstract: Abstract 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 shown to possess many desirable properties. Further, we show that the approximation process is strongly convergent in a certain Banach space with an explicit algebraic convergence rate. Finally, the global existence and regularity estimates of the unique solution process are proven by means of the strong convergence of the approximation process, which fills a gap on the global existence of the mild solution for stochastic Cahn–Hilliard equation when the diffusion coefficient satisfies a growth condition of order \(\alpha \in (\frac{1}{3},1).\) PubDate: 2022-09-15

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Abstract: Abstract The numerical analysis of stochastic parabolic partial differential equations of the form $$\begin{aligned} du + A(u)\, dt = f \,dt + g \, dW, \end{aligned}$$ is surveyed, where A is a nonlinear partial operator and W a Brownian motion. This manuscript unifies much of the theory developed over the last decade into a cohesive framework which integrates techniques for the approximation of deterministic partial differential equations with methods for the approximation of stochastic ordinary differential equations. The manuscript is intended to be accessible to audiences versed in either of these disciplines, and examples are presented to illustrate the applicability of the theory. PubDate: 2022-09-12

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Abstract: Abstract 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 Martin Dieckmann in 2013. We present also a new result on the convergence of “tamed Euler approximations" for SDEs with locally unbounded drifts, which we achieve by proving an estimate for appropriate exponential moments. PubDate: 2022-09-10

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Abstract: Abstract 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 \(\mathrm {e}^{-zH^{\nabla }_{Q}}\) in \(\Gamma _{L^2}(\mathscr {M},\mathscr {E})\) (where z runs through a complex sector which contains \([0,\infty )\) ), we prove an explicit Feynman–Kac type formula for \(\mathrm {e}^{-tH^{\nabla }_{Q}}\) , \(t>0\) , generalizing the standard self-adjoint theory where Q is a self-adjoint zeroth order operator. For compact \(\mathscr {M}\) ’s we combine this formula with Berezin integration to derive a Feynman–Kac type formula for an operator trace of the form $$\begin{aligned} \mathrm {Tr}\left( \widetilde{V}\int ^t_0\mathrm {e}^{-sH^{\nabla }_{V}}P\mathrm {e}^{-(t-s)H^{\nabla }_{V}}\mathrm {d}s\right) , \end{aligned}$$ where \(V,\widetilde{V}\) are of zeroth order and P is of order \(\le 1\) . These formulae are then used to obtain a probabilistic representations of the lower order terms of the equivariant Chern character (a differential graded extension of the JLO-cocycle) of a compact even-dimensional Riemannian spin manifold, which in combination with cyclic homology play a crucial role in the context of the Duistermaat–Heckmann localization formula on the loop space of such a manifold. PubDate: 2022-08-29

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Abstract: Abstract 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 estimate of the distance between the densities of the laws of the two random variables. These are done by using some integration by parts techniques in Malliavin calculus. PubDate: 2022-08-24

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Abstract: 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 lumped version of the discrete Laplacian. Hence, after a minor modification of the fully discrete numerical scheme the convergence in \(d=1\) follows along the lines of the original proof. The convergence proof of the time semi-discrete scheme, which relies on the continuous counterpart of the estimate [4, Lemma 4.4], remains valid in higher spatial dimension. The convergence of the fully discrete finite element scheme from [4] in any spatial dimension is shown in [3] by using a different approach. PubDate: 2022-08-04

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Abstract: Abstract 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 a common noise, so that the spatial dynamic is a random process. We prove that when the number N of individual goes to infinity, the conditional propagation of chaos holds : conditionally to the common noise, the individuals are asymptotically independent and the stochastic dynamic converges to a “random” nonlinear McKean-Vlasov process. As a consequence, the associated empirical measure converges to a measure, which is solution of a stochastic mean-field PDE driven by the common noise. PubDate: 2022-07-27

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Abstract: Abstract 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. PubDate: 2022-07-09 DOI: 10.1007/s40072-022-00265-7

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Abstract: Abstract 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 solutions and pathwise uniqueness. By adding vertical viscosity, i.e., considering the hydrostatic Navier-Stokes equations, we can relax the restriction on initial conditions to be only analytic in the horizontal variables with Sobolev regularity in the vertical variable, and allow the transport noise in the vertical direction. We establish the local existence of martingale solutions and pathwise uniqueness, and show that the solutions become analytic in the vertical variable instantaneously as \(t>0\) and the vertical analytic radius increases as long as the solutions exist. PubDate: 2022-07-04 DOI: 10.1007/s40072-022-00266-6

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Abstract: Abstract We consider a nonlocal evolution equation representing the continuum limit of a large ensemble of interacting particles on graphs forced by noise. The two principle ingredients of the continuum model are a nonlocal term and a Q-Wiener process describing the interactions among the particles in the network and stochastic forcing, respectively. The network connectivity is given by a square integrable function called a graphon. We prove that the initial value problem for the continuum model is well-posed. Further, we construct semidiscrete (discrete in space and continuous in time) and fully discrete schemes for the nonlocal model. The former is obtained by a discontinuous Galerkin method and the latter is based on further discretizing time using the Euler–Maruyama method. We prove convergence and estimate the rate of convergence in each case. For the semidiscrete scheme, the rate of convergence is expressed in terms of the regularity of the graphon, the Q-Wiener process, and the initial data. We work in generalized Lipschitz spaces, which allows us to treat models with data of lower regularity. This is important for applications as many interesting types of connectivity, including small-world and power-law, are expressed by graphons that are not smooth. The error analysis of the fully discrete scheme, on the other hand, reveals that for some models common in applied science, one has a higher speed of convergence than that predicted by the standard estimates for the Euler–Maruyama method. The rate of convergence analysis is supplemented with detailed numerical experiments, which are consistent with our analytical results. As a by-product, this work presents a rigorous justification for taking continuum limit for a large class of interacting dynamical systems on graphs subject to noise. PubDate: 2022-07-02 DOI: 10.1007/s40072-022-00262-w

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Abstract: Abstract 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. PubDate: 2022-07-01 DOI: 10.1007/s40072-022-00264-8

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Abstract: Abstract We introduce a mass conserving stochastic perturbation of the discrete nonlinear Schrödinger equation that models the action of a heat bath at a given temperature. We prove that the corresponding canonical Gibbs distribution is the unique invariant measure. In the one-dimensional cubic focusing case on the torus, we prove that in the limit for large time, continuous approximation, and low temperature, the solution converges to the steady wave of the continuous equation that minimizes the energy for a given mass. PubDate: 2022-06-24 DOI: 10.1007/s40072-022-00263-9

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Abstract: Abstract We establish convergence to an invariant measure as time tends to infinity, for a large class of (possibly non-Markovian) stochastic volatility models. Our arguments are based on a novel coupling idea for Markov chains which also extends to Markov chains in random environments in an efficient way. PubDate: 2022-06-20 DOI: 10.1007/s40072-022-00261-x

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Abstract: Abstract We establish the existence of martingale solutions to a class of stochastic conservation equations. The underlying models correspond to random perturbations of kinetic models for collective motion such as the Cucker F, Smale S (IEEE Transactions on Automatic Control 52(5):852–862, 2007), Cucker F, Smale S (Japanese Journal of Mathematics 2(1):197–227, 2007) and Motsch S, Tadmor E (Journal of Statistical Physics 144(5):923–947, 2011) models. By regularizing the coefficients, we first construct approximate solutions obtained as the mean-field limit of the corresponding particle systems. We then establish the compactness in law of this family of solutions by relying on a stochastic averaging lemma. This extends the results obtained in Karper T, Mellet A, Trivisa K (Springer Proceedings in Mathematics & Statistics, 2012), Karper T, Mellet A, Trivisa K (SIAM Journal on Mathematical Analysis 45(1):215–243, 2013) in the deterministic case. PubDate: 2022-06-17 DOI: 10.1007/s40072-022-00259-5

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Abstract: Abstract The filtering equations govern the evolution of the conditional distribution of a signal process given partial, and possibly noisy, observations arriving sequentially in time. Their numerical approximation plays a central role in many real-life applications, including numerical weather prediction [Llopis et al. (SIAM J Sci Comput 40(3):A1544–A1565, 2018), Galanis et al. (Geophysicae 24(10): 2451–2460, 2006)], finance [Brigo and Hanzon (Insurance Math Econom 22(1):53–64, 1998), Date and Ponomareva (IMA J Manag Math 22(3): 195–211, 2011), Crisan and Rozovskii (The Oxford handbook of nonlinear filtering, 2011)] and engineering [Myötyri et al. (Reliability Eng Syst Saf 91(2):200–208, 2005)]. One of the classical approaches to approximate the solution of the filtering equations is to use a PDE inspired method, called the splitting-up method, initiated by Gyongy, Krylov, LeGland, among other contributors, see e.g., Gyöngy and Krylov (Stochastic inequalities and applications, Progr. Probab. 56:301–321, 2003), Le Gland(Stochastic partial differential equations and their applications (Charlotte, NC, 1991), Lect. Notes Control Inf. Sci. 176:177–187, 1992). This method, and other PDE based approaches, have particular applicability for solving low-dimensional problems. In this work we combine this method with a neural network representation inspired by [Han et al. (Proc Natl acad Sci 115(34):8505–8510, 2018)]. The new methodology is used to produce an approximation of the unnormalised conditional distribution of the signal process. We further develop a recursive normalisation procedure to recover the normalised conditional distribution of the signal process. The new scheme can be iterated over multiple time steps whilst keeping its asymptotic unbiasedness property intact. We test the neural network approximations with numerical approximation results for the Kalman and Benes filter. PubDate: 2022-06-09 DOI: 10.1007/s40072-022-00260-y

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Abstract: Abstract We obtain existence results for the solution u of nonlocal semilinear parabolic PDEs on \({{\mathbb {R}}}^d\) with polynomial nonlinearities in \((u, \nabla u)\) , using a tree-based probabilistic representation. This probabilistic representation applies to the solution of the equation itself, as well as to its partial derivatives by associating one of d marks to the initial tree branch. Partial derivatives are dealt with by integration by parts and subordination of Brownian motion. Numerical illustrations are provided in examples for the fractional Laplacian in dimension up to 10, and for the fractional Burgers equation in dimension two. PubDate: 2022-06-01 DOI: 10.1007/s40072-021-00220-y

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Abstract: Abstract We prove a new Burkholder–Rosenthal type inequality for discrete-time processes taking values in a 2-smooth Banach space. As a first application we prove that if \((S(t,s))_{0\leqslant s\le t\leqslant T}\) is a \(C_0\) -evolution family of contractions on a 2-smooth Banach space X and \((W_t)_{t\in [0,T]}\) is a cylindrical Brownian motion on a probability space \((\Omega ,{\mathbb {P}})\) adapted to some given filtration, then for every \(0<p<\infty \) there exists a constant \(C_{p,X}\) such that for all progressively measurable processes \(g: [0,T]\times \Omega \rightarrow X\) the process \((\int _0^t S(t,s)g_s\,\mathrm{d} W_s)_{t\in [0,T]}\) has a continuous modification and $$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}\Big \Vert \int _0^t S(t,s)g_s\,\mathrm{d} W_s \Big \Vert ^p\leqslant C_{p,X}^p {\mathbb {E}} \Bigl (\int _0^T \Vert g_t\Vert ^2_{\gamma (H,X)}\,\mathrm{d} t\Bigr )^{p/2}. \end{aligned}$$ Moreover, for \(2\leqslant p<\infty \) one may take \(C_{p,X} = 10 D \sqrt{p},\) where D is the constant in the definition of 2-smoothness for X. The order \(O(\sqrt{p})\) coincides with that of Burkholder’s inequality and is therefore optimal as \(p\rightarrow \infty \) . Our result improves and unifies several existing maximal estimates and is even new in case X is a Hilbert space. Similar results are obtained if the driving martingale \(g_t\,\mathrm{d} W_t\) is replaced by more general X-valued martingales \(\,\mathrm{d} M_t\) . Moreover, our methods allow for random evolution systems, a setting which appears to be completely new as far as maximal inequalities are concerned. As a second application, for a large class of time discretisation schemes (including splitting, implicit Euler, Crank-Nicholson, and other rational schemes) we obtain stability and pathwise uniform convergence of time discretisation schemes for solutions of linear SPDEs $$\begin{aligned} \,\mathrm{d} u_t = A(t)u_t\,\mathrm{d} t + g_t\,\mathrm{d} W_t, \quad u_0 = 0, \end{aligned}$$ where the family \((A(t))_{t\in [0,T]}\) is assumed to generate a \(C_0\) -evolution family \((S(t,s))_{0\leqslant s\leqslant t\leqslant T}\) of contractions on a 2-smooth Banach spaces X. Under spatial smoothness assumptions on the inhomogeneity g, contractivity is not needed and explicit decay rates are obtained. In the parabolic setting this sharpens several know estimates in the literature; beyond the parabolic setting this seems to provide the first systematic approach to pathwise uniform convergence to time discretisation schemes. PubDate: 2022-06-01 DOI: 10.1007/s40072-021-00204-y

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Abstract: Abstract Fix \(d\in \{1,2\}\) , we consider a d-dimensional stochastic wave equation driven by a Gaussian noise, which is temporally white and colored in space such that the spatial correlation function is integrable and satisfies Dalang’s condition. In this setting, we provide quantitative central limit theorems for the spatial average of the solution over a Euclidean ball, as the radius of the ball diverges to infinity. We also establish functional central limit theorems. A fundamental ingredient in our analysis is the pointwise \(L^p\) -estimate for the Malliavin derivative of the solution, which is of independent interest. This paper is another addendum to the recent research line of averaging stochastic partial differential equations. PubDate: 2022-06-01 DOI: 10.1007/s40072-021-00209-7

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Abstract: Abstract This paper is concerned about stochastic convective Brinkman–Forchheimer (SCBF) equations subjected to multiplicative pure jump noise in bounded or periodic domains. Our first goal is to establish the existence of a pathwise unique strong solution satisfying the energy equality (Itô’s formula) to SCBF equations. We resolve the issue of the global solvability of SCBF equations, by using a monotonicity property of the linear and nonlinear operators and a stochastic generalization of the Minty–Browder technique. The major difficulty is that an Itô’s formula in infinite dimensions is not available for such systems. This difficulty is overcame by the technique of approximating functions using the elements of eigenspaces of the Stokes operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces simultaneously. Due to some technical difficulties, we discuss the global in time regularity results of such strong solutions in periodic domains only. Once the system is well-posed, we look for the asymptotic behavior of strong solutions. For large effective viscosity, the exponential stability results (in the mean square and pathwise sense) for stationary solutions is established. Moreover, a stabilization result of SCBF equations by using a multiplicative pure jump noise is also obtained. Finally, we prove the existence of a unique ergodic and strongly mixing invariant measure for SCBF equations subject to multiplicative pure jump noise, by using the exponential stability of strong solutions. PubDate: 2022-06-01 DOI: 10.1007/s40072-021-00207-9

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Abstract: Abstract A characterisation of the spaces \({\mathcal {G}}_K\) and \({\mathcal {G}}_K'\) introduced in Grothaus et al. (Methods Funct Anal Topol 3(2):46–64, 1997) and Potthoff and Timpel (Potential Anal 4(6):637–654, 1995) is given. A first characterisation of these spaces provided in Grothaus et al. (Methods Funct Anal Topol 3(2):46–64, 1997) uses the concepts of holomorphy on infinite dimensional spaces. We, instead, give a characterisation in terms of U-functionals, i.e., classic holomorphic function on the one dimensional field of complex numbers. We apply our new characterisation to derive new results concerning a stochastic transport equation and the stochastic heat equation with multiplicative noise. PubDate: 2022-06-01 DOI: 10.1007/s40072-021-00200-2