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Abstract: Abstract 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, Tokyo, 1995) in this case the homogenized operator is deterministic. The paper focuses on the diffusion approximation of solutions in the case of non-diffusive scaling, when the oscillation in spatial variables is faster than that in temporal variable. Our goal is to study the asymptotic behaviour of the normalized difference between solutions of the original and the homogenized problems. PubDate: 2024-02-13

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Abstract: Abstract 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 and explore certain second order properties of the solution. These are then utilized to deduce lower and upper bounds for probabilities that the path process hits bounded Borel sets in terms of the \(\mathfrak {g}_q\) -capacity and \(g_q\) -Hausdorff measure, respectively, which reveal the critical dimension for hitting points. Furthermore, by introducing the harmonizable representation of the solution and utilizing it to construct a family of approximating random fields which have certain smoothness properties, we prove that all points are polar in the critical dimension. This provides a compelling evidence supporting the conjecture raised in Hinojosa-Calleja and Sanz-Solé (Stoch Part Differ Equ Anal Comput 10(3):735–756, 2022. https://doi.org/10.1007/s40072-021-00234-6). PubDate: 2023-12-08

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Abstract: Abstract We develop a provably efficient importance sampling scheme that estimates exit probabilities of solutions to small-noise stochastic reaction–diffusion equations from scaled neighborhoods of a stable equilibrium. The moderate deviation scaling allows for a local approximation of the nonlinear dynamics by their linearized version. In addition, we identify a finite-dimensional subspace where exits take place with high probability. Using stochastic control and variational methods we show that our scheme performs well both in the zero noise limit and pre-asymptotically. Simulation studies for stochastically perturbed bistable dynamics illustrate the theoretical results. PubDate: 2023-12-08

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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: 2023-12-01 DOI: 10.1007/s40072-022-00263-9

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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: 2023-12-01 DOI: 10.1007/s40072-022-00266-6

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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: 2023-12-01 DOI: 10.1007/s40072-022-00269-3

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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: 2023-12-01 DOI: 10.1007/s40072-022-00264-8

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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: 2023-12-01 DOI: 10.1007/s40072-022-00271-9

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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: 2023-12-01 DOI: 10.1007/s40072-022-00262-w

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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: 2023-12-01 DOI: 10.1007/s40072-022-00267-5

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Abstract: Abstract We propose a derivative-free Milstein type scheme to approximate the mild solution of stochastic partial differential equations (SPDEs) that do not need to fulfill a commutativity condition for the noise term. The newly developed derivative-free Milstein type scheme differs significantly from schemes that are appropriate for the case of commutative noise. As a key result, the new derivative-free Milstein type scheme needs only two stages that are specifically tailored based on a technique that, compared to the original Milstein scheme, allows for a reduction of the computational complexity by one order of magnitude. Moreover, the proposed derivative-free Milstein scheme can flexibly be combined with some approximation method for the involved iterated stochastic integrals. As the main result, we prove the strong \(L^2\) -convergence of the introduced derivative-free Milstein type scheme, especially if it is combined with any suitable approximation algorithm for the necessary iterated stochastic integrals. We carry out a rigorous analysis of the error versus computational cost and derive the effective order of convergence for the derivative-free Milstein type scheme in the case that the truncated Fourier series algorithm for the approximation of the iterated stochastic integrals is applied. As a further novelty, we show that the use of approximations of iterated stochastic integrals based on truncated Fourier series together with the proposed derivative-free Milstein type scheme improves the effective order of convergence compared to that of the Euler scheme and the original Milstein scheme. This result is contrary to well known results in the finite dimensional SDE case where the use of merely truncated Fourier series does not improve the effective order of convergence in the \(L^2\) -sense compared to that of the Euler scheme. PubDate: 2023-12-01 DOI: 10.1007/s40072-022-00274-6

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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: 2023-12-01 DOI: 10.1007/s40072-022-00272-8

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Abstract: Abstract This paper is concerned with the problem of regularization by noise of systems of reaction–diffusion equations with mass control. It is known that strong solutions to such systems of PDEs may blow-up in finite time. Moreover, for many systems of practical interest, establishing whether the blow-up occurs or not is an open question. Here we prove that a suitable multiplicative noise of transport type has a regularizing effect. More precisely, for both a sufficiently noise intensity and a high spectrum, the blow-up of strong solutions is delayed up to an arbitrary large time. Global existence is shown for the case of exponentially decreasing mass. The proofs combine and extend recent developments in regularization by noise and in the \(L^p(L^q)\) -approach to stochastic PDEs, highlighting new connections between the two areas. PubDate: 2023-11-28

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Abstract: Abstract 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 to a one-dimensional nonlinear stochastic differential equation with a sticky boundary at zero. This new class of equations is then analyzed carefully. In particular, we show that the dominating equation has a phase transition. In the regime where the Dirac measure at zero is the only invariant probability measure, we prove exponential convergence to equilibrium both for the one-dimensional equation, and for the original nonlinear SDE. Similarly, propagation of chaos is shown by a componentwise sticky coupling and comparison with a system of one dimensional nonlinear SDEs with sticky boundaries at zero. The approach applies to equations without confinement potential and to interaction terms that are not of gradient type. PubDate: 2023-11-06 DOI: 10.1007/s40072-023-00315-8

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Abstract: Abstract 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. Solutions enjoy some Sobolev regularity in space but are not Leray–Hopf. PubDate: 2023-10-30 DOI: 10.1007/s40072-023-00318-5

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Abstract: 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 and weak solutions are defined; these concepts are shown to be equivalent and to lead to well-posed problems. Temporal and spatial regularity of the solution process X are investigated, the former being measured by mean-square or pathwise smoothness and the latter by using domains of fractional powers of A. In addition, the covariance of X and its long-time behavior are analyzed. These abstract results are applied to the cases when \(A:= L^\beta \) and \(Q:={\widetilde{L}}^{-\alpha }\) are fractional powers of symmetric, strongly elliptic second-order differential operators defined on (i) bounded Euclidean domains or (ii) smooth, compact surfaces. In these cases, the Gaussian solution processes can be seen as generalizations of merely spatial (Whittle–)Matérn fields to space–time. PubDate: 2023-10-30 DOI: 10.1007/s40072-023-00316-7

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Abstract: Abstract 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 of this result is that there exists no Banach space of distributions which carries the Gaussian free field on the 3D torus and to which the DeTurck–Yang–Mills heat flow extends continuously, which complements recent well-posedness results of Cao–Chatterjee and the author with Chandra–Hairer–Shen. Another consequence is that the (deterministic) non-linear heat equation exhibits norm inflation, and is thus locally ill-posed, at every point in the Besov space \(B^{-1/2}_{\infty ,\infty }\) ; the space \(B^{-1/2}_{\infty ,\infty }\) is an endpoint since the equation is locally well-posed for \(B^{\eta }_{\infty ,\infty }\) for every \(\eta >-\frac{1}{2}\) . PubDate: 2023-10-15 DOI: 10.1007/s40072-023-00317-6

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Abstract: Abstract 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 in the Sobolev space \(H^3\) . Our approach combines a cut-off approximation scheme, a stochastic compactness arguments and a general version of Yamada–Watanabe theorem. This leads to the existence of a local strong pathwise solution. PubDate: 2023-10-09 DOI: 10.1007/s40072-023-00314-9

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Abstract: Abstract 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 \((Y_s)_{s\in [0,t]}\) exists and belongs to \(L_p\) if the conditional density of \(X_0\) given \(Y_0\) exists and belongs to \(L_p\) . PubDate: 2023-10-03 DOI: 10.1007/s40072-023-00311-y