Stochastics and Partial Differential Equations-Analysis and Computations最新文献

We present a form of stratified MCMC algorithm built with non-reversible stochastic dynamics in mind. It can also be viewed as a generalization of the exact milestoning method or form of NEUS. We prove the convergence of the method under certain assumptions, with expressions for the convergence rate in terms of the process’s behavior within each stratum and large-scale behavior between strata. We show that the algorithm has a unique fixed point which corresponds to the invariant measure of the process without stratification. We will show how the convergence speeds of two versions of the algorithm, one with an extra eigenvalue problem step and one without, related to the mixing rate of a discrete process on the strata, and the mixing probability of the process being sampled within each stratum. The eigenvalue problem version also relates to local and global perturbation results of discrete Markov chains, such as those given by Van Koten, Weare et. al.

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 scale for this (C^{1,alpha }) regularity. As an application to stochastic homogenization, we partially generalize results by Gloria et al. (Anal PDE 14(8):2497–2537, 2021) on the growth of the corrector, the decay of its gradient, and a quantitative two-scale expansion to the degenerate setting. On a technical level, we demand the ensemble of coefficient fields to be stationary and subject to a spectral gap inequality, and we impose moment bounds on a and (a^{-1}). We also introduce the ellipticity radius (r_e) which encodes the minimal scale where these moments are close to their positive expectation value.

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 rate. It is the same property which allows for uniform bounds for the derivatives of the solution of the related Kolmogorov equation, and then leads to weak rate ({{mathcal {O}}}(tau )) in the presence of multiplicative noise. Hence, we obtain twice the rate of convergence known for the strong error in the presence of multiplicative noise.

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.

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).

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.

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.