Search ResultsShowing 1-20 of 38
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arXiv:2501.12944 (Published 2025-01-22)
Stability of travelling wave solutions to reaction-diffusion equations driven by additive noise with Hölder continuous paths
Comments: 36 pagesIn this paper we investigate stability of travelling wave solutions to a class of reaction-diffusion equations perturbed by infinite-dimensional additive noise with H\"older continuous paths, covering in particular fractional Wiener processes with general Hurst parameter. In the latter example, we obtain explicit error bounds on the maximal distance from the solution of the stochastic reaction-diffusion equation to the orbit of travelling wave fronts in terms of the Hurst parameter and the spatial regularity for small noise amplitude. Our bounds can be optimised for short times in terms of the Hurst parameter and for large times in terms of the spatial regularity of the noise covariance of the driving fractional Wiener process.
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arXiv:2409.11330 (Published 2024-09-17)
Parameter dependent rough SDEs with applications to rough PDEs
Categories: math.PRIn this paper we generalize Krylov's theory on parameter-dependent stochastic differential equations to the framework of rough stochastic differential equations (rough SDEs), as initially introduced by Friz, Hocquet and L\^e. We consider a stochastic equation of the form $$ dX_t^\zeta = b_t(\zeta,X_t^\zeta) \ dt + \sigma_t(\zeta,X_t^\zeta) \ dB_t + \beta_t (\zeta,X_t^\zeta) d\mathbf{W}_t,$$ where $\zeta$ is a parameter, $B$ denotes a Brownian motion and $\mathbf{W}$ is a deterministic H\"older rough path. We investigate the conditions under which the solution $X$ exhibits continuity and/or differentiability with respect to the parameter $\zeta$ in the $\mathscr{L}$-sense, as defined by Krylov. As an application, we present an existence-and-uniqueness result for a class of rough partial differential equations (rough PDEs) of the form $$-du_t = L_t u_t dt + \Gamma_t u_t d\mathbf{W}_t, \quad u_T =g.$$ We show that the solution admits a Feynman--Kac type representation in terms of the solution of an appropriate rough SDE, where the initial time and the initial state play the role of parameters.
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arXiv:2408.15920 (Published 2024-08-28)
Filtering SPDEs with Spatio-Temporal Point Process Observations
In this paper, we develop the mathematical framework for filtering problems arising from biophysical applications where data is collected from confocal laser scanning microscopy recordings of the space-time evolution of intracellular wave dynamics of biophysical quantities. In these applications, signals are described by stochastic partial differential equations (SPDEs) and observations can be modelled as functionals of marked point processes whose intensities depend on the underlying signal. We derive both the unnormalized and normalized filtering equations for these systems, demonstrate the asymptotic consistency and approximations of finite dimensional observation schemes respectively partial observations. Our theoretical results are validated through extensive simulations using synthetic and real data. These findings contribute to a deeper understanding of filtering with point process observations and provide a robust framework for future research in this area.
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arXiv:2407.14884 (Published 2024-07-20)
The Lions Derivative in Infinite Dimensions and Higher Order Expansion of Mean-Field SPDEs
Categories: math.PRIn this paper we present a new interpretation of the Lions derivative as the Radon-Nikodym derivative of a vector measure, which provides a canonical extension of the Lions derivative for functions taking values in infinite dimensional Hilbert spaces. This is of particular relevance for the analysis of Hilbert space valued Mean-Field equations. As an illustration we derive a mild Ito-formula for Mean-Field SPDEs, which provides the basis for a higher order Taylor expansion and higher order numerical schemes.
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arXiv:2301.11959 (Published 2023-01-27)
Approximation of Optimal Feedback Controls for Stochastic Reaction-Diffusion Equations
We approximate optimal feedback controls for stochastic reaction-diffusion equations by using a general type of approximation that allows us to reduce the control problem to an optimization over deterministic controls. Similar to ([Stannat, Wessels], Deterministic Control of stochastic reaction-diffusion equations, Evolution Equations & Control Theory (2020)) we derive necessary optimality conditions and prove the existence of an optimal control for the reduced problem. Furthermore we derive explicit convergence rates and numerically investigate a gradient descent algorithm for the approximation of the optimal feedback control using a radial basis approximation as a particular example.
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arXiv:2301.11926 (Published 2023-01-25)
Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations
Comments: 12 figuresWe present a numerical algorithm that allows the approximation of optimal controls for stochastic reaction-diffusion equations with additive noise by first reducing the problem to controls of feedback form and then approximating the feedback function using finitely based approximations. Using structural assumptions on the finitely based approximations, rates for the approximation error of the cost can be obtained. Our algorithm significantly reduces the computational complexity of finding controls with asymptotically optimal cost. Numerical experiments using artificial neural networks as well as radial basis function networks illustrate the performance of our algorithm. Our approach can also be applied to stochastic control problems for high dimensional stochastic differential equations and more general stochastic partial differential equations.
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arXiv:2205.14253 (Published 2022-05-27)
Analysis of the Ensemble Kalman--Bucy Filter for correlated observation noise
Categories: math.PREnsemble Kalman--Bucy filters (EnKBFs) are an important tool in Data Assimilation that aim to approximate the posterior distribution for continuous time filtering problems using an ensemble of interacting particles. In this work we extend a previously derived unifying framework for consistent representations of the posterior distribution to correlated observation noise and use these representations to derive an EnKBF suitable for this setting. Existence and uniqueness results for both the EnKBF and its mean field limit are provided. In the correlated noise case the evolution of the ensemble depends also on the pseudoinverse of its empirical covariance matrix, which has to be controlled for global well posedness. The existence and uniqueness of solutions to its limiting McKean-Vlasov equation does not seem to be covered by the existing literature. Finally the convergence to the mean-field limit is proven. The results can also be extended to other versions of EnKBFs.
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Necessary and Sufficient Conditions for Optimal Control of Semilinear Stochastic Partial Differential Equations
Comments: 29 pages; added a necessary assumption to Theorem 4.1 and improved the presentation; added Remark 4.3 regarding more general differential operatorsUsing a recently introduced representation of the second order adjoint state as the solution of a function-valued backward stochastic partial differential equation (SPDE), we calculate the viscosity super- and subdifferential of the value function evaluated along an optimal trajectory for controlled semilinear SPDEs. This establishes the well-known connection between Pontryagin's maximum principle and dynamic programming within the framework of viscosity solutions. As a corollary, we derive that the correction term in the stochastic Hamiltonian arising in non-smooth stochastic control problems is non-positive. These results directly lead us to a stochastic verification theorem for fully nonlinear Hamilton-Jacobi-Bellman equations in the framework of viscosity solutions.
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arXiv:2109.02761 (Published 2021-09-06)
Analysis of the feedback particle filter with diffusion map based approximation of the gain
Categories: math.PRControl-type particle filters have been receiving increasing attention over the last decade as a means of obtaining sample based approximations to the sequential Bayesian filtering problem in the nonlinear setting. Here we analyse one such type, namely the feedback particle filter and a recently proposed approximation of the associated gain function based on diffusion maps. The key purpose is to provide analytic insights on the form of the approximate gain, which are of interest in their own right. These are then used to establish a roadmap to obtaining well-posedness and convergence of the finite $N$ system to its mean field limit. A number of possible future research directions are also discussed.
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Peng's Maximum Principle for Stochastic Partial Differential Equations
Comments: 19 pages; accepted for publication in SIAM Journal on Control and OptimizationWe extend Peng's maximum principle for semilinear stochastic partial differential equations (SPDEs) in one space-dimension with non-convex control domains and control-dependent diffusion coefficients to the case of general cost functionals with Nemytskii-type coefficients. Our analysis is based on a new approach to the characterization of the second order adjoint state as the solution of a function-valued backward SPDE.
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arXiv:2103.03194 (Published 2021-03-04)
Stability and moment estimates for the stochastic singular $Φ$-Laplace equation
Comments: 23 pages, 54 referencesWe study stability, long-time behavior and moment estimates for stochastic evolution equations with additive Wiener noise and with singular drift given by a divergence type quasilinear diffusion operator which may not necessarily exhibit a homogeneous diffusivity. Our results cover the singular $p$-Laplace and, more generally, singular $\Phi$-Laplace equations with zero Dirichlet boundary conditions. We obtain improved moment estimates and quantitative convergence rates of the ergodic semigroup to the unique invariant measure, classified in a systematic way according to the degree of local degeneracy of the potential at the origin. We obtain new concentration results for the invariant measure and establish maximal dissipativity of the associated Kolmogorov operator. In particular, we recover the results for the curve shortening flow in the plane by Es-Sarhir, von Renesse and Stannat, NoDEA 16(9), 2012.
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arXiv:2012.14410 (Published 2020-12-28)
Analytic theory of Itô-stochastic differential equations with non-smooth coefficients
Comments: 101 pagesCategories: math.PRWe present a detailed analysis of time-homogeneous It\^o-stochastic differential equations with low local regularity assumptions on the coefficients. In particular the drift coefficient may only satisfy a local integrability condition. We discuss non-explosion, irreducibility, Krylov type estimates, regularity of the transition function and resolvent, moment inequalities, recurrence, transience, long time behavior of the transition function, existence and uniqueness of invariant measures, as well as pathwise uniqueness, strong solutions and uniqueness in law. This analysis shows in particular that sharp conditions can in this situation be derived similarly to the case of classical stochastic differential equations with local Lipschitz coefficients and closes hereby a gap in the existing literature.
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arXiv:2011.10516 (Published 2020-11-20)
Mean field limit of Ensemble Square Root Filters -- discrete and continuous time
Categories: math.PRConsider the class of Ensemble Square Root filtering algorithms for the numerical approximation of the posterior distribution of nonlinear Markovian signals partially observed with linear observations corrupted with independent measurement noise. We analyze the asymptotic behavior of these algorithms in the large ensemble limit both in discrete and continuous time. We identify limiting mean-field processes on the level of the ensemble members, prove corresponding propagation of chaos results and derive associated convergence rates in terms of the ensemble size. In continuous time we also identify the stochastic partial differential equation driving the distribution of the mean-field process and perform a comparison with the Kushner-Stratonovich equation.
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arXiv:2007.12658 (Published 2020-07-24)
McKean-Vlasov SDEs in nonlinear filtering
Various particle filters have been proposed over the last couple of decades with the common feature that the update step is governed by a type of control law. This feature makes them an attractive alternative to traditional sequential Monte Carlo which scales poorly with the state dimension due to weight degeneracy. This article proposes a unifying framework that allows to systematically derive the McKean-Vlasov representations of these filters for the discrete time and continuous time observation case, taking inspiration from the smooth approximation of the data considered in Crisan & Xiong (2010) and Clark & Crisan (2005). We consider three filters that have been proposed in the literature and use this framework to derive It\^{o} representations of their limiting forms as the approximation parameter $\delta \rightarrow 0$. All filters require the solution of a Poisson equation defined on $\mathbb{R}^{d}$, for which existence and uniqueness of solutions can be a non-trivial issue. We additionally establish conditions on the signal-observation system that ensures well-posedness of the weighted Poisson equation arising in one of the filters.
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arXiv:1912.10854 (Published 2019-12-23)
Fluctuation limits for mean-field interacting nonlinear Hawkes processes
Categories: math.PRWe investigate the asymptotic behaviour of networks of interacting non-linear Hawkes processes modeling a homogeneous population of neurons in the large population limit. In particular, we prove a functional central limit theorem for the mean spike-activity thereby characterizing the asymptotic fluctuations in terms of a stochastic Volterra integral equation. Our approach differs from previous approaches in making use of the associated resolvent in order to represent the fluctutations as Skorokhod continuous mappings of weakly converging martingales. Since the Lipschitz properties of the resolvent are explicit, our analysis in principle also allows to derive approximation errors in terms of driving martingales.
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arXiv:1910.12493 (Published 2019-10-28)
On the continuous time limit of Ensemble Square Root Filters
Comments: 31 pagesCategories: math.PRWe provide a continuous time limit analysis for the class of Ensemble Square Root Filter algorithms with deterministic model perturbations. In the particular linear case, we specify general conditions on the model perturbations implying convergence of the empirical mean and covariance matrix towards their respective counterparts of the Kalman-Bucy Filter. As a second main result we identify additional assumptions for the convergence of the whole ensemble towards solutions of the Ensemble Kalman-Bucy filtering equations introduced in [1]. The latter result can be generalized to nonlinear Lipschitz-continuous model operators. A striking implication of our results is the fact that the limiting equations for the ensemble members are universal for a large class of Ensemble Square Root Filters. This yields a mathematically rigorous justification for the analysis of these algorithms with the help of the Ensemble Kalman-Bucy Filter. [1] de Wiljes, Jana, Reich, Sebastian, Stannat, Wilhelm, Long-Time Stability and Accuracy of the Ensemble Kalman-Bucy Filter for Fully Observed Processes and Small Measurement Noise. SIAM Journal on Applied Dynamical Systems, Vol. 17, No. 2, 1152-1181, 2018
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arXiv:1905.09074 (Published 2019-05-22)
Deterministic Control of Stochastic Reaction-Diffusion Equations
Comments: 20 pages, 10 figuresWe consider the control of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise via deterministic controls. Existence of optimal controls and necessary conditions for optimality are derived. Using adjoint calculus, we obtain a representation for the gradient of the cost functional. The restriction to deterministic controls avoids the necessity of introducing a backward SPDE. Based on this novel representation, we present a probabilistic nonlinear conjugate gradient descent method to approximate the optimal control, and apply our results to the stochastic Schl\"ogl model. We also present some analysis in the case where the optimal control for the stochastic system differs from the optimal control for the deterministic system.
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arXiv:1904.04774 (Published 2019-04-09)
Drift Estimation for Stochastic Reaction-Diffusion Systems
A parameter estimation problem for a class of semilinear stochastic evolution equations is considered. Conditions for consistency and asymptotic normality are given in terms of growth and continuity properties of the nonlinear part. Emphasis is put on the case of stochastic reaction-diffusion systems. Robustness results for statistical inference under model uncertainty are provided.
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arXiv:1901.07778 (Published 2019-01-23)
Weak Solutions to Vlasov-McKean Equations under Lyapunov-type Conditions
Categories: math.PRWe present a Lyapunov type approach to the problem of existence and uniqueness of general law-dependent stochastic differential equations. In the existing literature most results concerning existence and uniqueness are obtained under regularity assumptions of the coefficients w.r.t the Wasserstein distance. Some existence and uniqueness results for irregular coefficients have been obtained by considering the total variation distance. Here we extend this approach to the control of the solution in some weighted total variation distance, that allows us now to derive a rather general weak uniqueness result, merely assuming measurability and certain integrability on the drift coefficient and some non-degeneracy on the dispersion coefficient. We also present an abstract weak existence result for the solution of law-dependent stochastic differential equations with merely measurable coefficients, based on an approximation with law-dependent stochastic differential equations with regular coefficients under Lyapunov type assumptions.
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arXiv:1901.05204 (Published 2019-01-16)
On the continuous time limit of the Ensemble Kalman Filter
Comments: 33 pagesCategories: math.PRWe present recent results on the existence of a continuous time limit for Ensemble Kalman Filter algorithms. In the setting of continuous signal and observation processes, we apply the original Ensemble Kalman Filter algorithm proposed by [1] as well as a recent variant [2] to the respective discretizations and show that in the limit of decreasing stepsize the filter equations converge to an ensemble of interacting (stochastic) differential equations in the ensemble-mean-square sense. Our analysis also allows for the derivation of convergence rates with respect to the stepsize. An application of our analysis is the rigorous derivation of continuous ensemble filtering algorithms consistent with discrete approximation schemes. Conversely, the continuous time limit allows for a better qualitative and quantitative analysis of the time-discrete counterparts using the rich theory of dynamical systems in continuous time. [1] Burgers, G., van Leeuwen, P. J., Evensen, G. (1998). Analysis scheme in the ensemble Kalman filter. Monthly weather review, 126(6), 1719-1724. [2] de Wiljes, J., Reich, S., Stannat, W. (2018). Long-Time Stability and Accuracy of the Ensemble Kalman-Bucy Filter for Fully Observed Processes and Small Measurement Noise. SIAM Journal on Applied Dynamical Systems, 17(2), 1152-1181.