Stochastics and Dynamics

Linear filtering problem for infinite-dimensional Gaussian processes is studied, the observation process being finite-dimensional. Integral equations for the filter and for covariance of the error are derived. General results are applied to linear SPDEs driven by Gauss–Volterra process observed at finitely many points of the domain.

We investigate the asymptotic behavior of a class of non-autonomous stochastic FitzHugh–Nagumo systems driven by additive white noise on unbounded thin domains. For this aim, we first show the existence and uniqueness of random attractors for the considered equations and their limit equations. Then, we establish the upper semicontinuity of these attractors when the thin domains collapse into a lower-dimensional unbounded domain.

We consider a parameter estimation problem for one-dimensional stochastic heat equations, when data is sampled discretely in time or spatial component. We prove that, the real valued parameter next to the Laplacian (the drift), and the constant parameter in front of the noise (the volatility) can be consistently estimated under somewhat surprisingly minimal information. Namely, it is enough to observe the solution at a fixed time and on a discrete spatial grid, or at a fixed space point and at discrete time instances of a finite interval, assuming that the mesh-size goes to zero. The proposed estimators have the same form and asymptotic properties regardless of the nature of the domain –bounded domain or whole space. The derivation of the estimators and the proofs of their asymptotic properties are based on computations of power variations of some relevant stochastic processes. We use elements of Malliavin calculus to establish the asymptotic normality properties in the case of bounded domain. We also discuss the joint estimation problem of the drift and volatility coefficient. We conclude with some numerical experiments that illustrate the obtained theoretical results.

In this paper, we prove a central limit theorem and a moderate deviation principle for a perturbed stochastic Cahn–Hilliard equation defined on $[0,T]\times {[0,\pi ]}^d$ with $d\in \{1,2,3\}$. This equation is driven by a space-time white noise. The weak convergence approach plays an important role.

A new modification of the minimum-contrast estimator (the weighted MCE) of drift parameter in a linear stochastic evolution equation with additive fractional noise is introduced in the setting of the spectral approach (Fourier coordinates of the solution are observed). The reweighing technique, which utilizes the self-similarity property, achieves strong consistency and asymptotic normality of the estimator as number of coordinates increases and time horizon is fixed (the space consistency). In this respect, this modification outperforms the standard (non-weighted) MCE. Compared to other drift estimators studied within spectral approach (e.g., maximum likelihood, trajectory fitting), the weighted MCE is rather universal. It covers discrete time as well as continuous-time observations and it is applicable to processes with any value of Hurst index $H\in (0,1)$. To the author’s best knowledge, this is so far the first space-consistent estimator studied for $H<1/2$.

In this paper, we prove a moderate deviation principle for the multivalued stochastic differential equations whose proof are based on recently well-developed weak convergence approach. As an application, we obtain the moderate deviation principle for reflected Brownian motion.

Existence of expansivity for group action $\phi :G\times X\to X$ depends on algebraic properties of $G$ and the topology of $X$. We give an expansive action of a solvable group on $S^1$ while there is no expansive action of a solvable group on a dendrite $X$. We prove that a continuous action $\phi :G\times X\to X$ on a compact metric space $X$ is expansive if and only if there exists an open cover ${𝒰}$ such that for any other open cover ${𝒱}$, ${\wedge }_{g\in F}\phi (g^{-1},{𝒰})\prec {𝒱},$ for some finite set $F\subseteq G$. In this paper, we introduce the notion of topological expansivity of a group action $\phi :G\times X\to X$ on a $T_1$-paracompact space $X$. If a $T_1$-paracompact space $X$ admits topologically expansive action, then $X$ is Hausdorff space. We also show that a continuous action $\phi :G\times X\to X$ of a finitely generated group $G$ on a compact Hausdorff uniform space $(X,{𝒰})$ is expansive with an expansive neighborhood $D\in {𝒰}$ if and only if for every $E\in {𝒰}$ there is an entourage $N\in {𝒰}$ such that for every two $N$-pseudo orbit $F_0,F_1:G\to X$ if $(F_0(g),F_1(g))\in D$ for all $g\in G$, then $(F_0(g),F_1(g))\in E$ for all $g\in G$.

Finally, we introduce measure $1$-expansive actions on a uniform space. The set of all $1$-expansive measures with common expansive neighborhood for a group action $\phi :G\times X\to X$ is a convex, closed and ${\phi }_{\star }$-invariant subset of the set of all Borel probability measures on $X$. Also, we show that a group action $\phi :G\times X\to X$ is expansive if all Borel probability measures are $1$-expansive or all Dirac measures $m_z$, $z\in X$, have a common expansive neighborhood.

We consider a linearized dynamical system modeling the flow rate of water along the rivers and hillslopes of an arbitrary watershed. The system is perturbed by a random rainfall in the form of a compound Poisson process. The model describes the evolution, at daily time scales, of an interconnected network of linear reservoirs and takes into account the differences in flow celerity between hillslopes and streams as well as their spatial variation. The resulting stochastic process is a piece-wise deterministic Markov process of the Orstein–Uhlembeck type. We provide an explicit formula for the Laplace transform of the invariant density of streamflow in terms of the geophysical parameters of the river network and the statistical properties of the precipitation field. As an application, we include novel formulas for the invariant moments of the streamflow at the watershed’s outlet, as well as the asymptotic behavior of extreme discharge events, and conditions for the statistical scaling of streamflow with respect to Horton order.

Let $D$ be a bounded Lipschitz domain of ${ℝ}^d$. We consider the complement value problem