Stochastics and Dynamics

Two deterministic Sharkovsky-type theorems on the circle due to L. Block [Proc. Amer. Math. Soc. 82 (1981) 481–486] and X. Zhao [Fixed Point Theory Appl., Article ID 194875 (2008)] are randomized. The randomization of Block’s theorem brings an additional information about the forcing alternatives. Some further possibilities of the randomized Zhao’s theorem are discussed for multivalued maps.

We study multivalued stochastic differential equations (MSDEs) with maximal monotone operators driven by semimartingales with jumps. We discuss in detail some methods of approximation of solutions of MSDEs based on discretization of processes and Yosida approximation of the monotone operator. We also study the general problem of stability of solutions of MSDEs with respect to the convergence of driving semimartingales.

We consider a linear equation ${\partial }_{t}u={ℒ}u$, where ${ℒ}$ is a generator of a semigroup of linear operators on a certain Hilbert space related to an initial condition $u(0)$ being a generalised stationary random field on ${ℝ}^d$. We show the existence and uniqueness of generalised solutions to such initial value problems. Then we investigate their scaling limits.

In this paper, we consider the infinite horizon nonlinear optimal control of forward–backward stochastic system governed by Teugels martingales associated with Lévy processes and one dimensional independent Brownian motion. Our aim is to establish the sufficient and necessary conditions for optimality of the above stochastic system under the convexity assumptions. Finally an application is given to illustrate the problem of optimal control of stochastic system.

Given $X$, $Y$, $Z$ and $W,$ compact metric spaces, we consider two iterated function systems $\{{\tau }_x:Z\to Z,x\in X\}$ and $\{{\tau }_y:W\to W,y\in Y\}$, where ${\tau }_x$ and ${\tau }_y$ are contractions. Let $\mathrm{\Pi }(\cdot ,\cdot ,\tau )$ be the set of probabilities $\pi \in {𝒫}(X\times Y\times Z\times W)$ with $(X,Z)$-marginal being holonomic with respect to ${\tau }_x$ and $(Y,W)$-marginal being holonomic with respect to ${\tau }_y$. Given $\mu \in {𝒫}(X)$ and $\nu \in {𝒫}(Y)$, let $\mathrm{\Pi }(\mu ,\nu ,\tau )$ be the set of probabilities in $\mathrm{\Pi }(\cdot ,\cdot ,\tau )$ having $X$-marginal $\mu$ and $Y$-marginal $\nu$. Let $H_{\alpha }(\pi )$ be the relative entropy of $\pi$ with respect to $\alpha$ and $H_{\beta }(\pi )$ be the relative entropy of $\pi$ with respect to $\beta$. Given a cost function $c\in C(X\times Y\times Z\times W)$, let $P_{\alpha ,\beta }(c)={sup}_{\pi \in \mathrm{\Pi }(\cdot ,\cdot ,\tau )}\int cd\pi +H_{\alpha }(\pi )+H_{\beta }(\pi )$. We will prove the duality equation:

This paper is divided into two parts. The first deals with some limit theorems to certain extensions of fractional Brownian motion like: bifractional Brownian motion, subfractional Brownian motion and weighted fractional Brownian motion. In the second part we give the similar results of their continuous additive functionals; more precisely, local time and its fractional derivatives involving slowly varying function.

Bond percolation on Cayley graphs provides examples of random graphs. Other examples arise from the dynamical study of proper repetitive subgraphs of Cayley graphs. In this paper we demonstrate that these two families have mutually singular laws as a corollary of a general lemma about countable Borel equivalence relations on first countable Hausdorff spaces.

We consider $\psi$-mixing dynamical systems $({𝒳},T,{{ℬ}}_{{𝒳}},\mu )$ and we find conditions on families of sets $\{{{𝒰}}_n\subset {𝒳}:n\in \mathbb{N}\}$ so that $\mu ({{𝒰}}_n){\tau }_n$ tends in law to an exponential random variable, where ${\tau }_n$ is the entry time to ${{𝒰}}_n.$