Stochastics and Dynamics

We consider stochastic 2D-Stokes equations with unbounded delay in fractional power spaces and moments of order $p\ge 2$ driven by a tempered fractional Brownian motion (TFBM) $B^{\sigma ,\lambda }(t)$ with $-1/2<\sigma <0$ and $\lambda >0$. First, the global existence and uniqueness of mild solutions are established by using a new technical lemma for stochastic integrals with respect to TFBM in the sense of $p$th moment. Moreover, based on the relations between the stochastic integrals with respect to TFBM and fractional Brownian motion, we show the continuity of mild solutions in the case of $\lambda \to 0$, $\sigma \in (-1/2,0)$ or $\lambda >0$, $\sigma \to {\sigma }_0\in (-1/2,0)$. In particular, we obtain $p$th moment Hölder regularity in time and $p$th polynomial stability of mild solutions. This paper can be regarded as a first step to study the challenging model: stochastic 2D-Navier–Stokes equations with unbounded delay driven by tempered fractional Gaussian noise.

In this paper, we study the existence and continuous dependence on coefficients of mild solutions for first-order McKean–Vlasov integrodifferential equations with delay driven by a cylindrical Wiener process using resolvent operator theory and Wasserstein distance. Under the situation that the nonlinear term depends on the probability distribution of the state, the existence and uniqueness of solutions are established. An example illustrating the general results is included.

We prove the existence and uniqueness of global solutions to the semilinear stochastic heat equation on an unbounded spatial domain with forcing terms that grow superlinearly and satisfy an Osgood condition $\int 1/|f(u)|du=+\infty$ along with additional restrictions. For example, consider the forcing $f(u)=ulog(e^e+|u|)log(log(e^e+\left|u\right|))$. A new dynamic weighting procedure is introduced to control the solutions, which are unbounded in space.

We prove some new results on reflected BSDEs and doubly reflected BSDEs driven by a multi-dimensional RCLL martingale. The goal is to develop a general multi-asset framework encompassing a wide spectrum of nonlinear financial models, including as particular cases the setups studied by Peng and Xu [BSDEs with random default time and their applications to default risk, working paper, preprint (2009), arXiv:0910.2091] and Dumitrescu et al. [BSDEs with default jump, in Computation and Combinatorics in Dynamics, Stochastics and Control, Abel Symposia, Vol. 13, eds. E. Celledoni, G. Di Nunno, K. Ebrahimi-Fard and H. Munthe-Kaas (Springer, Cham, 2018), pp. 233–263] who examined BSDEs driven by a one-dimensional Brownian motion and a purely discontinuous martingale with a single jump. Our results are not covered by existing literature on reflected and doubly reflected BSDEs driven by a Brownian motion and a Poisson random measure.

In this paper, the following $d$-dimensional distribution-dependent stochastic differential equation driven by a pure jump process is studied:

We study a backward SDE driven by a jump Markov process (BSDEJ for short) whose generator may be locally Lipschitz or of logarithmic growth in $(y,z(\cdot ))$-variables. The existence, uniqueness and stability theorems to such BSDEJs are established. We essentially approximate the initial problem by constructing a suitable sequence of BSDEJs with globally Lipschitz generators for which the existence and uniqueness of solutions hold. By passing to the limits, we show the existence and uniqueness of solutions to the original problems. We apply our main results to prove the existence of a unique solution to the Kolmogorov equation of the Markov process.

In this paper, we study the central limit theorem (CLT) and its weak invariance principle (WIP) for sums of stationary random fields non-necessarily adapted, under different normalizations. To do so, we first state sufficient conditions for the validity of a suitable ortho-martingale approximation. Then, with the help of this approximation, we derive projective criteria under which the CLT as well as the WIP hold. These projective criteria are in the spirit of Hannan’s condition and are well adapted to linear random fields with ortho-martingale innovations and which exhibit long memory.

In this paper, we introduce the concepts of induced topological entropy and induced measure-theoretic entropy for random dynamical systems. By providing the topological version of induced measure-theoretic entropy, we establish the Katok entropy formula for a continuous bundle random dynamical system over an invertible ergodic system. We also consider the induced version with lower and upper pointwise dimensions, and establish a measure-theoretic entropy formula for lower and upper induced pointwise dimensions for a continuous bundle random dynamical system over an invertible ergodic system.

In this paper, we consider a fractional Laplacian system with saturable nonlinearity. Under some assumptions on the parameters and potential functions, we obtain the existence and concentration behavior of the positive ground state solution by variational methods.