Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise

Abstract

Existence and uniqueness of the mild solutions for stochastic differential equations for Hilbert valued stochastic processes are discussed, with the multiplicative noise term given by an integral with respect to a general compensated Poisson random measure. Parts of the results allow for coefficients which can depend on the entire past path of the solution process. In the Markov case Yosida approximations are also discussed, as well as continuous dependence on initial data, and coefficients. The case of coefficients that besides the dependence on the solution process have also an additional random dependence is also included in our treatment. All results are proven for processes with values in separable Hilbert spaces. Differentiable dependence on the initial condition is proven by adapting a method of S. Cerrai.

Uniqueness of Invariant Measures of Infinite Dimensional Stochastic Differential Equations Driven by Lévy Noises

“…For the definition of the cylindrical Wiener process and the stochastic integral with respect to W(t), we refer the reader to [10] and references therein for details. Here we will briefly recall the stochastic integral with respect to the compensated Poisson random measure q(t, ·) and some properties, for details we refer the reader to [1,25,28]. Suppose f : R + × H × → H is a measurable and F t -adapted progress satisfying…”

“…For recently deep studies of Ornstein-Uhlenbeck processes with jumps, we refer readers to the works of D. Applebaum [6] and M. Röckner and his collaborators [13,23]. For general case, please see the research papers [1,18,20,25] for that driven by compensated Poisson random measures and the monograph [28] for general Lévy noises by semigroup approaches.…”

“…If we only want to show the existence and uniqueness of the solution X(t) satisfying a weak condition sup t∈[0,T] E |X(t)| 2 < ∞, then we can easily do by fixed point theorems, see [9,10] for that relative to Wiener noises and [1,18,20] for Poisson random measures. However, since in the above theorem we require X(t) ∈ H 2,T , we have to consider maximal inequalities of stochastic convolutions in mean square sense, see Theorems B.1 and B.2.…”

Classical robots perturbed by Lévy processes: analysis and Lévy disturbance rejection methods

“…In [8], existence and uniqueness theory for general stochastic differential equations driven by a continuous superposition of Poisson fields with the coefficients in the superposition being causal functionals of the state vector which takes values in a separable Hilbert space has been developed. This theory can be applied to the robot problem when the dynamical equations have the form…”

Existence and Exponential Mixing of Infinite White $\alpha$-Stable Systems with Unbounded Interactions

“…The SPDEs driven by Lévy noises were intensively studied in the past several decades ( [24], [3], [25], [28], [7], [5], [22], [21], · · · ). The noises can be Wiener( [11], [12]) Poisson ( [5]), α-stable types ( [27], [33]) and so on.…”