Open Access. © 2018 Mark A. McKibben and Micah Webster, published by De Gruyter. This work is licensed under the Creative
Commons Attribution-NonCommercial-NoDerivs 4.0 License.
Nonauton. Dyn. Syst. 2018; 5:59–75
Research Article Open Access
Mark A. McKibben* and Micah Webster
Fractional Measure-dependent Nonlinear
Second-order Stochastic Evolution Equations
with Poisson Jumps
https://doi.org/10.1515/msds-2018-0005
Received August 21, 2017; accepted April 7, 2018
Abstract: This paper focuses on a nonlinear second-order stochastic evolution equations driven by a frac-
tional Brownian motion (fBm) with Poisson jumps and which is dependent upon a family of probability mea-
sures. The global existence of mild solutions is established under various growth conditions, and a related
stability result is discussed. An example is presented to illustrate the applicability of the theory.
Keywords: Stochastic evolution equation; fractional Brownian motion; second-order equation; Poisson
jumps; cosine family
MSC: 60H05; 60H15; 60H20; 35R09
1 Introduction
Second-order stochastic evolution equations have been the subject of numerous papers over the past sev-
eral decades (See [20, 23, 24, 26, 27, 29]). In this paper, we extend existing work in this area by considering
a stochastic second-order evolution equations driven by a fractional Brownian motion (fBm) in which the
forcing terms are dependent on the probability law of the state function and in which Poisson jumps are
present. Specifically, let D be a bounded domain in R
N
with smooth boundary ∂D . Consider the following
initial boundary value problem:
∂
∂x(t , z)
∂t
+
∑
n
j , k=1
∂
∂z
j
a
jk
(·)
∂x(t , z)
∂z
j
∂t + C(z)x(t , z)∂t + B
∂x(t , z)
∂t
∂t
=
F
1
2
(
t , z , x(t , z)
)
+
L
2
(D)
F
2
2
(t , z , y)µ(t , z)(dy)
∂t + F
3
(t , z)dβ
h
(t)
+
Z
x(t−, z)u
˜
N(dt , du) a.e. on (0, T) × D
x(0, z)= ξ
1
(z), a.e. on D
∂x(0, z)
∂t
= ξ
2
(z) a.e. on D
x(t , z) = 0, a.e. on (0, T) × ∂D
(1.1)
where z = 〈z
1
, ... , z
n
〉∈ D ; x : [0, T] × D → R, F
1
2
: [0, T] × D × R → R; F
2
2
: [0, T] × D × L
2
(D ) → L
2
(D );
F
3
: [0, T] × D → L(R, L
2
(D )); a
jk
: D → R(1 ≤ j , k ≤ n); C : D → R; B : L
2
(D ) → L
2
(D ); and µ(t , ·) ∈
℘
λ
2 (L
2
(D )) is the probability law of x(t , ·). In addition, {β
h
(t):0 ≤ t ≤ T} is a real fBm and {p(t): t ∈ [0, T]} is
a Poisson point process, independent of β
h
(t), taking values in [0, ∞) with a σ-finite characteristic measure
*Corresponding Author: Mark A. McKibben: Department of Mathematics, West Chester University, 25 University Avenue,
West Chester, PA, 19383, E-mail: mmckibben@wcupa.edu
Micah Webster: Center for Data, Mathematical, and Computational Sciences, Goucher College, 1021 Dulaney Valley Road,
Baltimore, MD, 21286, E-mail: micah.webster@goucher.edu