On Solvability of Integro-Differential Equations
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On Solvability of Integro-Differential Equations 1 · Istvan 2 ´ ´ Gyongy ¨ Marta De Leon-Contreras
· Sizhou Wu3
Received: 14 August 2019 / Accepted: 7 July 2020 / © The Author(s) 2020
Abstract A class of (possibly) degenerate integro-differential equations of parabolic type is considered, which includes the Kolmogorov equations for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential spaces and in SobolevSlobodeckij spaces. Generalisations to stochastic integro-differential equations, arising in filtering theory of jump diffusions, will be given in a forthcoming paper. Keywords Integro-differential equation · Bessel potential spaces · Interpolation couples Mathematics Subject Classification (2010) Primary 45K05 · 35R09; Secondary 47G20
1 Introduction We consider the equation ∂ u(t, x) = Au(t, x) + f (t, x) ∂t
(1.1)
on HT = [0, T ] × Rd for a given T > 0, with initial condition u(0, x) = ψ(x) for x ∈ Rd , where A is an integro-differential operator of the form A = L + M + N + R, with a “zero-order” linear operator R, a second order differential operator
L(t) = a ij (t, x)Dij + bi (t, x)Di + c(t, x)
Istv´an Gy¨ongy
[email protected] Marta De Le´on-Contreras [email protected] Sizhou Wu [email protected] 1
Departamento de Matem´aticas, Facultad de Ciencias, Universidad Aut´onoma de Madrid, Madrid, Spain
2
School of Mathematics and Maxwell Institute, University of Edinburgh, Scotland, UK
3
School of Mathematics, University of Edinburgh, King’s Buildings, Edinburgh, EH9 3JZ, UK
M. De Le´on-Contreras et al.
and nonlocal linear operators M and N defined by M(t)ϕ(x) = (ϕ(x + ηt,z (x)) − ϕ(x) − ηt,z (x)∇ϕ(x))μ(dz), Z
(1.2)
N (t)ϕ(x) =
(ϕ(x + ξt,z (x)) − ϕ(x))ν(dz)
(1.3)
Z
for a suitable class of real-valued functions ϕ(x) on Rd . Here a ij , bi and c are real-valued bounded functions defined on HT , μ and ν are σ -finite measures on a measurable space (Z, Z ). The functions η and ξ are Rd -valued mapping defined on HT × Z. Under “zeroorder operators” we mean bounded linear operators R mapping the Sobolev spaces Wpk into themselves for k = 0, 1, 2, .., n for some n. Examples include integral operators R(t) defined by
R(t)ϕ(x) =
ϕ(x + ζt,z (x))λ(dz)
(1.4)
Z
with appropriate functions ζ on HT × Z and finite measures λ on Z . Our aim is to investigate the solvability of Eq. 1.1 in Bessel potential spaces Hpm and Sobolev-Slobodeckij spaces Wpm for p ≥ 2 and m ∈ [1, ∞). Such kind of equations arise, for example, as Kolmogorov equations for Markov processes given by stochastic differential equations, driven by Wiener processes and Poisson random measures, see e.g., [1, 2, 12, 13] and [17]. They play important roles in studying random phenomena modelled by Markov processes with jumps, in physics, biology, engineering and finance, see e.g., [3, 8, 33, 38] and the references therein. There is a huge literature on the solvability of these equations, but in most of the publications some kind of non-degeneracy conditions on the equations, or specific assumption
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