2016
DOI: 10.1007/s10959-015-0662-4
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Lévy-driven Volterra Equations in Space and Time

Abstract: We investigate nonlinear stochastic Volterra equations in space and time that are driven by Lévy bases. Under a Lipschitz condition on the nonlinear term, we give existence and uniqueness criteria in weighted function spaces that depend on integrability properties of the kernel and the characteristics of the Lévy basis. Particular attention is devoted to equations with stationary solutions, or more generally, to equations with infinite memory, that is, where the time domain of integration starts at minus infin… Show more

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Cited by 21 publications
(43 citation statements)
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References 33 publications
(105 reference statements)
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“…Indeed, because σ(Y 0 (t 0 , x 0 )) = 0, there exists by (1.1), which together contradict (3.1)). Therefore, we have E[|σ(Y (t 1 , x 1 ))|] > 0, and because the solution to (1.1) is L r -continuous on (0, ∞) × R d for all r < 1 + 2/d (see [10,Theorem 4.7(2)]), there exist ǫ, δ > 0 such that Y (s, y))|] > ǫ. (5.31) (1) We only carry out the proof for c = 0; the arguments are similar for c > 0 and we leave the details to the reader.…”
Section: Proof Of Theorem 35mentioning
confidence: 99%
“…Indeed, because σ(Y 0 (t 0 , x 0 )) = 0, there exists by (1.1), which together contradict (3.1)). Therefore, we have E[|σ(Y (t 1 , x 1 ))|] > 0, and because the solution to (1.1) is L r -continuous on (0, ∞) × R d for all r < 1 + 2/d (see [10,Theorem 4.7(2)]), there exist ǫ, δ > 0 such that Y (s, y))|] > ǫ. (5.31) (1) We only carry out the proof for c = 0; the arguments are similar for c > 0 and we leave the details to the reader.…”
Section: Proof Of Theorem 35mentioning
confidence: 99%
“…Next, by [3, (B.5)], we know that G(t; x, y) Cg(t, x − y) for any (t, x, y) ∈ [0, T ] × [0, π] 2 , with g as in (1.4). Consequently, (1) to (4) of Assumption B of [11] are satisfied, and we can apply [11,Theorem 3.5] to obtain the existence of a unique mild solution to (1.1) satisfying (2.3) for all p ∈ (0, 2]. In order to extend this to all p ∈ (2, 3), we notice that the only step in the proof of [11,Theorem 3.5] that uses p 2 is the moment estimate (6.9) given in [11, Lemma 6.1(2)] with respect to the martingale part L M .…”
Section: The Stochastic Heat Equation On An Intervalmentioning
confidence: 99%
“…A predictable random field u = (u(t, x) : (t, x) ∈ [0, T ] × D) is called a mild solution to (1.1) if for all (t, x) ∈ [0, T ] × D, is the solution to the homogeneous version of (1.1). In (1.2) and (1.3), G D denotes the Green's function of the heat operator on D, which for D = R d equals the Gaussian density (1.4) g(t, x) = (4πt) − d 2 e − |x| 2 4t 1 t 0 (when t = 0, we interpret g(0, x) as the Dirac delta function δ 0 (x)), while on a bounded domain D with smooth boundary it has the spectral representation and uniqueness of solutions for equations like (1.1) with Lévy noise have been investigated in [1,2,11,12,31,34]. Already in the linear case with σ(x) ≡ 1, due to the singularity of the Green's kernel on the diagonal x = y near t = 0, each jump of the noise creates a Dirac mass for the solution.…”
Section: Introductionmentioning
confidence: 99%
“…While the papers by Albeverio et al (1998) and Applebaum and Wu (2000) remain in the L 2 -framework of Walsh (1986), Saint Loubert Bié (1998 is one of the first to treat Lévydriven stochastic PDEs in L p -spaces with p < 2. The results are extended in Chong (2016) to Volterra equations with Lévy noise, on finite as well as on infinite time domains. In both Saint Loubert Bié (1998) and Chong (2016), one crucial assumption the Lévy noise has to meet is that its Lévy measure, say λ, must satisfy R |z| p λ(dz) < ∞ (1.4) for some p < 1 + 2/d.…”
Section: Introductionmentioning
confidence: 99%