Well-posedness for a Class of Dissipative Stochastic Evolution Equations with Wiener and Poisson Noise

Marinelli, Carlo

doi:10.1007/978-3-0348-0545-2_8

Cited by 8 publications

(7 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…x , which implies that condition (7), hence also the assumptions of theorem 3.6, are satisfied if we can find α and η such that 1 − α − η > 1/2. This is possible if m is sufficiently large, so that B ∈ γ(L 2 , L q ) with q large and d/(2q) is smaller than, say, 1/4.…”

Section: Non-degeneracy Of the Malliavin Derivativementioning

confidence: 91%

See 1 more Smart Citation

Absolute continuity of solutions to reaction-diffusion equations with multiplicative noise

Marinelli¹,

Quer-Sardanyons²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on L 2 (G), where G is an open bounded domain in R d with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function which is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces.

show abstract

Section: Non-degeneracy Of the Malliavin Derivativementioning

confidence: 91%

“…Remark 2.3. Further well-posedness results in L q spaces for semilinear parabolic SPDEs of accretive type, with more natural assumptions on the nonlinear drift term f , can be found in [7,8,10,11,12]. See also [2] for related results in spaces of continuous functions.…”

Section: Preliminariesmentioning

confidence: 99%

Absolute continuity of solutions to reaction-diffusion equations with multiplicative noise

Marinelli¹,

Quer-Sardanyons²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Similar stopping time arguments have been used by many authors including Gyöngy and Rovira for Burgers-type equations [14]. There have been other recent investigations of SPDEs on finite spatial domains and Banach-space-valued stochastic processes more generally that are exposed to nonlinearities that are not Lipschitz continuous [9,13,[18][19][20].…”

Section: Introductionmentioning

confidence: 91%

Existence and uniqueness for the mild solution of the stochastic heat equation with non-Lipschitz drift on an unbounded spatial domain

Salins¹

2020

Preprint

View full text Add to dashboard Cite

We prove the existence and uniqueness of the mild solution for a nonlinear stochastic heat equation defined on an unbounded spatial domain. The nonlinearity is not assumed to be globally, or even locally, Lipschitz continuous. Instead the nonlinearity is assumed to satisfy a one-sided Lipschitz condition. First, a strengthened version of the Kolmogorov continuity theorem is introduced to prove that the stochastic convolutions of the fundamental solution of the heat equation and a spatially homogeneous noise grow no faster than polynomially. Second, a deterministic mapping that maps the stochastic convolution to the solution of the stochastic heat equation is proven to be Lipschitz continuous on polynomially weighted spaces of continuous functions. These two ingredients enable the formulation of a Picard iteration scheme to prove the existence and uniqueness of the mild solution.

show abstract

“…Remark 2.3 Further well-posedness results in L q spaces for semilinear parabolic SPDEs of accretive type, with more natural assumptions on the nonlinear drift term f , can be found in [7,8,[10][11][12]. See also [2] for related results in spaces of continuous functions.…”

Section: Remark 22 Inmentioning

confidence: 99%

Absolute Continuity of Solutions to Reaction-Diffusion Equations with Multiplicative Noise

Marinelli

Quer-Sardanyons

2021

Potential Anal

Self Cite

View full text Add to dashboard Cite

We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on L2(G), where G is an open bounded domain in $\mathbb {R}^{d}$ ℝ d with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function that is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces.

show abstract

Well-posedness for a Class of Dissipative Stochastic Evolution Equations with Wiener and Poisson Noise

Cited by 8 publications

References 6 publications

Absolute continuity of solutions to reaction-diffusion equations with multiplicative noise

Absolute continuity of solutions to reaction-diffusion equations with multiplicative noise

Existence and uniqueness for the mild solution of the stochastic heat equation with non-Lipschitz drift on an unbounded spatial domain

Absolute Continuity of Solutions to Reaction-Diffusion Equations with Multiplicative Noise

Contact Info

Product

Resources

About