A version of the Hörmander–Malliavin theorem in 2-smooth Banach spaces

Shamarova, Evelina

doi:10.1142/s0219025714500040

Cited by 3 publications

(3 citation statements)

References 16 publications

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“…After we have finished the present paper, we learned that Shamarova [33] used the Malliavin calculus in 2-smooth Banach space to prove that any finite dimensional projection of the law of the solution of a SEE with coefficients satisfying the Hörmander conditions is absolutely continuous with respect to the Lebesgue measure. Although her assumptions are a little bit restrictive, it would be interesting to check whether her approach could be used in our framework to prove the strong Feller property.…”

Section: Zdzis Law Brzeźniak and Paul André Razafimandimbymentioning

confidence: 99%

“…Although her assumptions are a little bit restrictive, it would be interesting to check whether her approach could be used in our framework to prove the strong Feller property. We should note that the development of Malliavin calculus in Banach spaces is still at its infancy, however, we refer to [22], [23], [32] and [33] amongst others to some significant results that have been obtained. With the help of these results, we hope that in near future we will be able to develop and generalize the approach in [13] and [14], so that we will be able to analyze SEE with degenerate noise in Banach setting.…”

Section: Zdzis Law Brzeźniak and Paul André Razafimandimbymentioning

confidence: 99%

See 1 more Smart Citation

Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces

Brzeźniak¹,

Razafimandimby²

2016

DCDS-B

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The main goal of this paper is to generalize to Banach spaces the well-known results for diffusions on Hilbert spaces obtained in [Peszat, S. and Zabczyk, J. (1995). Strong Feller property and irreducibility for diffusions on Hilbert spaces. Ann. Probab. 23(1): 157-172.]. More precisely, we are aiming to prove the strong Feller property and irreducibility of the solutions to a stochastic evolution equations (SEEs) in Banach spaces. We give sufficient conditions on the path space and the coefficients of the SEEs for these aforementioned properties to hold. We apply our result to investigate the long-time behavior of a stochastic nonlinear heat equations on L p-space with p ą 4. Our result implies the uniqueness of the invariant measure, if it exists, for the stochastic nonlinear heat equations on L p-space with p ą 4.

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Section: Zdzis Law Brzeźniak and Paul André Razafimandimbymentioning

confidence: 99%

Section: Zdzis Law Brzeźniak and Paul André Razafimandimbymentioning

confidence: 99%

Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces

Brzeźniak¹,

Razafimandimby²

2016

DCDS-B

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“…In this case, the Jacobian becomes invertible. Shamarova [33] studies the existence of densities for a stochastic evolution equation driven by Brownian motion in 2-smooth Banach spaces. Recently, based on a pathwise Fubini theorem for rough path integrals, Gerasimovics and Hairer [17] overcome the lack of invertibility of the Jacobian for SPDEs driven by Brownian motion.…”

Section: Introductionmentioning

confidence: 99%

Existence of densities for stochastic evolution equations driven by fractional Brownian motion

Nascimento

Ohashi

2019

Preprint

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In this work, we prove a version of Hörmander's theorem for a stochastic evolution equation driven by a trace-class fractional Brownian motion with Hurst exponent 1 2 < H < 1 and an analytic semigroup on a given separable Hilbert space. In contrast to the classical finite-dimensional case, the Jacobian operator in typical solutions of parabolic stochastic PDEs is not invertible which causes a severe difficulty in expressing the Malliavin matrix in terms of an adapted process. Under Hörmander's bracket condition on the vector fields and the additional assumption that the range of the semigroup is dense, we prove the law of finite-dimensional projections of such solutions has a density w.r.t Lebesgue measure. The argument is based on rough path techniques and a suitable analysis on the Gaussian space of the fractional Brownian motion.

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Existence of densities for stochastic evolution equations driven by fractional Brownian motion

Nascimento

Ohashi

2020

Stoch. Dyn.

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In this work, we prove a version of Hörmander’s theorem for a stochastic evolution equation driven by a trace-class fractional Brownian motion with Hurst exponent [Formula: see text] and an analytic semigroup on a given separable Hilbert space. In contrast to the classical finite-dimensional case, the Jacobian operator in typical solutions of parabolic stochastic PDEs is not invertible which causes a severe difficulty in expressing the Malliavin matrix in terms of an adapted process. Under a Hörmander’s bracket condition and some algebraic constraints on the vector fields combined with the range of the semigroup, we prove that the law of finite-dimensional projections of such solutions has a density with respect to Lebesgue measure. The argument is based on rough path techniques and a suitable analysis on the Gaussian space of the fractional Brownian motion.

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A version of the Hörmander–Malliavin theorem in 2-smooth Banach spaces

Cited by 3 publications

References 16 publications

Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces

Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces

Existence of densities for stochastic evolution equations driven by fractional Brownian motion

Existence of densities for stochastic evolution equations driven by fractional Brownian motion

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