## MALLIAVIN CALCULUS FOR INFINITE-DIMENSIONAL SYSTEMS WITH ADDITIVE NOISE (2007)

Citations: | 5 - 2 self |

### BibTeX

@MISC{Bakhtin07malliavincalculus,

author = {Yuri Bakhtin and Jonathan and C. Mattingly},

title = {MALLIAVIN CALCULUS FOR INFINITE-DIMENSIONAL SYSTEMS WITH ADDITIVE NOISE},

year = {2007}

}

### OpenURL

### Abstract

ABSTRACT. We consider an infinite-dimensional dynamical system with polynomial nonlinearity and additive noise given by a finite number of Wiener processes. By studying how randomness is spread by the dynamics, we develop in this setting a partial counterpart of Hörmander’s classical theory of Hypoelliptic operators. We study the distributions of finite-dimensional projections of the solutions and give conditions that provide existence and smoothness of densities of these distributions with respect to the Lebesgue measure. We also apply our results to concrete SPDEs such as a Stochastic Reaction Diffusion Equation and the Stochastic 2D Navier–Stokes System. 1.

### Citations

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(Show Context)
Citation Context ...e implied by the following standard result from Malliavin calculus (see [Nua95, p.86, Section 2.1]; it is straightforward to check that the definitions of the Malliavin derivative and matrix given in =-=[Nua95]-=- are equivalent to ours): Theorem 7.1. Suppose the following conditions are satisfied for a finite-dimensional random vector Y : i) E|D(Y )(h)| 2 ≤ ∞, for all h ∈ L 2( [0,T], R d) . ii) The Malliavin ... |

103 |
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Citation Context ...eorem 3.1 and 3.3 hold for equation (27). Proof. We begin by proving that B(Ku,u) ∈ Poly 1(V 1 ,H). To do so we use the basic facts that |B(u,v)|0 ≤ C|u|1||v|1 and that |Ku|1 = |u|0 (see for instance =-=[CF88]-=-). Then |B(Ku,v)|0 ≤ C|u|0|v|1, which proves the first result. Assumptions 1 and 2 then follow from Corollary 4.2 or from Proposition 2.1, Proposition 2.2 of [MP06]. The existence of solutions to (27)... |

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Citation Context ...ier Stokes equations under the type of finite-dimensional forcing considered in this note ( see [HM06]). The results of this paper are a major step towards proving similar results for other SPDEs. In =-=[EH01]-=-, the ergodicity of a degenerately forced SPDE was also proven using techniques from Malliavin calculus. In contrast to our setting, there infinitely many directions were forced stochastically. Howeve... |

25 |
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Citation Context ...|u|0|v|1, which proves the first result. Assumptions 1 and 2 then follow from Corollary 4.2 or from Proposition 2.1, Proposition 2.2 of [MP06]. The existence of solutions to (27) can also be found in =-=[Fla94]-=-. Assumptions 3, 4 and 5 follow from Corollary14 YURI BAKHTIN AND JONATHAN C. MATTINGLY A.2 and Lemma B.1 of [MP06]. The fact that w(t) ∈ D∞ V1 (see Section 5 for the definition) is also proved in Le... |

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Citation Context ...(x) = 0 for all f ∈ I then x = 0 or 1. We now turn to proving that the density is smooth. In the sequel, we are going to restrict ourselves to initial data in H0 = H ∩ C([0,1]). It is well known (See =-=[Cer99]-=- Proposition 3.2 or [EH01]) that for all p ≥ 1 and u0 ∈ H0 (24) E|u(t)| p p 0 ≤ C(p,t)(1 + |u0| 0 ) and (25) E sup sup t∈[0,T] x∈[0,1] |u(t,x)| p ≤ C(T,p,u0) < ∞ . 4.1.1. Verification of Assumption 4.... |

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7 |
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Citation Context ...ns satisfy Hörmander’s condition for hypoellipticity independent of the order of the truncation. And thus in some sense, Hörmander’s condition holds, at least formally, for the whole SPDE (m = ∞). In =-=[BT05]-=-, the authors treat the case when the infinite-dimensional (m = ∞) evolution generates a fully invertible flow and prove conditions guaranteeing the existence of a density of the finite-dimensional ma... |

6 |
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Citation Context ...s more straightforward to use it to obtain the quantitative estimates needed to prove smoothness. While this paper was in its final stages of completion, the authors became aware of a recent preprint =-=[AKSS]-=- where it is proven that finite-dimensional projections of a randomly forced PDE’s Markov transition kernel are absolutely continuous4 YURI BAKHTIN AND JONATHAN C. MATTINGLY with respect to Lebesgue ... |

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5 |
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Citation Context ... 12 YURI BAKHTIN AND JONATHAN C. MATTINGLY u(x,t) = ∫ Rm ρt(x,y)u0(y)dy, where ρ is a smooth, positive function of (t,x,y). The function ρt(x,y) is called the density of xt starting from x0 = x (see =-=[Bas98]-=-). The fact that ρt is smooth and positive is a direct consequence of the randomness spreading through all of the degrees of freedom. If dim span{F1, · · · ,Fd} < m, then the preceding conclusions do ... |

5 |
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Citation Context ...ts. D. Ocone [Oco88] first used related ideas in the infinite-dimensional case when the equations were linear in the solution and the noise; and hence, explicit an formula exists for the solution. In =-=[MP06]-=-, the 2D Navier-Stokes equations are considered with additive noise. The techniques used there are very close to those used here. However, there the scope is more limited. The calculations are done in... |

4 |
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Citation Context ... Malliavin Calculus in the finitedimensional setting or the infinite-dimensional extensions given in [BT05]. However we will see that we can modify the proofs to produce the desired results. D. Ocone =-=[Oco88]-=- first used related ideas in the infinite-dimensional case when the equations were linear in the solution and the noise; and hence, explicit an formula exists for the solution. In [MP06], the 2D Navie... |

3 |
Dynamics of evolutionary equations, volume 143 of Applied Mathematical Sciences
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(Show Context)
Citation Context ...T . Furthermore, as a function of s. Ks,tφ ∈ C([t0,t];H) ∩ C([t0,t];V 2 ) ∩ L 2 ([t0,t];V 1 ) Proof. Most of the results follow from results about deterministic, time inhomogeneous equations found in =-=[SY02]-=-. As is often done (see for example [Fla94, DPZ96]), we begin by setting u(t) = X(t) + GW(t). Then X(t) satisfies a standard PDE ∂ X(t) = F(X(t),t), ∂t where the random right hand side is given by F(x... |

2 |
Ergodicity of the degenerate stochstic 2D Navier–Stokes equation
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(Show Context)
Citation Context ... this paper is a critical ingredient in the recent proof of unique ergodicity of the two-dimensional Navier Stokes equations under the type of finite-dimensional forcing considered in this note ( see =-=[HM06]-=-). The results of this paper are a major step towards proving similar results for other SPDEs. In [EH01], the ergodicity of a degenerately forced SPDE was also proven using techniques from Malliavin c... |

2 |
and Étienne Pardoux. Malliavin calculus and the randomly forced Navier Stokes equation
- Mattingly
- 2003
(Show Context)
Citation Context ...OISE 3 Ocone [Oco88] first used related ideas in the infinite dimensional case when the equations were linear in the solution and the noise; and hence, explicit an formula exists for the solution. In =-=[MP06]-=-, the 2D Navier-Stokes equations are considered with additive noise. The techniques used there are very close to those used here. However, there the scope is more limited. The calculations are done in... |

1 |
Stochastic Boussinesq Equations and the infinite dimensional Malliavin Calculus
- Wu
(Show Context)
Citation Context ...close to that of an associated linear equation. The type of analysis used there does not seem to be possible in our setting. Independently, and contemporaneously to this work M. Wu completed a thesis =-=[Wu06]-=- which carried out the program from [MP06, HM06] to prove the unique ergodicity of a degenerately forced Boussinesq equation. Since this equation has a quadratic nonlinearity, he was able to use the t... |

1 |
A Shirikyan. On finite-dimentional projections of distributions for solutions of randomly forced pde’s
- Agrachev, Kuksin, et al.
- 2006
(Show Context)
Citation Context ...t is more straightforward to obtain with it the quantitative estimates needed to prove smoothness. While this paper was in its final stages of completion the authors became aware of a recent preprint =-=[AKSS]-=- where it is proven that finite dimensional projections of a randomly forced PDE’s Markov transition kernel are absolutely continuous with respect to Lebesgue measure if a certain controllability cond... |