In this revised and extended version of his course notes from a 1-year-course at Scuola Normale Superiore, Pisa, the author provides an introduction- for an audience knowing basic functional analysis ...and measure theory but not necessarily probability theory - to analysis in a separable Hilbert spact of infinite dimension. Moreover, some detailes have been added as well as some new material on dynamical systems with dissipative nonlinearities and asymptotic behavior for gradient systems.
BV functions in Hilbert spaces Da Prato, Giuseppe; Lunardi, Alessandra
Mathematische Annalen,
12/2021, Letnik:
381, Številka:
3-4
Journal Article
Recenzirano
Odprti dostop
We study the basic theory of
BV
functions in a Hilbert space
X
endowed with a (not necessarily Gaussian) probability measure
ν
. We present necessary and sufficient conditions in order that a ...function
u
∈
L
p
(
X
,
ν
)
is of bounded variation. We also discuss the De Giorgi approach to
BV
functions through the behavior as
t
→
0
of
∫
X
‖
∇
T
(
t
)
u
‖
d
ν
, for a smoothing semigroup
T
(
t
). Particular attention is devoted to the case where
u
is the indicator function of a sublevel set
{
x
:
g
(
x
)
<
r
}
of a real Borel function
g
. We give several examples, for different measures
ν
such as weighted Gaussian measures, infinite products of non Gaussian measures, and invariant measures of some stochastic PDEs such as reaction-diffusion equations and Burgers equation.
We prove a new probabilistic formula for the gradient of the Dirichlet semigroup associated with a class of hypoelliptic operators in a bounded subset of
R
d
.
This paper addresses the existence and uniqueness of strong solutions to stochastic porous media equations dX - ΔΨ(X)dt = B(X)dW(t) in bounded domains of $\mathbb{R}^{d}$ with Dirichlet boundary ...conditions. Here Ψ is a maximal monotone graph in $\mathbb{R} x \mathbb{R}$ (possibly multivalued) with the domain and range all of $\mathbb{R}$ . Compared with the existing literature on stochastic porous media equations, no growth condition on Ψ is assumed and the diffusion coefficient Ψ might be multivalued and discontinuous. The latter case is encountered in stochastic models for self-organized criticality or phase transition.
The paper proves the existence of solutions to the magneto-hydrodynamics equations driven by random exterior forcing terms both in the velocity and in the magnetic field. The existence and uniqueness ...of an invariant measure is also obtained via coupling methods. PUBLICATION ABSTRACT
The existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and Wiener forcing term is proven. The finite time ...extinction of solutions with high probability is also proven in 1-
D
. The results are relevant for self-organized criticality behavior of stochastic nonlinear diffusion equations with critical states.
The 2D Euler equations with random initial condition has been investigates by Albeverio and Cruzeiro (Commun Math Phys 129:431–444,
1990
) and other authors. Here we prove existence of solutions for ...the associated continuity equation in Hilbert spaces, in a quite general class with LlogL densities with respect to the enstrophy measure.
We construct surface measures in a Hilbert space endowed with a probability measure \nu . The theory fits for invariant measures of some stochastic partial differential equations such as Burgers and ...reaction-diffusion equations. Other examples are weighted Gaussian measures and special product measures \nu of non-Gaussian measures. In any case we prove integration by parts formulae on sublevel sets of good functions (including spheres and hyperplanes) that involve surface integrals.
Mathematical analysis--surface measures in infinite dimension Prato, Giuseppe Da; Lunardi, Alessandra; Tubaro, Luciano
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni,
09/2014, Letnik:
25, Številka:
3
Journal Article
Recenzirano
We construct surface measures associated to Gaussian measures in separable Banach spaces, and we prove several properties including an integration by parts formula. KEY WORDS: Infinite dimensional ...analysis, surface measures, Gaussian measures. MATHEMATICS SUBJECT CLASSIFICATION: 28C20.
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