1,720,970 research outputs found
Regularizing Properties of (Non-Gaussian) Transition Semigroups in Hilbert Spaces
No full textLet X be a separable Hilbert space with norm ∥ ⋅ ∥ and let T > 0. Let Q be a linear, self-adjoint, positive, trace class operator on X, let F: X→ X be a (smooth enough) function and let W(t) be a X-valued cylindrical Wiener process. For α ∈ [0, 1/2] we consider the operator A: = − (1 / 2) Q2α−1: Q1−2α(X) ⊆ X→ X. We are interested in the mild solution X(t, x) of the semilinear stochastic partial differential equation{dX(t,x)=(AX(t,x)+F(X(t,x)))dt+QαdW(t),t∈(0,T];X(0,x)=x∈X, and in its associated transition semigroupP(t)φ(x):=E[φ(X(t,x))],φ∈Bb(X),t∈[0,T],x∈X; where Bb(X) is the space of the bounded and Borel measurable functions. We will show that under suitable hypotheses on Q and F, P(t) enjoys regularizing properties, along a continuously embedded subspace of X. More precisely there exists K := K(F, T) > 0 such that for every φ∈ Bb(X) , x∈ X, t ∈ (0, T] and h∈ Qα(X) it holds| P(t) φ(x+ h) − P(t) φ(x) | ≤ Kt− 1 / 2∥ Q−αh∥
Lp-Lq estimates for transition semigroups associated to dissipative stochastic systems
No full textIn a separable Hilbert space, we study supercontractivity and ultracontractivity properties for a transition semigroup associated with a stochastic partial differential equation. This is done in terms of exponential integrability of Lipschitz functions and some logarithmic Sobolev-type inequalities with respect to invariant measures. The abstract characterization results concerning the improvement of summability can be applied to transition semigroups associated to stochastic reaction-diffusion equations. (c) 2025 The Author(s). Published by Elsevier Masson SAS. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/)
On generators of transition semigroups associated to semilinear stochastic partial differential equations
Full text linkLet be a real separable Hilbert space. Let be a linear,
self-adjoint, positive, trace class operator on , let
be a (smooth enough) function and let
be a -valued cylindrical Wiener process. For
we consider the operator
.
We are interested in the mild solution of the semilinear stochastic
partial differential equation \begin{gather} \left\{\begin{array}{ll}
dX(t,x)=\big(AX(t,x)+F(X(t,x))\big)dt+ Q^{\alpha}dW(t), & t>0;\\ X(0,x)=x\in
\mathcal{X}, \end{array} \right. \end{gather} and in its associated transition
semigroup \begin{align} P(t)\varphi(x):=E[\varphi(X(t,x))], \qquad \varphi\in
B_b(\mathcal{X}),\ t\geq 0,\ x\in \mathcal{X}; \end{align} where
is the space of the real-valued, bounded and Borel
measurable functions on . In this paper we study the behavior of
the semigroup in the space , where is the
unique invariant probability measure of \eqref{Tropical}, when is
dissipative and has polynomial growth. Then we prove the logarithmic Sobolev
and the Poincar\'e inequalities and we study the maximal Sobolev regularity for
the stationary equation where is the infinitesimal generator of in
L2 -theory for transition semigroups associated to dissipative systems
No full textLet X be a real separable Hilbert space. Let C be a linear, bounded, non-negative self-adjoint operator on X and let A be the infinitesimal generator of a strongly continuous semigroup in X. Let { W(t) } t≥ be a X-valued cylindrical Wiener process on a filtered (normal) probability space (Ω,F,{Ft}t≥0,P). Let F: Dom (F) ⊆ X→ X be a smooth enough function. We are interested in the generalized mild solution { X(t, x) } t≥ of the semilinear stochastic partial differential equation {dX(t,x)=(AX(t,x)+F(X(t,x)))dt+CdW(t),t>0;X(0,x)=x∈X.We consider the transition semigroup defined by P(t)φ(x):=E[φ(X(t,x))],φ∈Bb(X),t≥0,x∈X.If O is an open set of X, we consider the Dirichlet semigroup defined by PO(t)φ(x):=E[φ(X(t,x))I{ω∈Ω:τx(ω)>t}],φ∈Bb(O),x∈O,t>0where τx is the exit time defined by τx=inf{s>0:X(s,x)∈Oc}.We study the infinitesimal generator of P(t), PO(t) in L2(X, ν) , L2(O, ν) respectively, where ν is the unique invariant measure of P(t)
Log-Sobolev inequalities and hypercontractivity for Ornstein – Uhlenbeck evolution operators in infinite dimension
No full textIn an infinite-dimensional separable Hilbert space X, we study the realizations of Ornstein–Uhlenbeck evolution operators Ps,t in the spaces Lp(X,γt), {γt}t∈R being a suitable evolution system of measures for Ps,t. We prove hypercontractivity results, relying on suitable Log-Sobolev estimates. Among the examples, we consider the transition evolution operator associated with a non-autonomous stochastic parabolic PDE
Schauder estimates for stationary and evolution equations associated to stochastic reaction-diffusion equations driven by colored noise
No full textWe consider stochastic reaction-diffusion equations with colored noise on the space of real-valued and continuous functions on a compact subset of Double-struck capital Rd for d=1,2,3. We prove Schauder-type estimates, which will depend on the color of the noise, for the stationary and evolution problems associated with the corresponding transition semigroup
Schauder regularity results in separable Hilbert spaces
No full textWe prove Schauder type estimates for solutions of stationary and evolution equations driven by weak generators of transition semigroups associated to a semilinear stochastic partial differential equations with values in a separable Hilbert space
Differentiability in infinite dimension and the Malliavin calculus
Full text linkIn this paper we study two notions of differentiability introduced by P. Cannarsa and G. Da Prato (see [28]) and L. Gross (see [56]) in both the framework of infinite dimensional analysis and the framework of Malliavin calculus
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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