We prove that a family of linear bounded evolution operators (G(t, s)) t≥s∈I can be associated, in the space of vector-valued bounded and continuous functions, to a class of systems of elliptic operators A with unbounded coefficients defined in I × R d (where I is a right-halfline or I = R) all having the same principal part. We establish some continuity and representation properties of (G(t, s)) t≥s∈I and a sufficient condition for the evolution operator to be compact in C b (R d ; R m ). We prove also a uniform weighted gradient estimate and some of its more relevant consequence.2000 Mathematics Subject Classification. 35K45; 35K58, 47B07, 60H10, 91A15.Remark 2.4. (i) Hypothesis 2.2(i) can be replaced with the weaker requirement that K η,ε is bounded from below in J × R d , uniformly with respect to η ∈ ∂B 1 , for any bounded interval J ⊂ I. Indeed, in this case, for any J as above, let c J > 0 be such that K η,ε ≥ −c J in J × R d for any η ∈ ∂B 1 . The change of unknowns v(t, x) := e −cJ (t−s)/4 u(t, x) transforms the elliptic operator A into the operator A − c J /4, which satisfies Hypothesis 2.2(i) and, clearly, the uniqueness of v is equivalent to the uniqueness of u. (ii) In the scalar case when the elliptic operator in (1.1) is A = Tr(QD 2 ) + b, ∇ + c and c is bounded from above (otherwise, Proposition 2.5 fails in general), taking ε = 1 and κ = c, one easily realizes that Hypothesis 2.2(i) is trivially satisfied. Moreover, Hypothesis (2.2)(ii) reduces to require the existence of a Lyapunov function for the operator A+ c, for any bounded interval J ⊂ I. This condition seems to be much more general than that typically assumed
In this paper we consider nonautonomous elliptic operators A with nontrivial potential term defined in I × R d , where I is a right-halfline (possibly I = R). We prove that we can associate an evolution operator (G(t, s)) with A in the space of all bounded and continuous functions on R d . We also study the compactness properties of the operator G(t, s). Finally, we provide sufficient conditions guaranteeing that each operator G(t, s) preserves the usual L p -spaces and C 0 (R d ).
We consider a class of nonautonomous parabolic first-order coupled systems in the Lebesgue space +∞). Sufficient conditions for the associated evolution operator G(t, s) in C b (R d ; R m ) to extend to a strongly continuous operator in L p (R d ; R m ) are given. Some L p -L q estimates are also established together with L p gradient estimates.2000 Mathematics Subject Classification. 35K45, 47D06.
Let X be a separable Hilbert space endowed with a non-degenerate centred Gaussian measure γ and let λ 1 be the maximum eigenvalue of the covariance operator associated with γ. The associated Cameron-Martin space is denoted by H. For a sufficiently regular convex function U : X → R and a convex set Ω ⊆ X, we set ν := e −U γ and we consider the semigroup (T Ω (t)) t≥0 generated by the self-adjoint operator defined via the quadratic formwhere ϕ, ψ belong to D 1,2 (Ω, ν), the Sobolev space defined as the domain of the closure in L 2 (Ω, ν) of D H , the gradient operator along the directions of H.A suitable approximation procedure allows us to prove some pointwise gradient estimates for (T Ω (t)) t≥0 . In particular, we show thatfor any p ∈ [1, +∞) and f ∈ D 1,p (Ω, ν). We deduce some relevant consequences of the previous estimate, such as the logarithmic Sobolev inequality and the Poincaré inequality in Ω for the measure ν and some improving summability properties for (T Ω (t)) t≥0 . In addition we prove that if f belongs to L p (Ω, ν) for some p ∈ (1, ∞), thenwhere Kp is a positive constant depending only on p. Finally we investigate on the asymptotic behaviour of the semigroup (T Ω (t)) t≥0 as t goes to infinity. GRADIENT ESTIMATES ON INFINITE DIMENSIONAL CONVEX DOMAINS3 therein) and coupling methods (see for example [15,16,39]). On the other hand, in infinite dimensional Wiener spaces some partial results are also available. In the case of a Gaussian measure γ and Ω = X, the classical Mehler's representation formulawhere the equality has to be meant componentwise, (see [10, Proposition 1.5.6]). Again for the Gaussian measure γ on a convex subset Ω, in [11, Theorem 3.1] it is proved that |D H T (t)f | H ≤ e −t T (t)|D H f | H for any smooth function f . In this case, the idea consists in approximating the parabolic problem with a sequence of finite dimensional parabolic problems and using the factorisation of the Gaussian measure. Clearly, this approach does not work in our case since our measure in general does not decompose as a product of measures on orthogonal subspaces. Finally, the case of a weighted Gaussian measure is also considered in [20] where a version of (3) is proved when Ω = X and the H-derivative is replaced by the Fréchet one. We point out that, in this latter case, the proof of the gradient estimate is based on purely stochastic techniques. Hence, taking account of the existing literature, estimate (3) represents a generalisation of all the above results and the purely analytical proof we proposed, inspired by an idea due to Bakry andÉmery (see [4] and [38]), is a novelty in the proofs of gradient estimates.As announced, the pointwise gradient estimate (3) has several interesting consequences. First of all it yields that the semigroup T Ω (t) is smoothing, in the sense that it is bounded from L p (Ω, ν) into D 1,p (Ω, ν), for any p ∈ (1, ∞) and t > 0 as the estimatereveals. Due to the fact that the Sobolev embedding theorems fail to hold when we replace the Lebesgue measure with another general m...
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