$$L^2$$-theory for transition semigroups associated to dissipative systems

Schauder estimates for stationary and evolution equations associated to stochastic reaction-diffusion equations driven by colored noise

“…We conclude this introduction with some observations. At the cost of some technical modifications in the proofs, we belive to be possible to prove the same results contained in this paper in the more abstract setting considered in [1,7,36]. But this goes beyond the scope of the present paper.…”

Schauder regularity results in separable Hilbert spaces

“…For every x ∈ X, by Hypotheses 2.1 and [23, Theorem 7.5], (1.1) has a unique mild solution {X(t, x)} t≥0 . Recall that the map x → X(•, x) is Lipschitz continuous uniformly with respect to the time variable (see [6,Proposition 3.13] or [47,Proposition 3.7] for a proof). More precisely, for every T > 0, there exists a positive constant η = η(T ) such that sup…”

“…If {δ h 1 (t, x)} t≥0 , {δ h,k 2 (t, x)} t≥0 and {δ h,k,j 3 (t, x)} t≥0 are equal to zero, then (4.5), (4.6) and (4.7) are obvious, so we can fix t > 0, x ∈ X and h, k, j ∈ H R such that the processes {δ h 1 (t, x)} t≥0 , {δ h,k 2 (t, x)} t≥0 and {δ h,k,j 3 (t, x)} t≥0 are non-zero. We assume that the processes {δ h 1 (t, x)} t≥0 , {δ h,k 2 (t, x)} t≥0 and {δ h,k,j 3 (t, x)} t≥0 are strict solutions of (4.2), (4.3) and (4.4) respectively, otherwise we proceed as in [6] or [13, Proposition 6.2.2] approximating {δ h 1 (t, x)} t≥0 , {δ h,k 2 (t, x)} t≥0 and {δ h,k,j 3 (t, x)} t≥0 by means of sequences of more regular processes. We start by proving (4.5).…”

Equivalence of Sobolev norms with respect to weighted Gaussian measures

“…2 u := g, (A. 8) where W 1,2 (X, ν ε ) is the domain of the closure of D H : F C ∞ b (X) → L 2 (X, ν ε ; H) in L 2 (X, ν ε ). We denote by (T ε ) t≥0 the analytic symmetric strongly continuous semigroup of contractions generated by L ε on L 2 (X, ν ε ).…”

“…The interest in weighted Gaussian measures ν in infinite dimension increases in last years, since in general they represent a class of infinite dimensional measures which are not decomposable along directions of H. Further, by means of the Malliavin derivative D H , on L p (X, ν) it is possible to define a strongly continuous semigroup (T p (t)) t≥0 whose infinitesimal generator L p is a perturbation of the Ornstein-Uhlenbeck operator. Features of (T p (t)) t≥0 and of L p , such as maximal Sobolev regularity for solution to the elliptic problem λu − L p u = f , with λ > 0 and f ∈ L 2 (X, ν), smoothness properties of the semigroup (T (t)) t≥0 and some functional inequalities are investigated both from an analytic and a probabilistic point of view also in more general setting, see [2,1,5,4,8,9,10,13,14,16,17]. We also refer to [20] for an in-depth analysis of the Sobolev spaces W k,p (X, ν).…”