Results 1  10
of
13
Elliptic and parabolic secondorder PDEs with growing coefficients
"... Abstract. We consider a secondorder parabolic equation in R d+1 with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally Hölder continuous in the space variables. We show that global Schauder estimates hold even in this ca ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
Abstract. We consider a secondorder parabolic equation in R d+1 with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally Hölder continuous in the space variables. We show that global Schauder estimates hold even in this case. The proof introduces a new localization procedure. Our results show that the constant appearing in the classical Schauder estimates is in fact independent of the L∞norms of the lower order coefficients. We also give a proof of uniqueness which is of independent interest even in the case of bounded coefficients.
Discreteness of the Spectrum for Some Differential Operators With Unbounded Coefficients in R^n
"... We give sufficient conditions for the discreteness of the spectrum of differential operators of the form Au = \Gamma\Deltau +hrF;rui in L 2 (R n ) where d(x) = e \GammaF (x) dx and for Schrödinger operators in L 2 (R n ). Our conditions are also necessary in the case of polynomial coeffic ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
We give sufficient conditions for the discreteness of the spectrum of differential operators of the form Au = \Gamma\Deltau +hrF;rui in L 2 (R n ) where d(x) = e \GammaF (x) dx and for Schrödinger operators in L 2 (R n ). Our conditions are also necessary in the case of polynomial coefficients.
Pathwise uniqueness for singular SDE’s driven by stable processes
"... We prove pathwise uniqueness for stochastic differential equations driven by nondegenerate symmetric αstable Lévy processes with values in R d having a bounded and βHölder continuous drift term. We assume β> 1 − α/2 and α ∈ [1, 2). The proof requires analytic regularity results for the associa ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We prove pathwise uniqueness for stochastic differential equations driven by nondegenerate symmetric αstable Lévy processes with values in R d having a bounded and βHölder continuous drift term. We assume β> 1 − α/2 and α ∈ [1, 2). The proof requires analytic regularity results for the associated integrodifferential operators of Kolmogorov type. We also study differentiability of solutions with respect to initial conditions and the homeomorphism property.
NORM DISCONTINUITY AND SPECTRAL PROPERTIES OF ORNSTEINUHLENBECK SEMIGROUPS
, 2005
"... Let E be a real Banach space. We study the OrnsteinUhlenbeck semigroup P = {P(t)}t≥0 associated with the OrnsteinUhlenbeck operator Lf(x) = 1 2 Tr QD2 f(x) + 〈Ax, Df(x)〉, x ∈ E. Here Q ∈ L (E ∗ , E) is a positive symmetric operator and A is the generator of a C0semigroup S = {S(t)}t≥0 on E. Unde ..."
Abstract
 Add to MetaCart
Let E be a real Banach space. We study the OrnsteinUhlenbeck semigroup P = {P(t)}t≥0 associated with the OrnsteinUhlenbeck operator Lf(x) = 1 2 Tr QD2 f(x) + 〈Ax, Df(x)〉, x ∈ E. Here Q ∈ L (E ∗ , E) is a positive symmetric operator and A is the generator of a C0semigroup S = {S(t)}t≥0 on E. Under the assumption that P admits an invariant measure µ ∞ we prove that if S is eventually compact and the spectrum of its generator is nonempty, then ‖P(t) − P(s) ‖ L(L 1 (E,µ∞)) = 2 for all t, s ≥ 0 with t ̸ = s. This result is new even when E = R n. We also study the behaviour of P in the space BUC(E). We show that if A ̸ = 0 there exists t0> 0 such that ‖P(t) − P(s) ‖ L(BUC(E)) = 2 for all 0 ≤ t, s ≤ t0 with t ̸ = s. Moreover, under a nondegeneracy assumption or a strong Feller assumption, the following dichotomy holds: either ‖P(t) − P(s) ‖ L(BUC(E)) = 2 for all t, s ≥ 0, t ̸ = s, or S is the direct sum of a nilpotent semigroup and a finitedimensional periodic semigroup. Finally we investigate the spectrum of L in the spaces L 1 (E, µ∞) and BUC(E).
Global L p estimates for degenerate OrnsteinUhlenbeck operators
, 2009
"... We consider a class of degenerate OrnsteinUhlenbeck operators in R N, of the kind p0X A aij @ 2 NX xix + j i;j=1 i;j=1 ..."
Abstract
 Add to MetaCart
We consider a class of degenerate OrnsteinUhlenbeck operators in R N, of the kind p0X A aij @ 2 NX xix + j i;j=1 i;j=1
The Domain of the OrnsteinUhlenbeck Operator on an L^pSpace with Invariant Measure
, 2001
"... We show that the domain of the OrnsteinUhlenbeck operator on ; dx) equals the weighted Sobolev space W ; dx), where dx is the corresponding invariant measure. Our approach relies on the operator sum method, namely the commutative and the non commutative DoreVenni theorems. ..."
Abstract
 Add to MetaCart
We show that the domain of the OrnsteinUhlenbeck operator on ; dx) equals the weighted Sobolev space W ; dx), where dx is the corresponding invariant measure. Our approach relies on the operator sum method, namely the commutative and the non commutative DoreVenni theorems.
Global L^p estimates for degenerate . . .
, 2011
"... We present a new approach to prove global L p estimates for degenerate OrnsteinUhlenbeck operators in R N. We then show how to pave the way to extend such a technique to classes of general Hörmander operators. Several historical notes related to CalderónZygmund’s singular integrals theory in Eucl ..."
Abstract
 Add to MetaCart
We present a new approach to prove global L p estimates for degenerate OrnsteinUhlenbeck operators in R N. We then show how to pave the way to extend such a technique to classes of general Hörmander operators. Several historical notes related to CalderónZygmund’s singular integrals theory in Euclidean and in nonEuclidean settings are also provided.
Smoothing properties of stochastic equations in Hilbert spaces
"... A nonlinear stochastic equation in a Hilbert space is considered, with constant but possibly degenerate diffusion term. Some smoothing properties for the associated transition semigroup are studied. In particular, strong Feller property and irreducibility are proved. The main tools are Malliavin cal ..."
Abstract
 Add to MetaCart
A nonlinear stochastic equation in a Hilbert space is considered, with constant but possibly degenerate diffusion term. Some smoothing properties for the associated transition semigroup are studied. In particular, strong Feller property and irreducibility are proved. The main tools are Malliavin calculus and Girsanov transformation. 1 Introduction In a Hilbert space H, let X(t; x); t 0, be the solution of the stochastic differential equation ae dX(t) = AX(t)dt + F (X(t))dt + Q 1=2 dW (t); t 0; X(0) = x 2 H; (1.1) where A; Q are linear operators in H, F : H ! H, and W is a Wiener process. For bounded measurable OE : H ! R, let us define the transition semigroup P t by (P t OE)(x) = E [OE(X(t; x))] ; x 2 H; t 0: We are concerned with regularity properties of the function P t OE. One of our main aims is to show that, under appropriate assumptions, the Fr'echet derivative (P t OE) 0 satisfies the estimate k(P t OE) 0 (x)k C t sup z2H jOE(z)j; x 2 H; t ? 0; (1.2) for a co...
L^pRegularity for Elliptic Operators with Unbounded Coefficients
, 2002
"... Under suitable conditions on the functions a 2 C N 2 ), F 2 C ), and V : R [0; 1), we show that the operator Au = r(aru) + F ru V u with domain W V (R ) = fu 2 ) : V u 2 L )g generates a positive analytic semigroup on L ), 1 < p < 1. ..."
Abstract
 Add to MetaCart
Under suitable conditions on the functions a 2 C N 2 ), F 2 C ), and V : R [0; 1), we show that the operator Au = r(aru) + F ru V u with domain W V (R ) = fu 2 ) : V u 2 L )g generates a positive analytic semigroup on L ), 1 < p < 1.