On Regularity of Transition Probabilities and Invariant Measures of Singular Diffusions Under Minimal Conditions (original) (raw)

Regularity of Invariant Measures: The Case of Non-constant Diffusion Part

Journal of Functional Analysis, 1996

We prove regularity (i.e., smoothness) of measures + on R d satisfying the equation L*+=0 where L is an operator of type Lu=tr(Au")+B } {u. Here A is a Lipschitz continuous, uniformly elliptic matrix-valued map and B is merely +-square integrable. We also treat a class of corresponding infinite dimensional cases where R d is replaced by a locally convex topological vector space X. In this cases + is proved to be absolutely continuous w.r.t. a Gaussian measure on X and the square root of the Radon Nikodym density belongs to the Malliavin test function space D 2, 1 .

Dirichlet regularity and degenerate diffusion

Transactions of the American Mathematical Society, 2010

Let Ω ⊂ R N be an open and bounded set and let m : Ω → (0, ∞) be measurable and locally bounded. We study a natural realization of the operator m in C 0 (Ω) := u ∈ C(Ω) : u |∂Ω = 0. If Ω is Dirichlet regular, then the operator generates a positive contraction semigroup on C 0 (Ω) whenever 1 m ∈ L p loc (Ω) for some p > N 2. If m(x) does not go fast enough to 0 as x → ∂Ω, then Dirichlet regularity is necessary. However, if |m(x)| ≤ c•dist(x, ∂Ω) 2 , then we show that m 0 generates a semigroup on C 0 (Ω) without any regularity assumptions on Ω. We show that the condition for degeneration of m near the boundary is optimal.

On uniqueness of invariant measures for finite- and infinite-dimensional diffusions

Communications on Pure and Applied Mathematics, 1999

We prove uniqueness of "invariant measures", i.e., solutions to the equation L = 0 where L = + B r on R n with B satisfying some mild integrability conditions and is a probability measure on R n . This solves an open problem posed by S.R.S. Varadhan in 1980. The same conditions are shown to imply that the closure of L on L 1 ( ) generates a strongly continuous semigroup having as its unique invariant measure. The question whether an extension of L generates a strongly continuous semigroup on L 1 ( ) and whether such an extension is unique is addressed separately and answered positively under even weaker local integrability conditions on B. The special case when B is a gradient of a function (i.e., the "symmetric case") is in particular studied and conditions are identi ed ensuring that L = 0 implies that L is symmetric on L 2 ( ) resp. L = 0 has a unique solution. We also prove in nite dimensional analogues of the latter two results and a new elliptic regularity theorem for invariant measures in in nite dimensions. 1 := 0). Then the restriction of (T t ) t>0 to L r ( ) is the unique strongly continuous semigroup on L r ( ) whose generator extends L. This generator is, in fact, the closure of (L; C 1 0 (R n )) on L r ( ). 6

Invariant measures associated to degenerate elliptic operators

Indiana University Mathematics Journal, 2010

This paper is devoted to the study of the existence and uniqueness of the invariant measure associated to the transition semigroup of a diffusion process in a bounded open subset of R n . For this purpose, we investigate first the invariance of a bounded open domain with piecewise smooth boundary showing that such a property holds true under the same conditions that insure the invariance of the closure of the domain. A uniqueness result for the invariant measure is obtained in the class of all probability measures that are absolutely continuous with respect to Lebesgue's measure. A sufficient condition for the existence of such a measure is also provided.

Stochastic Invariance and Degenerate Elliptic Operators

2008

This paper is devoted to the invariance of a bounded open domain in R n under a diffusion process with Lipschitz continuous data. We show that such a property holds under the same conditions that insure the invariance of a closed domain. This result is applied to get the wellposedness of a degenerate elliptic equation without imposing explicit boundary conditions, and to study the existence and uniqueness of the invariant measure for the associated transition semigroup.

Elliptic equations for measures on infinite dimensional spaces and applications

Probability Theory and Related Fields, 2001

We introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces. This concept allows one to consider equations whose coefficients are not globally integrable. By using a suitably extended Lyapunov function technique, we derive a priori estimates for the solutions of such equations and prove new existence results. As an application, we consider stochastic Burgers, reaction-diffusion, and Navier-Stokes equations and investigate the elliptic equations for the corresponding invariant measures. Our general theorems yield a priori estimates and existence results for such elliptic equations. We also obtain moment estimates for Gibbs distributions and prove an existence result applicable to a wide class of models. J n,j q j ≤ C 1 |n/2| + 1 −r + C 2 |n/2| + 1 −r ≤ C 3 q n , since J n,j ≤ J r (1 + |n − j|) −r .

Lp Regularity of the dirichlet problem for elliptic equations with singular drift

2006

Let L 0 and L 1 be two elliptic operators in nondivergence form, with coefficients A ℓ and drift terms b ℓ , ℓ = 0, 1 satisfying sup |Y −X|≤ δ(X) 2 |A 0 (Y) − A 1 (Y)| 2 + δ (X) 2 |b 0 (Y) − b 1 (Y)| 2 δ (X) dX is a Carleson measure in a Lipschitz domain Ω ⊂ R n+1 , n ≥ 1, (here δ (X) = dist (X, ∂Ω)). If the harmonic measure dω L 0 ∈ A∞, then dω L 1 ∈ A∞. This is an analog to Theorem 2.17 in [8] for divergence form operators. As an application of this, a new approximation argument and known results we are able to extend the results in [10] for divergence form operators while obtaining totally new results for nondivergence form operators. The theorems are sharp in all cases.