1,721,065 research outputs found
The fundamental solution for a degenerate parabolic Dirichlet problem
No full textWe study a homogeneous parabolic Dirichlet problem involving a possibly degenerate Ornstein-Uhlenbeck operator in a half-space H+ of Rn. We find an explicit formula for the fundamental solution. Under the Hörmander condition of hypoellipticity, we prove a global regularity result in spaces of continuous and bounded functions. We extend our explicit formula to the infinite-dimensional setting
Stochastic flow for SDEs with jumps and irregular drift term
Full text linkAbstract. We consider non-degenerate SDEs with a β-Hölder continuous and bounded drift
term and driven by a Lévy noise L which is of α-stable type. If β > 1 − α2 and α ∈ [1, 2), we
show pathwise uniqueness and existence of a stochastic flow. We follow the approach of [Priola,
Osaka J. Math. 2012] improving the assumptions on the noise L. In our previous paper L was
assumed to be non-degenerate, α-stable and symmetric. Here we can also recover relativistic and
truncated stable processes and some classes of tempered stable processes
On weak uniqueness for some degenerate SDEs by global Lp estimates
No full textWe prove uniqueness in law for possibly degenerate SDEs having a linear part in the drift term. Diffusion coefficients corresponding to non-degenerate directions of the noise are assumed to be continuous. When the diffusion part is constant we recover the classical degenerate Ornstein-Uhlenbeck process which only has to satisfy the Hörmander hypoellipticity condition. In the proof we also use global Lp-estimates for hypoelliptic Ornstein-Uhlenbeck operators recently proved in Bramanti et al. (Math. Z. 266, 789–816 2010) and adapt the localization procedure introduced by Stroock and Varadhan. Appendix contains a quite general localization principle for martingale problems
Davie's type uniqueness for a class of SDEs with jumps
Full text linkA result of A.M. Davie [Int. Math. Res. Not. 2007] states that a multidimensional stochastic equation , , driven by a Wiener process with a coefficient which is only bounded and measurable has a unique solution for almost all choices of the driving Wiener path.
We consider a similar problem when is replaced by a L\'evy process and
is -H\"older continuous in the space variable, . We assume that
has a finite moment of order , for some . Using also a new c\`adl\`ag regularity result for strong solutions, we prove that strong existence and uniqueness for the SDE together with -Lipschitz continuity of the strong solution with respect to imply a Davie's type uniqueness result for almost all choices of the L\'evy path. We apply this result to a class of SDEs driven by non-degenerate -stable L\'evy processes,
-parabolic regularity and non-degenerate Ornstein-Uhlenbeck type operators
No full textWe prove Lp -parabolic a-priori estimates for solutions to purely second order parabolic PDEs on
on R^d+1 when the coefficients cij are time dependent locally bounded functions on R. We slightly
generalize the usual parabolicity assumption and show that still Lp -estimates hold
for the second spatial derivatives of u. We also investigate the dependence of the
constant appearing in such estimates from the parabolicity constant. The proof
requires the use of the stochastic integral when p is different from 2. Finally we
extend our estimates to parabolic equations involving non-degenerate Ornstein-
Uhlenbeck type operators
Well-posedness of semilinear stochastic wave equations with Hölder continuous coefficients
No full textWe prove that semilinear stochastic abstract wave equations, including wave and plate equations, are well-posed in the strong sense with an -H\"older continuous drift coefficient, if .
The uniqueness may fail for the corresponding
deterministic PDE and well-posedness is restored
by adding an external random forcing of
white noise type. This shows a
kind of regularization by noise
for the semilinear wave equation.
To prove the result we introduce an
approach based on backward stochastic differential equations.
We also establish regularizing properties of the transition semigroup associated to the
stochastic wave equation by using control theoretic results
On the Cauchy problem for non-local Ornstein--Uhlenbeck operators
No full textWe study the Cauchy problem involving non-local Ornstein–Uhlenbeck operators in finite and infinite dimensions. We prove classical solvability without requiring that the Lévy measure corresponding to the large jumps part has a first finite moment. Moreover, we determine a core of regular functions which is invariant for the associated transition Markov semigroup. Such a core allows to characterize the marginal laws of the Ornstein–Uhlenbeck stochastic process as unique solutions to Fokker–Planck–Kolmogorov equations for measures
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