Bounded variation approximation of LpL_pLp​ dyadic martingales and solutions to elliptic equations (original) (raw)

On the trace operator for functions of bounded 𝔸-variation

Analysis & PDE, 2020

In this paper, we consider the space BV A (Ω) of functions of bounded A-variation. For a given first order linear homogeneous differential operator with constant coefficients A, this is the space of L 1-functions u : Ω → R N such that the distributional differential expression Au is a finite (vectorial) Radon measure. We show that for Lipschitz domains Ω ⊂ R n , BV A (Ω)-functions have an L 1 (∂Ω)-trace if and only if A is C-elliptic (or, equivalently, if the kernel of A is finite dimensional). The existence of an L 1 (∂Ω)-trace was previously only known for the special cases that Au coincides either with the full or the symmetric gradient of the function u (and hence covered the special cases BV or BD). As a main novelty, we do not use the fundamental theorem of calculus to construct the trace operator (an approach which is only available in the BV-and BD-setting) but rather compare projections onto the nullspace as we approach the boundary. As a sample application, we study the Dirichlet problem for quasiconvex variational functionals with linear growth depending on Au.

On quadratic variation of martingales

Proceedings - Mathematical Sciences, 2014

We give a construction of an explicit mapping : D([0, ∞), R) → D([0, ∞), R), where D([0, ∞), R) denotes the class of real valued r.c.l.l. functions on [0, ∞) such that for a locally square integrable martingale (M t) with r.c.l.l. paths, (M.(ω)) = A.(ω) gives the quadratic variation process (written usually as [M, M] t) of (M t). We also show that this process (A t) is the unique increasing process (B t) such that M 2 t − B t is a local martingale, B 0 = 0 and P((B) t = [(M) t ] 2 , 0 < t < ∞) = 1. Apart from elementary properties of martingales, the only result used is the Doob's maximal inequality. This result can be the starting point of the development of the stochastic integral with respect to r.c.l.l. martingales.

The p-Variation and an Extension of the Class of Semimartingales

2003

This is an overview of results concerning the class of càdlàg stochastic processes X such that for t 0, X(t) = X(0) + M(t) + A(t), where M is a local martingale and A is an adapted, càdlàg stochastic process for some p < 2 having almost all sample functions of bounded p-variation.

Approximate and exact extensions of Lebesgue boundary functions

2014

We prove continuity and surjectivity of the trace map onto L p (R n), from a space of functions of locally bounded variation, defined by the Carleson functional. The extension map is constructed through a stopping time argument. This extends earlier work by Varopoulos in the BMO case, related to the Corona theorem. We also prove L p Carleson approximability results for solutions to elliptic non-smooth divergence form equations, which generalize results in the case p = ∞ by Hofmann, Kenig, Mayboroda and Pipher.

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 .

Criteria for the finiteness of the strong p-variation for Lévy-type processes

Stochastic Analysis and Applications

Using generalized Blumenthal-Getoor indices, we obtain criteria for the finiteness of the p-variation of Lévy-type processes. This class of stochastic processes includes solutions of Skorokhod-type stochastic differential equations (SDEs), certain Feller processes and solutions of Lévy driven SDEs. The class of processes is wider than in earlier contributions and using fine continuity we are able to handle general measurable subsets of R d as state spaces. Furthermore, in contrast to previous contributions on the subject, we introduce a local index in order to complement the upper index. This local index yields a sufficient condition for the infiniteness of the p-variation. We discuss various examples in order to demonstrate the applicability of the method.

Trace theorems for functions of bounded variation in metric spaces

In this paper we show existence of traces of functions of bounded variation on the boundary of a certain class of domains in metric measure spaces equipped with a doubling measure supporting a 1-Poincaré inequality, and obtain L 1 estimates of the trace functions. In contrast with the treatment of traces given in other papers on this subject, the traces we consider do not require knowledge of the function in the exterior of the domain. We also establish a Maz'yatype inequality for functions of bounded variation that vanish on a set of positive capacity.