On the Stochastic Magnus Expansion and Its Application to SPDEs

We derive the stochastic version of the Magnus expansion for linear systems of stochastic differential equations (SDEs). The main novelty with respect to the related literature is that we consider SDEs in the Itô sense, with progressively measurable coefficients, for which an explicit Itô-Stratonovich conversion is not available. We prove convergence of the Magnus expansion up to a stopping time $$\tau $$ τ and provide a novel asymptotic estimate of the cumulative distribution function of $$\tau $$ τ . As an application, we propose a new method for the numerical solution of stochastic partial differential equations (SPDEs) based on spatial discretization and application of the stochastic Magnus expansion. A notable feature of the method is that it is fully parallelizable. We also present numerical tests in order to asses the accuracy of the numerical schemes.

The Regularity Problem for Uniformly Elliptic Operators in Weighted Spaces

This paper studies the regularity problem for block uniformly elliptic operators in divergence form with complex bounded measurable coefficients. We consider the case where the boundary data belongs to Lebesgue spaces with weights in the Muckenhoupt classes. Our results generalize those of S. Mayboroda (and those of P. Auscher and S. Stahlhut employing the first order method) who considered the unweighted case. To obtain our main results we use the weighted Hardy space theory associated with elliptic operators recently developed by the last two named authors. One of the novel contributions of this paper is the use of an “inhomogeneous” vertical square function which is shown to be controlled by the gradient of the function to which is applied in weighted Lebesgue spaces.

On off-diagonal decay properties of the generalized Stokes semigroup with bounded measurable coefficients

We investigate off-diagonal decay properties of the generalized Stokes semigroup with bounded measurable coefficients on $$\mathrm {L}^2_{\sigma } ({\mathbb {R}}^d)$$ L σ 2 ( R d ) . Such estimates are well-known for elliptic equations in the form of pointwise heat kernel bounds and for elliptic systems in the form of integrated off-diagonal estimates. On our way to unveil this off-diagonal behavior we prove resolvent estimates in Morrey spaces $$\mathrm {L}^{2 , \nu } ({\mathbb {R}}^d)$$ L 2 , ν ( R d ) with $$0 \le \nu < 2$$ 0 ≤ ν < 2 .

Penalization for a PDE with a nonlinear Neumann boundary condition and measurable coefficients

We consider a system of semilinear partial differential equations (PDEs) with measurable coefficients and a nonlinear Neumann boundary condition. We then construct a sequence of penalized PDEs, which converges to our initial problem. Since the coefficients we consider may be discontinuous, we use the notion of solution in the [Formula: see text]-viscosity sense. The method we use is based on backward stochastic differential equations and their [Formula: see text]-tightness. This work is motivated by the fact that many PDEs in physics have discontinuous coefficients. As a consequence, it follows that if the uniqueness holds, then the solution can be constructed by a penalization.

Space regularity for evolution operators modeled on Hörmander vector fields with time dependent measurable coefficients

We consider a heat-type operator $$\mathcal {L}$$ L structured on the left invariant 1-homogeneous vector fields which are generators of a Carnot group, with a uniformly positive matrix of bounded measurable coefficients depending only on time. We prove that if $$\mathcal {L}u$$ L u is smooth with respect to the space variables, the same is true for u, with quantitative regularity estimates in the scale of Sobolev spaces defined by right invariant vector fields. Moreover, the solution and its space derivatives of every order satisfy a 1/2-Hölder continuity estimate with respect to time. The result is proved both for weak solutions and for distributional solutions, in a suitable sense.

Strict Positivity for the Principal Eigenfunction of Elliptic Operators with Various Boundary Conditions

We consider elliptic operators with measurable coefficients and Robin boundary conditions on a bounded domain {\Omega\subset\mathbb{R}^{d}} and show that the first eigenfunction v satisfies {v(x)\geq\delta>0} for all {x\in\overline{\Omega}}, even if the boundary {\partial\Omega} is only Lipschitz continuous. Under such weak regularity assumptions the Hopf–Oleĭnik boundary lemma is not available; instead we use a new approach based on an abstract positivity improving condition for semigroups that map {L_{p}(\Omega)} into {C(\overline{\Omega})}. The same tool also yields corresponding results for Dirichlet or mixed boundary conditions. Finally, we show that our results can be used to derive strong minimum and maximum principles for parabolic and elliptic equations.