For a class of degenerate diffusion processes of rank 2, i.e. when only Poisson brackets of order one are needed to span the whole space, we obtain a parametrix representation of the density from which we derive some explicit Gaussian controls that characterize the additional singularity induced by the degeneracy. We then give a local limit theorem with the usual convergence rate for an associated Markov chain approximation. The key point is that the "weak" degeneracy allows to exploit the techniques first introduced by Konakov and Molchanov and then developed by Konakov and Mammen that rely on Gaussian approximations.
Parametrix techniques and martingale problem for some degenerate Kolmogorov's equations
证明了退化算子的鞅问题的唯一性。关键点是利用Bass和Perkins（From Probability to Geometry，vol. in honor of J.M Bismut（2009））引入的新技术之间的强平行性，以证明非退化椭圆算子框架中鞅问题的唯一性，以及Mc Kean和Singer（Journal of Diff.几何，1967年）参数化方法的密度膨胀，以前已扩展到退化设置，我们认为（见Delarue和Menozzi，杂志的功能。Analysis，2010）。
We adapt and extend Yosida's parametrix method, originally introduced for the construction of the fundamental solution to a parabolic operator on a Riemannian manifold, to derive Varadhan-type asymptotic estimates for the transition density of a degenerate diffusion under the weak Hörmander condition. This diffusion process, widely studied by Yor in a series of papers, finds direct application in the study of a class of path-dependent financial derivatives known as Asian options. We obtain the Varadhan formula \begin{equation} \frac{-2 \log p(t,x;T,y) } { Ψ(t,x;T,y) } \to 1, \qquad \text{as } …
We provide here some sharp Schauder estimates for degenerate PDEs of Kolmogorov type when the coefficients lie in some suitable anisotropic H{ö}lder spaces and the first order term is non-linear and unbounded. We proceed through a perturbative approach based on forward parametrix expansions. Due to the low regularizing properties of the degenerate variables, for the procedure to work, we heavily exploit duality results between appropriate Besov spaces. Our method can be seen as constructive and provides, even in the non-degenerate case, an alternative approach to Schauder estimates.
We provide here global Schauder-type estimates for a chain of integro-partial differential equations (IPDE) driven by a degenerate stable Ornstein-Uhlenbeck operator possibly perturbed by a deterministic drift, when the coefficients lie in some suitable anisotropic H{ö}lder spaces. Our approach mainly relies on a perturbative method based on forward parametrix expansions and, due to the low regularizing properties on the degenerate variables and to some integrability constraints linked to the stability index, it also exploits duality results between appropriate Besov Spaces. In particular, our…
In this article, we obtain a functional CLT for a class of degenerate diffusion processes with periodic coefficients, thus generalizing the already classical results in the context of uniformly elliptic diffusions. As an application, we also discuss periodic homogenization of a class of linear degenerate elliptic and parabolic PDEs.
In this paper we prove a new strong uniqueness result and a weak existence result for possibly {\it degenerate} multidimensional stochastic differential equations with Sobolev diffusion coefficients and rough drifts. In particular, examples with Hölder diffusion coefficients are provided to show our results.
This paper proves a Krylov-Safonov estimate for a multidimensional diffusion process whose diffusion coefficients are degenerate on the boundary. As applications the existence and uniqueness of invariant probability measures for the process and Hoelder estimates for the associated partial differential equation are obtained.
In this work, we show that for the martingale problem for a class of degenerate diffusions with bounded continuous drift and diffusion coefficients, the small noise limit of non-degenerate approximations leads to a unique Feller limit. The proof uses the theory of viscosity solutions applied to the associated backward Kolmogorov equations. Under appropriate conditions on drift and diffusion coefficients, we will establish a comparison principle and a one-one correspondence between Feller solutions to the martingale problem and continuous viscosity solutions of the associated Kolmogorov equatio…
We study a symmetric diffusion process on $\mathbb{R}^d$, $d\geq 2$, in divergence form in a stationary and ergodic random environment. The coefficients are assumed to be degenerate and unbounded but satisfy a moment condition. We derive upper off-diagonal estimates on the heat kernel of this process for general speed measure. Lower off-diagonal estimates are also shown for a natural choice of speed measure, under an additional mixing assumption on the environment. Using these estimates, a scaling limit for the Green's function is proven.
Integration by parts formulae are given for a class of measures on the space of paths of a smooth manifold $M$ determined by the laws of degenerate diffusions. The mother of such formulae, on the path space of diffeomorphism group of $M$ is shown to arise from a quasi-invariance property of measures determined by stochastic flows. From this the other formulae are derived by filtering out redundant noise using an associated LeJan-Watanabe connection.
For given strongly local Dirichlet forms with possibly degenerate symmetric (sub)-elliptic matrix, we show the existence of weak solutions to the stochastic differential equations (associated with the Dirichlet forms) starting from all points in $\R^d$. More precisely, using heat kernel estimates, stochastic calculus, and Dirichlet form theory, we obtain the pointwise existence of weak solutions to the stochastic differential equations which have possibly unbounded and discontinuous drift. We also present some conditions that the weak solutions become pathwise unique strong solutions and provi…
We study a class of R^d-valued continuous strong Markov processes that are generated, only locally, by an ultra-parabolic operator with coefficients that are regular w.r.t. the intrinsic geometry induced by the operator itself and not w.r.t. the Euclidean one. The first main result is a local Ito formula for functions that are not twice-differentiable in the classical sense, but only intrinsically w.r.t. to a set of vector fields, related to the generator, satisfying the Hormander condition. The second main contribution, which builds upon the first one, is an existence and regularity result fo…
In this work we adopt a combination of probabilistic approach and analytic methods to study the fundamental solutions to variations of the Wright-Fisher equation in one dimension. To be specific, we consider a diffusion equation on $\left(0,\infty\right)$ whose diffusion coefficient vanishes at the boundary 0, equipped with the Cauchy initial data and the Dirichlet boundary condition. One type of diffusion operator that has been extensively studied is the one whose diffusion coefficient vanishes linearly at 0. Our main goal is to extend the study to cases when the diffusion coefficient has a g…
We study the spectrum of the semiclassical Witten Laplacian $Δ_{f}$ associated to a smooth function $f$ on ${\mathbb R}^d$. We assume that $f$ is a confining Morse--Bott function. Under this assumption we show that $Δ_{f}$ admits exponentially small eigenvalues separated from the rest of the spectrum. Moreover, we establish Eyring-Kramers formula for these eigenvalues. Our approach is based on microlocal constructions of quasimodes near the critical submanifolds.
First essential m-dissipativity of an infinite-dimensional Ornstein-Uhlenbeck operator $N$, perturbed by the gradient of a potential, on a domain $\mathcal{F}C_b^{\infty}$ of finitely based, smooth and bounded functions, is shown. Our considerations allow unbounded diffusion operators as coefficients. We derive corresponding second order regularity estimates for solutions $f$ of the Kolmogorov equation $αf-Nf=g$, $α\in (0,\infty)$, generalizing some results of Da Prato and Lunardi. Second we prove essential m-dissipativity for generators $(L_Φ,\mathcal{F}C_b^{\infty})$ of infinite-dimensional …
In this paper, we construct the fundamental solution to a degenerate diffusion of Kolmogorov type and develop a time-discrete variational scheme for its adjoint equation. The so-called mean squared derivative cost function plays a crucial role occurring in both the fundamental solution and the variational scheme. The latter is implemented by minimizing a free energy functional with respect to the Kantorovich optimal transport cost functional associated with the mean squared derivative cost. We establish the convergence of the scheme to the solution of the adjoint equation generalizing previous…
