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K. D. Elworthy

Publications and source records attributed to K. D. Elworthy.

17 recordsLinked to original sources
arXiv2019

Equivariant diffusions on Principal bundles

Given a pair of second order diffusion operators, one on the total space of a principle bundle $N$ and the other on the base space $M$, intertwined by the projection $π:N\to M$, if the operator ${\mathcal A}$ on the base manifold has constant rank, we define a semi-connection on the principal bundle which allows to split the diffusion operator ${\mathcal B}$ on the total space into the sum of the horizontal lift of ${\mathcal A}$ and the other vertical. This allow to conclude a disintegration theorem for the law of ${\mathcal B}$. As an application, a decomposition of stochastic flow is given.

math.PR↗
arXiv2019

A class of integration by parts formulae in stochastic analysis I

An integration by parts formula is the foundation for stochastic analysis on path spaces over a (finite dimensional) Riemannian manifold or over $R^n$, from which we may deduce the operator $d$ is closable and define the Laplacian operator on path spaces. A useful formula on the Riemannian manifold is $$dP_tf(v)=(1/t)E f(x_t) \int_0^t \langle d\{x_s\}, v_s\rangle ,$$ for $P_t$ the heat semi-group, $x_t$ the BM, $v_t$ the derivative flow or its conditional expectation (which is a damped parallel translation), $d\{x_s\}$ is the martingale part of $x_t$. As a meta theorem, this leads to the Clark-Ocone formula (martingale representation theorem with specific integrand) and Logrithmic Sobolev inequalities. Interpreted appropriately, the latter formula is obviously a special case of the integration by parts formula. Here we show by the Markov property and by induction that the latter formula implies the integration by parts formula. WE also use Bismut's original approach to prove an integration by parts formula, using a connection with torsion and one on the free path space.

math.PR↗
arXiv2019

Integration by parts formulae for degenerate diffusion measures on path spaces and diffeomorphism groups

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.

math.PR↗
arXiv2019

Intertwining and the Markov uniqueness problem on path spaces

There are two open problem on the analysis of continuous paths on a Riemannian manifold, the Markov uniqueness and the independence of the closure of the differential operator $d$ on its initial domain. The operator $d$ acts naturally on $BC^1$ functions, one is concerned with its extensions to the $L^2$ spaces. With a suitable choice of an initial domain we denote by $D^{2,1}$ its closure under the graph norm. For the Wiener space, the domain of $d$ can be classified, as a consequence its extension is unique whether the initial domain is smooth cylindrical or in $BC^1$ etc. This has not shown to be the same when the measure is the probability distribution of any smooth elliptic diffusion. In an earlier paper, we have shown that the closure of $BC^\infty$ functions agree with that of smooth cylindrical functions, leaving an undesirable gap. The Markov uniqueness is essentially concerned with the problem whether there exists a unique Markov process on the path space whose Markov generator agrees with the infinite-dimensional Laplacian on $C^\infty$ cylindrical functions. Here we reduce Markov uniqueness to whether the pull back of $D^{2,1}$ by the ito map is $ D^{2,1}$ (i.e. a surjection). We also propose a possible approach for tackle this problem.

math.PR↗
arXiv2019

Formulae for the derivatives of heat semigroups

We use a basic martingale method to show a differentiation formula for the derivatives $$d(P_tf)(x_0)(v_0)={1\over t} E f(x_t) \int_0^t \langle Y(x_s)(v_s),dB_t\rangle_{R^m}.$$ These are proved first on $R^n$, then on manifolds. Afterwards for solutions of heat equations on differential forms, and a second order formula.

math.PR↗
arXiv2019

Bismut type formulae for differential forms

Formulae are given for $dP_t ϕ$, $d^*P_tϕ$ and $ΔP_tϕ$ for $P_t$ the heat semigroup acting on a q-form $ϕ$. The formulae are Brownian motion expectations of $ϕ$ composed with random translations determined by Weitzenbock curvarure terms. Derivatives of the curvature are not involved.

math.PR↗
arXiv2019

Geometric stochastic analysis on path spaces

An approach to analysis on path spaces of Riemannian manifolds is described. The spaces are furnished with `Brownian motion' measure which lies on continuous paths, though differentiation is restricted to directions given by tangent paths of finite energy. An introduction describes the background for paths on ${\mathbb R}^m$ and Malliavin calculus. For manifold valued paths the approach is to use `Itô' maps of suitable stochastic differential equations as charts . `Suitability' involves the connection determined by the stochastic differential equation. Some fundamental open problems concerning the calculus and the resulting `Laplacian' are described. A theory for more general diffusion measures is also briefly indicated. The same method is applied as an approach to getting over the fundamental difficulty of defining exterior differentiation as a closed operator, with success for one \& two forms leading to a Hodge -Kodaira operator and decomposition for such forms. Finally there is a brief description of some related results for loop spaces.

math.PR↗
arXiv2019

Some family of q-vector fields on path spaces

A great open problem is: can one learn the topology of the non-smooth path spaces with an L2 Hodge-deRham theory This one hopes to establish through a suitable complex of differential forms. Since the space is a Banach manifolds, and the Hodge theory is based on Hilbert spaces, it is trick to find such a complex. The relatively simpler Bismut tangent spaces and their tensor products, whose dual spaces are natural candidates for the complex, cannot be used, because these spaces are not necessarily closed under the Lie bracket operation if there is the effect of curvature. In this article we seek out a class of nice vector fields whose brackets behaves are calculable and behave nicely the damped tensor vector fields, using an Itô mp, which is also used by the authors in `Special Itô maps and an L2 Hodge theory for one forms on path spaces. Stochastic processes, physics and geometry: new interplays, I (Leipzig, 1999), 145-162, CMS Conf. Proc., 28, Amer. Math. Soc., Providence, RI, 2000').

math.PR↗
arXiv2019

Concerning the geometry of stochastic differential equations and stochastic flows

Following Le Jan and Watanabe we define a connection associated with a non-degenrrate diffusion operators. This connection is characterized here and shown to be the Levi-Civita connection for gradient systems. This both explains why such systems have useful properties and allows us to extend these properties to more general systems. Topics described here include: moment estimates for $Tξ_t$, a Weitzenböck formula for the generator of the semigroup on p-forms induced by the flow, a Bismut type formula for $d\log p_t$ in terms of an arbitrary metric connection, and a generalized Bochner vanishing theorem. A comprehensive theory on this, its generalization to semi-elliptic case and applications is published in the book `On the geometry of diffusion operators and stochastic flows'. Related to this is also the book `The geometry of filtering'. This article is easier to read.

math.PR↗
arXiv2015

Generalized Ito Formulae and Space-Time Lebesgue-Stieltjes Integrals of Local Times

Generalised Ito formulae are proved for time dependent functions of continuous real valued semi-martingales. The conditions involve left space and time first derivatives, with the left space derivative required to have locally bounded 2-dimensional variation. In particular a class of functions with discontinuous first derivative is included. An estimate of Krylov allows further weakening of these conditions when the semi-martingale is a diffusion.

math.PR↗
arXiv2012

The vanishing of L2 harmonic one-forms on based path spaces

We prove the triviality of the first L2 cohomology class of based path spaces of Riemannian manifolds furnished with Brownian motion measure, and the consequent vanishing of L2 harmonic one-forms. We give explicit formulae for closed and co-closed one-forms expressed as differentials of functions and co-differentials of L2 two-forms, respectively; these are considered as extended Clark-Ocone formulae. A feature of the proof is the use of the temporal structure of path spaces to relate a rough exterior derivative operator on one-forms to the exterior differentiation operator used to construct the de Rham complex and the self-adjoint Laplacian on L2 one-forms. This Laplacian is shown to have a spectral gap.

math.PR↗
arXiv2008

The Geometry of Filtering

Geometry arising from two diffusion operators (smooth semi-elliptic, second order differential operators) on different spaces but intertwined by a smooth map is described. Particular cases arise from Riemannian submersions when the operators are Laplace-Beltrami operators, from equivariant operators on the total space of a principal bundle, and for the operators on the diffeomorphism group arising from stochastic flows. Classical non-linear filtering problems also lead to such conffigurations. A basic tool is the, possibly, non-linear "semi-connection" induced by this set up, leading to a canonical decomposition of the operator on the domain space. Topics discussed include: generalised Wietzenbock curvatures arising in the equivariant case, skew -product decompositions of diffusion processes, conditioned processes, classical filtering, decomposition of stochastic flows, and connections determined by stochastic differential equations.

math.DG↗
arXiv2006

An L2 theory for differential forms on path spaces I

An L2 theory of differential forms is proposed for the Banach manifold of continuous paths on Riemannian manifolds M furnished with its Brownian motion measure. Differentiation must be restricted to certain Hilbert space directions, the H-tangent vectors. To obtain a closed exterior differential operator the relevant spaces of differential forms, the H-forms, are perturbed by the curvature of M. A Hodge decomposition is given for L2 H-one-forms, and the structure of H-two -forms is described. The dual operator d* is analysed in terms of a natural connection on the H-tangent spaces. Malliavin calculus is a basic tool.

math.PR↗
arXiv2006

Ito maps and analysis on path spaces

We consider versions of Malliavin calculus on path spaces of compact manifolds with diffusion measures, defining Gross-Sobolev spaces of differentiable functions and proving their intertwining with solution maps, I, of certain stochastic differential equations. This is shown to shed light on fundamental uniqueness questions for this calculus including uniqueness of the closed derivative operator $d$ and Markov uniqueness of the associated Dirichlet form. A continuity result for the divergence operator by Kree and Kree is extended to this situation. The regularity of conditional expectations of smooth functionals of classical Wiener space, given I, is considered and shown to have strong implications for these questions. A major role is played by the (possibly sub-Riemannian) connections induced by stochastic differential equations: Damped Markovian connections are used for the covariant derivatives.

math.PR↗
arXiv1997

Bounded and $L^2$ Harmonic Forms on Universal Covers

We relate the positivity of the curvature term in the Weitzenbock formula for the Laplacian on p-forms on a complete manifold to the existence of bounded and $L^2$ harmonic forms. In the case where the manifold is the universal cover of a compact manifold, we obtain topological and geometric information about the compact manifold. For example, we show that a compact manifold cannot admit one metric with pinched negative curvature and another metric with positive Weitzenbock term on two-forms. Many of these results can be thought of as differential form analogues of Myers' theorem. We also give pinching conditions on certain sums of sectional curvatures which imply the positivity of the curvature term, and hence yield vanishing theorems. In particular, we construct a compact manifold with planes of negative sectional curvature at each point and which satisfies the hypothesis of our vanishing theorems.

dg-ga↗
arXiv1995

Homotopy and Homology Vanishing Theorems and the Stability of Stochastic Flows

We relate stability properties (i.e. moment exponents) of a stochastic dynamical system on a compact manifold $M$ to the homotopy and integral homology groups of $M$. In the special case of gradient Brownian systems associated to isometric immersions of $M$ in Euclidean space, these moment exponents can be estimated in terms of the second fundamental form of the immersion. This yields topological obstructions to isometric immersions generalizing results of Lawson-Simons and others. Our work also places these authors' work into the general framework of Weitzenböck formulas.

dg-ga↗