We will be concerned with the existence of minimizing geodesies for certain Finsler structures on... more We will be concerned with the existence of minimizing geodesies for certain Finsler structures on Sobolev manifolds. A little background in infinitedimensional Riemannian geometry is useful for motivation. Essentially all local results of finite-dimensional Riemannian geometry (e.g. existence of convex neighborhoods) carry over to infinite-dimensions. A central theorem in global Riemannian geometry is the Hopf-Rinow Theorem. It gives equivalent conditions for completeness of a Riemannian manifold and further states that if M is complete and p, qeM then there is a geodesic from p to q whose length is equal to the distance from p to q. Such a geodesic is called a minimizing geodesic. In infinite dimensions, all that survives of the Hopf-Rinow Theorem is the statement: M is complete if and only if all geodesies from any point of M are infinitely extendable. Thus the existence of minimizing geodesies is of central concern in global infinite-dimensional Riemannian (and Finsler) geometry. Indeed, there are examples of complete Riemannian manifolds without minimizing geodesies. (i) Let e u e 2 ,... be an orthonormal basis of Hilbert space H. Let/ f = (1 4l/i)e t and let L be the lattice generated by 2f t and/j-/ i+1 for i-1,2,.... Then H/L is a complete Riemannian manifold. If p: H-> H/L is the projection then p(/ f) = p(fy for all i and j, but p(0) =£ p(fy. One easily sees, by lifting geodesies to if, that there are geodesies from p(0) to p(f t) of length 1 + 1/i for each i, but no geodesic of length < 1. (2) An ellipsoid in H with axes of lengths 1 + l/i has similar collections of geodesies. It is interesting to note that example (2) is contractible, but highly curved, while example (1) is flat but has infinitely generated fundamental group. The most interesting Banach manifolds (due to their close connection with finite-dimensional manifolds) are those which consist of functions from one finite-dimensional manifold to another. We will consider in particular, the Sobolev manifolds, and will show that for certain natural Finsler structures on these manifolds minimizing geodesies do exist. In §2 we treat the case of a Finsler metric induced by an imbedding in a Banach space. We say that a Finsler manifold Fis well imbedded in a Banach space B if B is reflexive and V is isometrically imbedded as a sequentially weakly closed submanifold of B. In particular, if V is a closed submanifold of Euclidean space with the induced metric then V is well imbedded. We will show that if M is an ^-dimensional manifold and Fis well imbedded in B, then the Sobolev manifold L%(M, V) is well imbedded in LP k {M, B) for 1 < p < oo and k > n/p.
Let π : P → M be a smooth principal bundle with structure group G. This means that there is a C ∞... more Let π : P → M be a smooth principal bundle with structure group G. This means that there is a C ∞ right multiplication P × G → P , u → u • g say, of the Lie group G such that π identifies the space of orbits of G with the manifold M and π is locally trivial in the sense that each point of M has an open neighbourhood U with a diffeomorphism over U , which is equivariant with respect to the right action of . Assume for simplicity that M is compact. Set n = dimM . The fibres, π -1 (x), x ∈ M are diffeomorphic to G and their tangent spaces V T u P (= kerT u π), u ∈ P , are the 'vertical' tangent spaces to P . A connection on P , (or on π) assigns a complementary 'horizontal' subspace HT u P to V T u P in T u P for each u, giving a smooth horizontal subbundle HT P of the tangent bundle T P to P . Given such a connection it is a classical result that for any C 1 curve: σ : [0, T ] → M and u 0 ∈ π -1 (σ(0)) there is a unique horizontal σ : [0, T ] → P which is a lift of σ, i.e. π(σ(t)) = σ(t) and has σ(0) = u 0 . In his startling ICM article [8] Itô showed how this construction could be extended to give horizontal lifts of the sample paths of diffusion processes. In fact he was particularly concerned with the case when M is given a Riemannian metric , x , x ∈ M , the diffusion is Brownian motion on M , and P is the orthonormal frame bundle π : OM → M . Recall that each u ∈ OM with u ∈ π -1 (x) can be considered as an isometry u : R n → T x M , , x and a
Limits for stochastic models: stochastic and quantum mechanical scattering on hyperbolic spaces, ... more Limits for stochastic models: stochastic and quantum mechanical scattering on hyperbolic spaces, E.A. Carlen and K.D. Elsworthy intermittency and phase transitions from some particle systems in random media, R. Carmona and S.A. Molchanov predator-prey systems, R. Durrett homogenization of reflecting barrier Brownian motions, H. Osada nonlinear diffusion limit for a system with nearest neighbour interactions-II, S.R.S. Varadhan. Part 2 Analysis on fractals: random walks, electrical resistance, and nested fractals, M.R. Barlow Hausdorff dimension of nonlinear Cantor sets, K. Handa self-avoiding process on the Sierpinski gasket, K. Hatton harmonic metric and Dirichlet form on the Sierpinski gasket, J. Kigami construction and some properties of a class of non-symmetric diffusion processes on the Sierpinski gasket, T. Kumagai Brownian motion penetrating the Sierpinski gasket, T. Lindstrom the eigenvalue problem for the Laplacian on the Sierpinski gasket, T. Shima.
... A. Truman Mathematics Department Heriot-Watt University Edinburgh EH14 4AS i. INTRODUCTION Th... more ... A. Truman Mathematics Department Heriot-Watt University Edinburgh EH14 4AS i. INTRODUCTION The Cameron-Martin formula for Wiener integrals giving their transformation properties under linear changes of integration variables is well-known [i]. Here we present an ...
Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1983
Note 1.2. For our systems the Markov property implies that if (X, z) has a flow for the time inte... more Note 1.2. For our systems the Markov property implies that if (X, z) has a flow for the time interval [0, T] then it has a flow for all time. 2. Preliminary Results 2.1. Behaviour of Solutions Under Transformations Proposition 2.1. Let U and V be open in Hilbert spaces H and K. Consider C 2 stochastic dynamical systems (X, z) on U and (Y, z) on V for which there is a C 2 Proof See It6 [18], Clarke [7], or Elworthy [13]. // When solutions exist for all (positive) time we will say (X, z) is complete, sometimes the terms non-explosive or conservative are used. 2.3. Piecewise Linear Approximations By the piecewise linear approximation to the system (X, z) in the time interval [0, T], with respect to the partition H given by 0 = t o < t 1 <... < t m = T we mean the system (X, z~) where z~ is the piecewise linear approximation to z, given by z~(t)=(tj-tj_l)-l[(t-tj_l)z(tj)+(tj-t)z(tj_l)] tj_l<=t<-_t j. Note that (X, z~) can be considered as a family of ordinary dynamical systems parametrized by co~ since the sample paths of z are all piecewise C 1.
Summary. Generalised Itô formulae are proved for time dependent functions of continuous real valu... more Summary. Generalised Itô 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. Keywords: Local time, continuous semi-martingale, generalized Itô’s formula, two-dimensional Lebesgue-Stieltjes integral.
Stochastic Partial Differential Equations and Related Fields, 2018
The decomposition of a class of diffusion processes, due to Elworthy–LeJan–Li is described. In pa... more The decomposition of a class of diffusion processes, due to Elworthy–LeJan–Li is described. In particular it applies to processes such as derivative processes coming from stochastic flows. How this decomposition leads automatically to Weitzenbock type formula for related operators acting on sections of associated vector bundles is described in detail clarifying the difference between the action of flows on vector fields and on forms noted recently by Shizan Fang and Dejun Luo. Remarks are made on the possible application to higher order derivative formulae and estimates for heat semigroups, and also to certain diffusions with sub-Riemannian generators using Baudoin’s generalised Levi-Civita semi-connection.
For a compact Riemannian manifold the space L 2 A of L 2 differential forms decomposes into the d... more For a compact Riemannian manifold the space L 2 A of L 2 differential forms decomposes into the direct sum of three spaces, the Hodge decomposition,
We will be concerned with the existence of minimizing geodesies for certain Finsler structures on... more We will be concerned with the existence of minimizing geodesies for certain Finsler structures on Sobolev manifolds. A little background in infinitedimensional Riemannian geometry is useful for motivation. Essentially all local results of finite-dimensional Riemannian geometry (e.g. existence of convex neighborhoods) carry over to infinite-dimensions. A central theorem in global Riemannian geometry is the Hopf-Rinow Theorem. It gives equivalent conditions for completeness of a Riemannian manifold and further states that if M is complete and p, qeM then there is a geodesic from p to q whose length is equal to the distance from p to q. Such a geodesic is called a minimizing geodesic. In infinite dimensions, all that survives of the Hopf-Rinow Theorem is the statement: M is complete if and only if all geodesies from any point of M are infinitely extendable. Thus the existence of minimizing geodesies is of central concern in global infinite-dimensional Riemannian (and Finsler) geometry. Indeed, there are examples of complete Riemannian manifolds without minimizing geodesies. (i) Let e u e 2 ,... be an orthonormal basis of Hilbert space H. Let/ f = (1 4l/i)e t and let L be the lattice generated by 2f t and/j-/ i+1 for i-1,2,.... Then H/L is a complete Riemannian manifold. If p: H-> H/L is the projection then p(/ f) = p(fy for all i and j, but p(0) =£ p(fy. One easily sees, by lifting geodesies to if, that there are geodesies from p(0) to p(f t) of length 1 + 1/i for each i, but no geodesic of length < 1. (2) An ellipsoid in H with axes of lengths 1 + l/i has similar collections of geodesies. It is interesting to note that example (2) is contractible, but highly curved, while example (1) is flat but has infinitely generated fundamental group. The most interesting Banach manifolds (due to their close connection with finite-dimensional manifolds) are those which consist of functions from one finite-dimensional manifold to another. We will consider in particular, the Sobolev manifolds, and will show that for certain natural Finsler structures on these manifolds minimizing geodesies do exist. In §2 we treat the case of a Finsler metric induced by an imbedding in a Banach space. We say that a Finsler manifold Fis well imbedded in a Banach space B if B is reflexive and V is isometrically imbedded as a sequentially weakly closed submanifold of B. In particular, if V is a closed submanifold of Euclidean space with the induced metric then V is well imbedded. We will show that if M is an ^-dimensional manifold and Fis well imbedded in B, then the Sobolev manifold L%(M, V) is well imbedded in LP k {M, B) for 1 < p < oo and k > n/p.
Let π : P → M be a smooth principal bundle with structure group G. This means that there is a C ∞... more Let π : P → M be a smooth principal bundle with structure group G. This means that there is a C ∞ right multiplication P × G → P , u → u • g say, of the Lie group G such that π identifies the space of orbits of G with the manifold M and π is locally trivial in the sense that each point of M has an open neighbourhood U with a diffeomorphism over U , which is equivariant with respect to the right action of . Assume for simplicity that M is compact. Set n = dimM . The fibres, π -1 (x), x ∈ M are diffeomorphic to G and their tangent spaces V T u P (= kerT u π), u ∈ P , are the 'vertical' tangent spaces to P . A connection on P , (or on π) assigns a complementary 'horizontal' subspace HT u P to V T u P in T u P for each u, giving a smooth horizontal subbundle HT P of the tangent bundle T P to P . Given such a connection it is a classical result that for any C 1 curve: σ : [0, T ] → M and u 0 ∈ π -1 (σ(0)) there is a unique horizontal σ : [0, T ] → P which is a lift of σ, i.e. π(σ(t)) = σ(t) and has σ(0) = u 0 . In his startling ICM article [8] Itô showed how this construction could be extended to give horizontal lifts of the sample paths of diffusion processes. In fact he was particularly concerned with the case when M is given a Riemannian metric , x , x ∈ M , the diffusion is Brownian motion on M , and P is the orthonormal frame bundle π : OM → M . Recall that each u ∈ OM with u ∈ π -1 (x) can be considered as an isometry u : R n → T x M , , x and a
Limits for stochastic models: stochastic and quantum mechanical scattering on hyperbolic spaces, ... more Limits for stochastic models: stochastic and quantum mechanical scattering on hyperbolic spaces, E.A. Carlen and K.D. Elsworthy intermittency and phase transitions from some particle systems in random media, R. Carmona and S.A. Molchanov predator-prey systems, R. Durrett homogenization of reflecting barrier Brownian motions, H. Osada nonlinear diffusion limit for a system with nearest neighbour interactions-II, S.R.S. Varadhan. Part 2 Analysis on fractals: random walks, electrical resistance, and nested fractals, M.R. Barlow Hausdorff dimension of nonlinear Cantor sets, K. Handa self-avoiding process on the Sierpinski gasket, K. Hatton harmonic metric and Dirichlet form on the Sierpinski gasket, J. Kigami construction and some properties of a class of non-symmetric diffusion processes on the Sierpinski gasket, T. Kumagai Brownian motion penetrating the Sierpinski gasket, T. Lindstrom the eigenvalue problem for the Laplacian on the Sierpinski gasket, T. Shima.
... A. Truman Mathematics Department Heriot-Watt University Edinburgh EH14 4AS i. INTRODUCTION Th... more ... A. Truman Mathematics Department Heriot-Watt University Edinburgh EH14 4AS i. INTRODUCTION The Cameron-Martin formula for Wiener integrals giving their transformation properties under linear changes of integration variables is well-known [i]. Here we present an ...
Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1983
Note 1.2. For our systems the Markov property implies that if (X, z) has a flow for the time inte... more Note 1.2. For our systems the Markov property implies that if (X, z) has a flow for the time interval [0, T] then it has a flow for all time. 2. Preliminary Results 2.1. Behaviour of Solutions Under Transformations Proposition 2.1. Let U and V be open in Hilbert spaces H and K. Consider C 2 stochastic dynamical systems (X, z) on U and (Y, z) on V for which there is a C 2 Proof See It6 [18], Clarke [7], or Elworthy [13]. // When solutions exist for all (positive) time we will say (X, z) is complete, sometimes the terms non-explosive or conservative are used. 2.3. Piecewise Linear Approximations By the piecewise linear approximation to the system (X, z) in the time interval [0, T], with respect to the partition H given by 0 = t o < t 1 <... < t m = T we mean the system (X, z~) where z~ is the piecewise linear approximation to z, given by z~(t)=(tj-tj_l)-l[(t-tj_l)z(tj)+(tj-t)z(tj_l)] tj_l<=t<-_t j. Note that (X, z~) can be considered as a family of ordinary dynamical systems parametrized by co~ since the sample paths of z are all piecewise C 1.
Summary. Generalised Itô formulae are proved for time dependent functions of continuous real valu... more Summary. Generalised Itô 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. Keywords: Local time, continuous semi-martingale, generalized Itô’s formula, two-dimensional Lebesgue-Stieltjes integral.
Stochastic Partial Differential Equations and Related Fields, 2018
The decomposition of a class of diffusion processes, due to Elworthy–LeJan–Li is described. In pa... more The decomposition of a class of diffusion processes, due to Elworthy–LeJan–Li is described. In particular it applies to processes such as derivative processes coming from stochastic flows. How this decomposition leads automatically to Weitzenbock type formula for related operators acting on sections of associated vector bundles is described in detail clarifying the difference between the action of flows on vector fields and on forms noted recently by Shizan Fang and Dejun Luo. Remarks are made on the possible application to higher order derivative formulae and estimates for heat semigroups, and also to certain diffusions with sub-Riemannian generators using Baudoin’s generalised Levi-Civita semi-connection.
For a compact Riemannian manifold the space L 2 A of L 2 differential forms decomposes into the d... more For a compact Riemannian manifold the space L 2 A of L 2 differential forms decomposes into the direct sum of three spaces, the Hodge decomposition,
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Papers by K.d. Elworthy