Publications
71 results found
Elworthy KD, Le Jan Y, Li X-M, 1996, Integration by parts formulae for degenerate diffusion measures on path spaces and diffeomorphism groups, Comptes Rendus de l’Académie des Sciences. Série I. Mathématique, Vol: 323, Pages: 921-926, ISSN: 0764-4442
Li X-M, Zhao HZ, 1996, Gradient estimates and the smooth convergence of approximate travelling waves for reaction-diffusion equations, Nonlinearity, Vol: 9, Pages: 459-477, ISSN: 0951-7715
Elworthy KD, Li X-M, 1996, A class of integration by parts formulae in stochastic analysis. I, Itô’s stochastic calculus and probability theory, Publisher: Springer, Tokyo, Pages: 15-30, ISBN: 978-4-431-68532-6
Consider a Stratonovich stochastic differential equationdχt=X(χt)odBt+A(χt)dt(1.1)with C∞ coefficients on a compact Riemannian manifold M, with associated differential generator A=12ΔM+Z and solution flow {ξt : t ≥ 0} of random smooth diffeomorphisms of M. Let Tξt: TM → TM be the induced map on the tangent bundle of M obtained by differentiating ξt with respect to the initial point. Following an observation by A. Thalmaier we extend the basic formula of [EL94] to obtainEdf(Tξ.(h.))=EF(ξ.(χ))∫T0⟨Tξs(h˙s),X(ξs(χ))dBs⟩(1.2)where F∈FC∞b(Cχ(M)), the space of smooth cylindrical functions on the space C x (M) of continuous paths γ : [0,T] → M with γ(0) = x, dF is its derivative, and h. is a suitable adapted process with sample paths in the Cameron-Martin space L 2,10([0,T];T x M).Set F xt = σ{ξs(x) : 0 ≤ s ≤ t} Taking conditional expectation with respect to.F xT, formula (1.2) yields integration by parts formulae on C x (M) of the formEdF(γ)(V¯¯¯¯h)=EF(γ)δV¯¯¯¯h(γ)(1.3)where V¯¯¯¯h is the vector field on C x(M)V¯¯¯¯h(γ)t−E{Tξt(ht)|ξ.(χ)=γ}and δV¯¯¯¯h:Cx(M)→ is given byδV¯¯¯¯h(γ)=IE{∫T0<Tξs(h˙s),X(ξs(x))dBs>|ξ.(x)=γ}
Elworthy KD, Li X-M, 1995, Derivative flows of stochastic differential equations: moment exponents and geometric properties, Stochastic analysis (Ithaca, NY, 1993), Publisher: Amer. Math. Soc., Providence, RI, Pages: 565-574
Li X-M, 1995, On extensions of Myers’ theorem, The Bulletin of the London Mathematical Society, Vol: 27, Pages: 392-396, ISSN: 0024-6093
Li X-M, 1994, Properties at infinity of diffusion semigroups and stochastic flows via weak uniform covers, Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis, Vol: 3, Pages: 339-357, ISSN: 0926-2601
A unified treatment is given of some results of H. Donnelly, P. Li and L. Schwartz concerning the behaviour of heat semigroups on open manifolds with given compactifications, on one hand, and the relationship with the behaviour at infinity of solutions of related stochastic differential equations on the other. A principal tool is the use of certain covers of the manifold: which also gives a non-explosion test. As a corollary we obtain known results about the behaviour of Brownian motions on a complete Riemannian manifold with Ricci curvature decaying at most quadratically in the distance function.
Elworthy KD, Li X-M, 1994, Formulae for the derivatives of heat semigroups, Journal of Functional Analysis, Vol: 125, Pages: 252-286, ISSN: 0022-1236
Li X-M, 1994, Strong $p$-completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds, Probability Theory and Related Fields, Vol: 100, Pages: 485-511, ISSN: 0178-8051
Elworthy KD, Li X-M, 1994, Differentiation of heat semigroups and applications, Probability theory and mathematical statistics: Proceeding of the Sixth Vilnius Conference (1993), Editors: Kubilius, Grigelionis, Pragarauskas, Statulevicius, Publisher: TEV, Vilnius, Pages: 239-251
Li X-M, 1994, Stochastic differential equations on noncompact manifolds: moment stability and its topological consequences, Probability Theory and Related Fields, Vol: 100, Pages: 417-428, ISSN: 0178-8051
Elworthy KD, Li X-M, Rosenberg S, 1993, Curvature and topology: spectral positivity, Methods and applications of global analysis, Publisher: Voronezh. Univ. Press, Voronezh, Pages: 45-156
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