### Citations

743 |
Zabczyk J: Stochastic Equations in Infinite Dimensions
- G
- 1992
(Show Context)
Citation Context ...ichlet boundary conditions. Here d refers to the Ito differential ( in time). We let the driving noise fW t : ts0g be a cylindrical Wiener process on L 2 ([0; ��]) and consider (12) in the mild se=-=nse [DPZ92b]. Ta-=-ke H = L 2 ([0; ��]) as state space. Then a Markov semigroup P t is induced on the space of bounded measurable maps f : H ! R. With the use of Yosida approximations it is shown in [DPEZ95] that if... |

443 |
Stochastic Flows and Stochastic Differential Equations, Cambridge
- Kunita
- 1990
(Show Context)
Citation Context ...sing the Levi-Civita connection. 5.1 Compact manifolds A. For compact M there is a version of the solution flow which is continuous in time onto the space of C 1 diffeomorphisms of M [Elw78], [CE83], =-=[Kun90]-=-. The derivatives now consist of linear maps T x 0 F t : T x 0 M ! T x t M and if v t = T x 0 F t (v 0 ) for v 0 2 T x 0 M it satisfies the covariant equation Dv t = rX(v t ) ffi dB t +rA(v t )dt (13)... |

210 |
Functional integration and partial differential equations, volume 109 of Annals of Mathematics Studies
- Freidlin
- 1985
(Show Context)
Citation Context ... 1 in t, is given by: f t (x 0 ) = Ef 0 i x t t j e R t 0 V t\Gammas( x t s)ds (9) for fx T t g the solution of dx T t = X T \Gammat (x T t )dB t + Z T \Gammat (x T t )dt starting from x 0 . See e.g. =-=[Fre85]-=-. Let ff T t (x 0 ) = e R t 0 V T \Gammas (x T s )ds ; and let fv T t j TF T t (v 0 ) : ts0g be the solution to dv T t = DX T \Gammat (x T t )(v T t ) dB t +DZ T \Gammat (x T t )(v T t )dt with v T 0 ... |

93 |
Stochastic Differential Equations on Manifolds, London
- Elworthy
- 1982
(Show Context)
Citation Context ... a Lie group with bi-invariant Riemannian metric its solutions can be represented as solutions of a left invariant s.d.e. (1) (i.e. m=dimM and the A and X i are left invariant vector fields) e.g. see =-=[Elw82]-=-. The solution flow is given by F t (x 0 ) = g t x 0 ; ts0; x 0 2 M where fg t : ts0g is the solution starting from the identity element 1. Equation(14) can be written [EL94] as d(P t f)(v 0 ) = 1 t E... |

79 |
Large Deviations and the Malliavin Calculus,
- Bismut
- 1984
(Show Context)
Citation Context ... it is well known that P t is smoothing: it maps bounded measurable maps to C 1 maps for each t ? 0. This is a classical result from p.d.e. theory, or could be proved by Malliavin calculus. Indeed in =-=[Bis84]-=-, though for Brownian motion on a compact manifold, Bismut obtained a formula for the derivative of P t f in terms of f , and not involving the derivative of f ; thus clearly demonstrating the smoothi... |

65 |
Formulae for the derivatives of heat semigroups,
- Elworthy, Li
- 1994
(Show Context)
Citation Context ...on by parts formula can also be used to give such Bismut type formulae in a variety of situations and with elementary proofs (and statements). Here we will prove the basic formula on R n , taken from =-=[EL94]-=- and various variations and applications which have been made of it: the details of these will appear elsewhere. In particular we look at (a). state space R n , global Lipschitz coefficients [EL94], (... |

50 |
Lectures on Stochastic Differential Equations and Malliavin Calculus. Tata
- Watanabe
- 1984
(Show Context)
Citation Context ... t = y oe +E ae e R t 0 V t\Gammas (x t s )ds 1 t Z t 0 (t \Gamma r)DV t\Gammar (x t r )(v t r )dr j x t t = y oe In fact the equation holds for all y by continuity of both sides in y [Bis84] [Nor93] =-=[Wat84]-=-. For the time independent case with V j 0 we have k(0; x; t; y) = p t t (x; y) giving the logarithmic gradient formula: r log k(0; \Gamma; t; y)(x 0 ) = 1 t E aeZ t 0 (TF s ) X(x s )dB s j x t = y oe... |

26 |
Stochastic Flows in Riemannian manifolds, Diffusion Problems and Related Problems in Analytics
- Elworthy
- 1992
(Show Context)
Citation Context ...efficients are time dependent [EL94] (b). infinite dimensional state space, stochastic evolution equations [DPEZ95], [PZ93] (c). non-linear K.P.P. equations [LZ] (d). state space a compact manifold M =-=[Elw92]-=- (e). more general coefficients on R n or non-compact M [EL94] (f). Sobolev estimates [EL94] (g). heat equations for differential forms [Elw88] [Li92] [EL94]. There is also a version in preparation fo... |

26 |
Feller Property and Irreducibility for Diffusions on Hilbert Spaces.
- Peszat, Zabczyk, et al.
- 1995
(Show Context)
Citation Context ...vatives of Ptf (ii) the semigroup given by A + V where V is a potential, and the coefficients are time dependent [EL94] (b). infinite dimensional state space, stochastic evolution equations [DPEZ95], =-=[PZ93]-=- (c). non-linear K.P.P. equations [LZ] (d). state space a compact manifold M [Elw92] (e). more general coefficients on Rn or non-compact M [EL94] (f). Sobolev estimates [EL94] (g). heat equations for ... |

24 |
Geometric aspects of diffusions on manifolds
- Elworthy
- 1988
(Show Context)
Citation Context ....P. equations [LZ] (d). state space a compact manifold M [Elw92] (e). more general coefficients on R n or non-compact M [EL94] (f). Sobolev estimates [EL94] (g). heat equations for differential forms =-=[Elw88]-=- [Li92] [EL94]. There is also a version in preparation for the non-linear heat equations for maps between Riemannian manifolds, but we do not describe that here. 2 The basic formula on R n A. Assume o... |

20 |
Conditional expectations for derivatives of certain stochastic flows.
- Elworthy, Yor
- 1993
(Show Context)
Citation Context ...e predictable projection of fTF t (v 0 ) : ts0g using the filtration of fx t : 0st ! 1g. Essentially if you filter out the redundant noise in fTF t (v 0 ) : ts0g then fW Z t (v 0 ) : ts0g is obtained =-=[EY93]-=-: E fTF t (v 0 ) j oefx s : 0ssstgg = W Z t (v 0 ) where the conditional expectation is defined by parallel translating back to T x 0 M then taking the classical conditional expectation of the resulti... |

16 |
Compact families of Wiener functionals,
- Prato, Malliavin, et al.
- 1992
(Show Context)
Citation Context ...ense of mapping L 2 functions into some Sobolev space L 2;1 , see section 6 below. There are now results on compactness of the inclusion of such L 2;1 into L 2 at least for Gaussian measures [Pes93], =-=[DPMN92]-=-, and so it seems that there are many cases for which the infinite dimensional Markov semigroup consists of compact operators on L 2 , [DPZ93]. 5 Stochastic differential equations on manifolds Any ell... |

13 |
Freı̆dlin.: Some properties of diffusion processes depending on a parameter, Dokl
- Blagoveščenskiı̆, I
- 1961
(Show Context)
Citation Context ...ochastic differential equation with coefficients assumed to be globally Lipschitz: dx t = X(x t )dB t + Z(x t )dt: (3) Let fx t j F t (x 0 ) : ts0g be the solution to (3) starting at x 0 2 R n . From =-=[BF61]-=- we can take a version of this to give a solution flow jointly continuous in space and time, and C 1 in space for each time t. Suppose there is a bounded C 1 map Y : R n ! L(R n ; R m ) such that Y (x... |

12 |
Flows of stochastic dynamical systems: the functional analytic approach
- CARVERHILL, ELWORTHY
- 1983
(Show Context)
Citation Context ... +A(x) using the Levi-Civita connection. 5.1 Compact manifolds A. For compact M there is a version of the solution flow which is continuous in time onto the space of C 1 diffeomorphisms of M [Elw78], =-=[CE83]-=-, [Kun90]. The derivatives now consist of linear maps T x 0 F t : T x 0 M ! T x t M and if v t = T x 0 F t (v 0 ) for v 0 2 T x 0 M it satisfies the covariant equation Dv t = rX(v t ) ffi dB t +rA(v t... |

12 |
Stochastic flows on noncompact manifolds,
- Li
- 1992
(Show Context)
Citation Context ...tions [LZ] (d). state space a compact manifold M [Elw92] (e). more general coefficients on R n or non-compact M [EL94] (f). Sobolev estimates [EL94] (g). heat equations for differential forms [Elw88] =-=[Li92]-=- [EL94]. There is also a version in preparation for the non-linear heat equations for maps between Riemannian manifolds, but we do not describe that here. 2 The basic formula on R n A. Assume our stat... |

11 |
Stochastic dynamical systems and their flows.
- Elworthy
- 1978
(Show Context)
Citation Context ...(X i (x)) +A(x) using the Levi-Civita connection. 5.1 Compact manifolds A. For compact M there is a version of the solution flow which is continuous in time onto the space of C 1 diffeomorphisms of M =-=[Elw78]-=-, [CE83], [Kun90]. The derivatives now consist of linear maps T x 0 F t : T x 0 M ! T x t M and if v t = T x 0 F t (v 0 ) for v 0 2 T x 0 M it satisfies the covariant equation Dv t = rX(v t ) ffi dB t... |

10 |
Path integral formulae for heat kernels and their derivatives,
- Norris
- 1993
(Show Context)
Citation Context ...vative of f ; thus clearly demonstrating the smoothing properties in this case. This then led to his celebrated formula for the logarithmic derivative of the heat kernel, r x log p t (x; y), see also =-=[Nor93]-=-. It turns out that an approach used by Elliot & Kohlmann [EK89] to prove Malliavin's integration by parts formula can also be used to give such Bismut type formulae in a variety of situations and wit... |

5 |
Strong Feller property and irreducibility for diffusions on Hilbert spaces
- Prato, Zabczyk
- 1993
(Show Context)
Citation Context ...h L 2;1 into L 2 at least for Gaussian measures [Pes93], [DPMN92], and so it seems that there are many cases for which the infinite dimensional Markov semigroup consists of compact operators on L 2 , =-=[DPZ93]-=-. 5 Stochastic differential equations on manifolds Any elliptic generator A coming from an s.d.e. (1) on R n , or more generally on a smooth manifold, can be written A = 1 2 \Delta + Z where Z is a ve... |

4 |
Regular densities of invariant measures for nonlinear stochastic equations
- Prato, Zabczyk
- 1995
(Show Context)
Citation Context ...ticular important since [Kha60], given irreducibility, it implies that the transition probabilities are mutually absolutely continuous and that there is at most one invariant probability measure, see =-=[DPZ92a]-=-. For the precise class of stochastic differential equations for which such a result holds we refer to [DPEZ95]. The basic point is that the non-linearity can be very irregular (e.g. u 7! \Gammau 2k+1... |

4 |
Integration by parts, homogeneous chaos expansions and smooth densities
- Elliott, Kohlmann
- 1989
(Show Context)
Citation Context ... in this case. This then led to his celebrated formula for the logarithmic derivative of the heat kernel, r x log p t (x; y), see also [Nor93]. It turns out that an approach used by Elliot & Kohlmann =-=[EK89]-=- to prove Malliavin's integration by parts formula can also be used to give such Bismut type formulae in a variety of situations and with elementary proofs (and statements). Here we will prove the bas... |

3 | Gradient estimates and the smooth convergence of approximate travelling waves for reactiondiffusion equations
- Li, Zhao
(Show Context)
Citation Context ...by A + V where V is a potential, and the coefficients are time dependent [EL94] (b). infinite dimensional state space, stochastic evolution equations [DPEZ95], [PZ93] (c). non-linear K.P.P. equations =-=[LZ]-=- (d). state space a compact manifold M [Elw92] (e). more general coefficients on R n or non-compact M [EL94] (f). Sobolev estimates [EL94] (g). heat equations for differential forms [Elw88] [Li92] [EL... |

3 |
The traveling wave solutions of scalar generalized KPP equation via classical mechanics and stochastic approaches, Stochastics and quantum mechanics
- Elworthy, Zhao
- 1992
(Show Context)
Citation Context ... ��: 8 ? ! ? : @f �� t @t = �� 2 2 \Deltaf �� t + 1 �� 2sc(1 \Gamma f �� t )f �� t f �� 0 (x) = e \Gamma ff �� 2 x 2 (21) for ff andsc constants. Then following Fre=-=idlin e.g. [Fre85], it was shown in [ZE92] that li-=-m ��!0 jf �� t (x)j = 0; if ffx 2 ?sct(1 + 2fft): We shall show that this convergence is C 1 in x as an example of more general results from [LZ]. First by (10), with B \Delta now one-dimensio... |

2 |
Feller property for stochastic semilinear equations. Stochastic Analysis and its Applications
- Strong
- 1995
(Show Context)
Citation Context ...her derivatives of P t f (ii) the semigroup given by A + V where V is a potential, and the coefficients are time dependent [EL94] (b). infinite dimensional state space, stochastic evolution equations =-=[DPEZ95]-=-, [PZ93] (c). non-linear K.P.P. equations [LZ] (d). state space a compact manifold M [Elw92] (e). more general coefficients on R n or non-compact M [EL94] (f). Sobolev estimates [EL94] (g). heat equat... |

2 |
Sobolev spaces of functions on an infinite-dimensional domain. Warwick Preprint 49/1993
- Peszat
- 1993
(Show Context)
Citation Context ... in the sense of mapping L 2 functions into some Sobolev space L 2;1 , see section 6 below. There are now results on compactness of the inclusion of such L 2;1 into L 2 at least for Gaussian measures =-=[Pes93]-=-, [DPMN92], and so it seems that there are many cases for which the infinite dimensional Markov semigroup consists of compact operators on L 2 , [DPZ93]. 5 Stochastic differential equations on manifol... |

1 |
Strong Feller property from semilinear stochastic evolution equation and stabilization of the solution to the Cauchy problem for parabolic equations. Theory of probability and its applications
- Khas'minskii
- 1960
(Show Context)
Citation Context ...t boundary conditions. In particular this shows that P t maps bounded measurable functions to continuous functions, if t ? 0; the 'strong Feller' property. This property is particular important since =-=[Kha60]-=-, given irreducibility, it implies that the transition probabilities are mutually absolutely continuous and that there is at most one invariant probability measure, see [DPZ92a]. For the precise class... |