Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural?

It is shown that the geometrically correct investigation of regularity of nonlinear differential flows on manifolds and related parabolic equations requires the introduction of a new type of variations with respect to the initial data. These variations are defined via a certain generalization of a c...

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Hauptverfasser:	Antoniouk, A.Val., Antoniouk, Vict.
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spelling	irk-123456789-1653792020-02-14T01:28:00Z Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural? Antoniouk, A.Val. Antoniouk, Vict. Статті It is shown that the geometrically correct investigation of regularity of nonlinear differential flows on manifolds and related parabolic equations requires the introduction of a new type of variations with respect to the initial data. These variations are defined via a certain generalization of a covariant Riemannian derivative to the case of diffeomorphisms. The appearance of curvature in the structure of high-order variational equations is discussed and a family of a priori nonlinear estimates of regularity of any order is obtained. By using the relationship between the differential equations on manifolds and semigroups, we study C∞ regular properties of solutions of the parabolic Cauchy problems with coefficients increasing at infinity. The obtained conditions of regularity generalize the classical coercivity and dissipation conditions to the case of a manifold and correlate (in a unified way) the behavior of diffusion and drift coefficients with the geometric properties of the manifold without traditional separation of curvature. Показано, що геометрично коректне дослідження регулярності нелінійних диференціальних потоків на багатовидах та асоційованих параболічних рівнянь вимагає введення нового типу варіацій за початковими умовами. Ці варіації означені за допомогою певного узагальнення коваріантної похідної Рімана на випадок дифеоморфізмів. Встановлено, яким чином кривина виникає в варіаційних рівняннях високого порядку, і одержано сім'ю апріорних нелінійних оцінок на регулярність довільного порядку. Використовуючи зв'язокміж диференціальними рівняннями на багатовидах і напівгрупами, досліджено C∞-гладкі властивості розв'язків параболічних задач Коші зі зростаючими на нескінченності коефіцієнтами. Отримані умови регулярності узагальнюють класичні умови коерцитивності та дисипативності на випадок багатовиду і пов'язують поведінку коефіцієнтів дифузії та зсуву з геометричними властивостями багатовиду, без традиційного відокремлення кривини. 2006 Article Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural? / A.Val. Antoniouk, A.Vict. Antoniouk // Український математичний журнал. — 2006. — Т. 58, № 8. — С. 1011–1034. — Бібліогр.: 27 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165379 en Український математичний журнал Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
topic	Статті Статті
spellingShingle	Статті Статті Antoniouk, A.Val. Antoniouk, Vict. Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural? Український математичний журнал
description	It is shown that the geometrically correct investigation of regularity of nonlinear differential flows on manifolds and related parabolic equations requires the introduction of a new type of variations with respect to the initial data. These variations are defined via a certain generalization of a covariant Riemannian derivative to the case of diffeomorphisms. The appearance of curvature in the structure of high-order variational equations is discussed and a family of a priori nonlinear estimates of regularity of any order is obtained. By using the relationship between the differential equations on manifolds and semigroups, we study C∞ regular properties of solutions of the parabolic Cauchy problems with coefficients increasing at infinity. The obtained conditions of regularity generalize the classical coercivity and dissipation conditions to the case of a manifold and correlate (in a unified way) the behavior of diffusion and drift coefficients with the geometric properties of the manifold without traditional separation of curvature.
format	Article
author	Antoniouk, A.Val. Antoniouk, Vict.
author_facet	Antoniouk, A.Val. Antoniouk, Vict.
author_sort	Antoniouk, A.Val.
title	Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural?
title_short	Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural?
title_full	Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural?
title_fullStr	Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural?
title_full_unstemmed	Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural?
title_sort	regularity of nonlinear flows on noncompact riemannian manifolds: differential geometry versus stochastic geometry or what kind of variations is natural?
publisher	Інститут математики НАН України
publishDate	2006
topic_facet	Статті
url	http://dspace.nbuv.gov.ua/handle/123456789/165379
citation_txt	Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural? / A.Val. Antoniouk, A.Vict. Antoniouk // Український математичний журнал. — 2006. — Т. 58, № 8. — С. 1011–1034. — Бібліогр.: 27 назв. — англ.
series	Український математичний журнал
work_keys_str_mv	AT antonioukaval regularityofnonlinearflowsonnoncompactriemannianmanifoldsdifferentialgeometryversusstochasticgeometryorwhatkindofvariationsisnatural AT antonioukvict regularityofnonlinearflowsonnoncompactriemannianmanifoldsdifferentialgeometryversusstochasticgeometryorwhatkindofvariationsisnatural
first_indexed	2025-07-14T18:22:49Z
last_indexed	2025-07-14T18:22:49Z
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fulltext	UDC 519.217.4, 517.955.4, 517.956.4, 517.958:536.2 A. Val. Antoniouk, A. Vict. Antoniouk (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv) REGULARITY OF NONLINEAR FLOWS ON NONCOMPACT RIEMANNIAN MANIFOLDS: DIFFERENTIAL GEOMETRY VERSUS STOCHASTIC GEOMETRY OR WHAT KIND OF VARIATIONS IS NATURAL?∗ REHULQRNIST\ NELINIJNYX POTOKIV NA NEKOMPAKTNYX RIMANOVYX MNOHOVYDAX: DYFERENCIAL\NA HEOMETRIQ PROTY STOXASTYÇNO} ABO QKI VARIACI} { PRYRODNYMY? We demonstrate that the geometrically correct study of the regularity of nonlinear differential flows on manifolds and related parabolic equations requires the introduction of a new type of variations with respect to the initial data. These variations are defined via a certain generalization of a covariant Riemannian derivative to the case of diffeomorphisms. We find how the curvature appears in the structure of higher-order variational equations and determine a family of a priori nonlinear estimates of any-order regularity. Using the relation between differential equa- tions on manifolds and semigroups, we investigate C∞-regular properties of solutions of the Cauchy parabolic problems with coefficients growing on infinity. The obtained conditions of regularity generalize the classical coercitivity and dissipativity conditions to the case of manifold and relate in unified way the behaviour of diffusion and drift coefficients with the geometric properties of the manifold without the traditional separation of curvature. Pokazano, wo heometryçno korektne doslidΩennq rehulqrnosti nelinijnyx dyferencial\nyx potokiv na bahatovydax ta asocijovanyx paraboliçnyx rivnqn\ vymaha[ vvedennq novoho typu variacij za poçat- kovymy umovamy. Ci variaci] oznaçeni za dopomohog pevnoho uzahal\nennq kovariantno] poxidno] Rimana na vypadok dyfeomorfizmiv. Vstanovleno, qkym çynom kryvyna vynyka[ v variacijnyx rivnqnnqx vysokoho porqdku, i oderΩano sim’gapriornyx nelinijnyx ocinok na rehulqrnist\ dovil\noho porqdku. Vykorystovugçy zv’qzokmiΩ dyferencial\nymy rivnqnnqmy na bahatovydax i napivhrupamy, doslidΩeno C∞ -hladki vlastyvosti rozv’qzkiv paraboliçnyx zadaç Koßi zi zrostagçymy na neskinçennosti koefici[ntamy. Otrymani umovy rehulqrnosti uzahal\nggt\ klasyçni umovy koercytyvnosti ta dysypatyvnosti na vypadok bahatovydu i pov’qzugt\ povedinku koefici[ntiv dyfuzi] ta zsuvu z heometryçnymy vlas- tyvostqmy bahatovydu, bez tradycijnoho vidokremlennq kryvyny. 1. Introduction. Up-to-date there are formed a lot of qualitively different approaches to the construction and study of differential equations on manifolds with random terms. In transfer from the linear space R d to manifold the main attention was to make consistent the geometrical structures of manifold with the purely stochastic effects, influenced by second order differentials, that arise in Itô formula for coordinate changes. The already known approaches include, in particular: purely stochastic, based on the definition of diffusion in a consistent with geometry way by implementation of Stratonovich integrals [1 – 3] or more complicate description of diffusion via Itô equations in local coordinates [4, 5]; in the second case arise special Itô bundles of nontensorial fields, related with diffusion coefficients; to make the pic- ture consistent, a special attention should be devoted to the normal charts, generated by exponential mappings, more geometric, related, for example, with the raise of diffusion from manifold M to the orthoframe bundle O(M) over it; the direct advantage of such approach is related with the globally existing horizontal vector fields and possibility to write diffusions with Laplacian generator for manifolds with non-zero Euler number; as it has been becoming clear for years, these geometric * This research was supported by Alexander von Humboldt Foundation. c© A. VAL. ANTONIOUK, A. VICT. ANTONIOUK, 2006 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 1011 1012 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK ideas influenced much the construction of advanced analysis on Wiener space and raise of profound analytical questions of stochastics in the Malliavin calculus [6, 7], or with the consideration of manifold as embedded into R d of higher dimension, e.g. [8, 9]; with interpretation of Itô differential and diffusion equations as defined on the bundle of second-order differential operators [10, 11]; putting forward Itô developments of equations via parallel transitions of orthoframes [12]; with more stress on properties of associated transitional probabilities [13], etc. One can continue this list, by speaking about peculiarities, related with other infinite- dimensional models [9, 14]. The procedure of correct correspondence between geometry and stochastics was suc- cessful in all cases. However, further question of consistency with the problematic of differential geometry, namely how the geometrically invariant differentials are constructed from invariant objects, (1.1) remained in shadow. Of course, one may try to consider the traditional derivatives in directions of vector fields or more advanced covariant and stochastic derivatives, e.g. [4 – 7], but as we will soon see, they all miss an important property of geometric invariance with respect to the diffusion process. Other approach to define the derivatives via stochastic parallel transport γ//yx of cor- responding derivatives or via Cartan orthoframes, e.g. [6, 7, 12, 13], does not provide a transparent definition of geometrically invariant derivatives. Such transport essentially depends on particular path of process y and therefore on the coefficients of equation. But the correct definition of higher-order derivatives should be quite general and does not depend on particular equation. Such definition is possible. Let us turn to the corresponding constructions. Consider the following situation. Sup- pose that some process yt (of diffusion or any other nature) enters coordinate vicinity U ⊂ M of manifold with coordinate functions ϕ = (ϕi)dimM i=1 , ϕ : U → R dimM, so that one can speak about the coordinates of process yit = (ϕi)◦yt when it stays in this vicinity. Now let D be some first order differentiation operation, correctly defined on process yt. What kind of differentiation it could be is not essential now, the principal moment is that the first order differentiation must obey chain rule D(f ◦ y) = (f ′ ◦ y)Dy. In particular, because the local coordinate changes yi ′ = ϕi′(yt) = (ϕi′ ◦ϕinv)(yi)dimM i=1 represent a special case of locally defined functions, one has rule Dym ′ = ∂ym ′ ∂ym Dym. Therefore, though process yt does not determine some coordinate system, like local co- ordinate mappings ϕ, ϕ′ do, the expression Dy becomes a vector field with respect to the “coordinate” changes (y) → (y′) of ”coordinate” variable y. By classical arguments of differential geometry, related with the standard construction of covariant derivatives, ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 REGULARITY OF NONLINEAR FLOWS ON NONCOMPACT RIEMANNIAN MANIFOLDS ... 1013 the only way to give a correct definition of the higher-order derivatives (D̃)iy should use additional terms with connection Γ(yt). The correct recurrent definition of the invariant higher-order variations will be D̃ym = Dym, D̃[ (D̃)iym ] = D[ (D̃)iy ] + Γ m p q(y) [ (D̃)iyp]Dyq. Like in the classical differential geometry, additional terms with Γ(y) in definition of higher-order derivatives D̃n guarantee the preservance of vector transformation law with respect to the (y) → (y′) coordinate transformations: (D̃)nym ′ = ∂ym ′ ∂ym (D̃)nym ∀n ≥ 1. In view of problem (1.1), such invariance with respect to the changes of local coor- dinates (y) → (y′) in vicinity, where comes process y, represents a new and purely geo- metric requirement of first priority. From another side, the relation between differentials in time of this change (y) → (y′), given by Itô formula, reflects the behaviour on time coordinate and is secondarily. It is purely stochastic and related with the nonvoidness of quadratic variation processes. Finally, let us remark that the above construction and the way to introduce the new type derivatives is independent on particular approach we choose to define the diffusion on manifold, actually it works for any differential equations (higher-order, etc.) on manifolds, because, by consideration above, symbol y ∈ M must have values in manifold, but noth- ing more. This is especially underlined in the article by the use of notation y instead of traditional Greek letters, like ξ, η, ζ, for stochastic processes. In this article we discuss more concrete case of general diffusion process yxt on non- compact manifold. We investigate its regular dependence on initial data x and provide the geometrically correct construction of higher-order variations. Consider the first order variation ∂(yxt )m ∂xk , that represents a vector field on index m for (y) → (y′) ”coordinate” transformations and covector field on index k for (x) → (x′) coordinate changes. From arguments above we can immediately conclude that the defini- tion of geometrically invariant higher-order variations must include terms with Γ(x) and Γ(y) to guarantee the preservance of tensorial character on both ”image” (y) → (y′) and ”domain” (x) → (x′) coordinate changes of mapping x → yxt . Definition 1.1. Higher-order variations ∇∇x γy x t , γ = {k1, . . . , kn}, of process yxt are defined by recurrent relations ∇∇x ky m = ∂(yxt )m ∂xk , ∇∇x k(∇∇x γy m) = ∇x k(∇∇x γy m) + Γ m p q(y x t )∇∇x γy p ∂yq ∂xk (1.2) where ∇x k(∇∇x γy m) represents a classical covariant derivative ∇x k(∇∇x γy m) = ∂x k (∇∇x γy m) − ∑ j∈γ Γ h k j(x)∇∇x γ\|j=h ym and ∇∇x γ\|j=h ym means substitution of index j by h. ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 1014 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK From the point of view of classical Riemannian geometry such definition of the higher- order invariant variation of y with terms Γ(x) and Γ(y) provides generalization of the classical covariant derivative. Unlike all already existing torsion, polynomial connection and other generalizations of variation, defined primarily at point x, it depends not only on initial point of differentiation x, but also on behaviour of process at point y. Remark 1.1. Due to the invariance of higher-order variations ∇∇x i1,...,isy m with re- spect to the (y) → (y′) and (x) → (x′) coordinate changes we can introduce the invari- ant norm of the higher-order variation ‖(∇∇x)jyxt ‖2 = gmn(yxt ) j∏ s=1 gisks(x)∇∇x i1,...,isy m∇∇x k1,...,ks yn. (1.3) After that the regularity problem becomes well-posed geometrically and the regular be- haviour of process yxt with respect to the initial data can be expressed in terms of some estimates on higher-order variations ∇∇x i1,...,isy m. In this article we also demonstrate how the geometry of manifold and its curvature is reflected in the structure of equations on new type variations. We also find conditions on existence and uniqueness of variational processes, which give a natural generalization of coercitivity and dissipativity conditions to the manifold case and can be used even for noncompact and infinite-dimensional manifolds. In particular, they relate the behaviour of geometry and diffusion in a unified way, without traditional separation of curvature. We also discuss the consequences of regular dependence of diffusion on initial data for the smooth properties of semigroups. The use of nonlinear symmetries of variational equations and a set of associated nonlinear estimates permits us to study the regular prop- erties of semigroup in the case of globally non-Lipschitz behaviour of nonlinear coeffi- cients on infinity. The development of advanced constructions of Malliavin calculus to the new type stochastic derivatives D̃, that generalize the classical Malliavin and Bismut derivatives, and applications to the raise of smoothness properties of diffusion semigroups is a subject of [15]. 2. Invariant representation of semigroup derivatives in terms of new variations. On noncompact connected oriented smooth Riemannian manifold M without boundary consider diffusion yxt , written in Stratonovich form yxt = x + t∫ 0 A0(yxs )ds + d∑ α=1 t∫ 0 Aα(yxs )δWα s . (2.1) Here A0, Aα, α = 1, . . . , d, are smooth vector fields, globally defined on M , initial data x ∈ M and Wα s denotes the R d-valued Wiener process. Equation (2.1) is understood in sense that for any smooth function with compact sup- port f ∈ C2 0 (M) the following equation: f(yxt ) = f(x) + t∫ 0 (A0f)(yxs )ds + d∑ α=1 t∫ 0 (Aαf)(yxs )δWα s (2.2) holds as usual equation in R 1. In particular, one can take functions f i(x) = xi to be local coordinates and find generator of yxt ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 REGULARITY OF NONLINEAR FLOWS ON NONCOMPACT RIEMANNIAN MANIFOLDS ... 1015 Lf = A0f + 1 2 ∑ α Aα(Aαf). (2.3) Below we are going to study how the properties of nonlinear diffusion Aα and drift A0 coefficients should be related with the geometric properties of manifold to lead to the regular dependence of process yxt on initial data and smooth properties of corresponding diffusion semigroup (Ptf)(x) = Ef(yxt ) (2.4) in some scales of continuously differentiable functions on manifold. To obtain the regular properties of semigroup one should consider its higher-order derivatives. Taking formally the first-order derivative of (2.4) we have ∇kPtf(x) = ∂ ∂xk Ef(yxt ) = E ∂f(yxt ) ∂ym ∂(yxt )m ∂xk . (2.5) This representation is invariant with respect to the local coordinates transformations (x) → → (x′), because in (2.5) the first-order variation ∂(yxt )m ∂xk of diffusion with respect to the initial data is covector field on index k with respect to coordinate transformations of domain (x) → → (x′); vector field on index m with respect to the choice of local coordinate vicinity for diffusion (y) → (y′). To find the higher-order representation of semigroup derivatives let us write the second order covariant derivative of semigroup ∇k∇jPtf(x) = { ∂ ∂xk ∂ ∂xj − Γ h k j(x) ∂ ∂xh } Ptf(x) = = E { ∂ ∂xk ∂ ∂xj − Γ h k j(x) ∂ ∂xh } f(yxt ) = = E { ∂f(y) ∂ym ∂2ym ∂xk∂xj + ∂2f(y) ∂ym∂yn ∂ym ∂xk ∂yn ∂yj − Γ h k j(x) ∂f(y) ∂ym ∂ym ∂xh } , (2.6) where the covariant derivative of a tensor field is defined in a standard way ∇x ku i1,...,ip j1,...,jq = ∂ ∂xk u i1,...,ip j1,...,jq + p∑ s=1 Γis k �(x)ui1,...,ip\|is=� j1,...,jq − q∑ s=1 Γ� k js (x)ui1,...,ip j1,...,jq\|js=� (2.7) u i1,...,ip\|is=� j1,...,jq means substitution of index is by !, the summation on repeating indexes is implemented, and Γ(x) are connection coefficients. Now let us form the covariant derivatives of f in the right-hand side of (2.6). Using that ∇y �f(y) = ∂ ∂y� f(y) and ∇y m∇y nf(y) = ∂ ∂ym ∂ ∂yn f(y) − Γ � m n(y) ∂ ∂y� f(y) ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 1016 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK we can continue (2.6) ∇k∇jPtf(x) = E { (∇y m∇y nf(y) + Γ � m n(y)∇y �f(y) ) ∂ym ∂xk ∂ym ∂xj + (2.8) +∇y mf(y) ( ∂2ym ∂xk∂xj − Γ h k j(x) ∂ym ∂xh )} = = E { ∇y m∇y nf(y) ∂ym ∂xk ∂yn ∂xj + +∇y mf(y) ( ∂2ym ∂xk∂xj − Γ h k j(x) ∂ym ∂xh + Γ m � n(y) ∂y� ∂xk ∂yn ∂xj )} . (2.9) Here we redenoted index ! in term with Γ(y). The first term ∇y∇yf ∂y ∂x ∂y ∂x is obviously invariant under transformations (x) → (x′) and (y) → (y′), but what one should do with the expression in brackets ∂2ym ∂xk∂xj − Γ h k j(x) ∂ym ∂xh + Γ m � n(y) ∂y� ∂xk ∂yn ∂xj ? (2.10) There are two ways to collect the terms in brackets. 1st way. One may form the covariant derivative on x variable from first and second terms ∂2ym ∂xk∂xj −Γ h k j(x) ∂ym ∂xh +Γ m � n(y) ∂y� ∂xk ∂yn ∂xj = ∇x k ( ∂ym ∂xj ) +Γ m � h(y) ∂y� ∂xk ∂yn ∂xj . (2.11) Such representation is obviously invariant with respect to transformations (x) → (x′). The third term with connection Γ(y) has transformation of coordinates law, that in- cludes the second-order derivatives of coordinate change, similar to Itô formula. There- fore the traditional interpretation of (2.11) was that in the stochastic case one should add terms with Γ(y) to the classical covariant derivative to compensate the influence of Itô for- mula. A concept of stochastic differential geometry as a mixture of classical differential geometry and Itô formula arose [5, 10, 11, 16]. 2nd way. It is not clear, whether representation (2.11) is invariant with respect to the transformations (y) → (y′) in the image. Let us work with (2.10) in other way, by collecting first and third terms together ∂2ym ∂xk∂xj − Γ h k j(x) ∂ym ∂xh + Γ m � n(y) ∂y� ∂xk ∂yn ∂xj = = ∂ ∂xk ( ∂ym ∂xj ) − Γ h k j(x) ∂ym ∂xh + Γ m � h(y) ∂y� ∂xk ∂yn ∂xj = = ∂y� ∂xk ∂ ∂y� ( ∂ym ∂xj ) − Γ h k j(x) ∂ym ∂xh + Γ m � h(y) ∂y� ∂xk ∂yn ∂xj = = ∂y� ∂xk ∇y � ( ∂ym ∂xj ) − Γ h k j(x) ∂ym ∂xh , (2.12) where we used that ∂ ∂xk = ∂y� ∂xk ∂ ∂y� . This representation, in comparison to (2.11), is obviously invariant with respect to the coordinate changes (y) → (y′). ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 REGULARITY OF NONLINEAR FLOWS ON NONCOMPACT RIEMANNIAN MANIFOLDS ... 1017 Therefore all terms in (2.10) define a second variation of process yxt and represent vector field on index m with respect to the ”Itô” changes of coordinates (y) → (y′); twice covariant field on indexes k, j with respect to the ”differential geometric” changes of coordinates (x) → (x′). If one knows how to define the second-order covariant derivative, then, by common procedures of differential geometry, its higher-order analogies could be easily written (Definition 1.1). The invariance of (1.2) with respect to (x) → (x′) transformations is obvious, for transformations in image (y) → (y′) one should argue like in (2.12), e.g. [17]. Remarks. 2.1. One should note that arguments above also work for the choice Aα ≡ ≡ 0, i.e., in the ordinary differential equations case, when stochastic terms do not appear and there are no complications, related with Itô formula. Therefore the introduction of higher-order variation is a pure question of differential geometry. 2.2. The last term with Γ(y) in (1.2) depends on solution yxt and ensures that the higher-order variation, similar to the first-order variation, remains a vector field with re- spect to transformations (y) → (y′). It compensates the inevitably arising derivatives on variable x of jacobians ∂(y′(x, t))n ∂(y(x, t))m of coordinate changes (y) → (y′). 2.3. The consideration of classical derivatives along vector fields, covariant deriva- tives of the first-order variation ∇x kn . . .∇x k2 ∂(yxt )m ∂xk1 , or similar objects, like in [4, 5], destroys the invariance with respect to transformations (y) ↔ (y′) and leads to the geometrically noninvariant objects. Such approach also hides the curvature in a set of geometrically noninvariant variational equations. Using variations ∇∇x γy x t , we can now write invariant representations of semigroup’s derivatives: Theorem 2.1. The covariant derivatives of semigroup action and initial function are related via new type variations by ∇x γPtf(x) = ∑ δ1∪...∪δs=γ E (∇y {j1,...,js}f)(yxt )∇∇x δ1y j1 . . .∇∇x δs yjs . (2.13) Here ∇x γ = ∇x k1 . . .∇x kn for γ = {k1, . . . , kn}. Proof. This representation is easily verified recurrently. Indeed, suppose it is true for all \|γ\| ≤ n. Similar to (2.8), one should consider next order derivative ∇x k∇x γPtf(x) = ∂x k∇x γPtf(x) − ∑ j∈γ Γ h k j(x)∇x γ\|j=h Ptf(x). Then one should substitute expressions (2.13), add and subtract Γ(y) to form the higher- order covariant derivatives of f , redenote summation indices and come to (2.13) for ∇x k∇x γPtf . ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 1018 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK 3. Tensors on domain (x) and image (y) coordinates and recurrent form of the higher-order variational equations. Being equipped with the new definition of variation with respect to the initial data and corresponding representations of semigroup’s derivatives, we can turn to the regularity problem. First we find the recurrent equations on higher-order variations. Differentiating (2.1) on initial data x we have δ ( ∂ym ∂xk ) = ( ∂ ∂xk Am α (y) ) δWα + ( ∂ ∂xk Am 0 (y) ) dt. (3.1) To proceed further it is necessary to give an invariant sense to the partial derivatives ∂ ∂x A(yxt ) in the above equation. To do this we need a certain generalization of Defini- tion 1.1 for tensors on (x) and (yxt ) coordinates, given by Definition 3.2. Definition 3.1. Object u (i/α) (j/β) forms a mixed tensor with respect to the coordinate changes (x) → (x′) and (φ) → (φ′) iff its coordinates u (i/α) (j/β) = u i1...ip/α1...αr j1...jq/β1...βs form T p,q x M tensor on multiindexes (i) = (i1, . . . , ip), (j) = (j1, . . . , jq) with respect to the local coordinates (xk) and form T r,sM tensor on multiindexes (α), (β) with respect to the local coordinates (φm). In other words, after the simultaneous change of local coordinate systems (xk) → → (xk′ ) and (φm) → (φm′ ) one has transformation law u (i/α) (j/β) = ∂x(i) ∂x(i′) ∂x(j′) ∂x(j) ∂φ(α) ∂φ(α′) ∂φ(β′) ∂φ(β) u (i′/α′) (j′/β′) (3.2) with jacobians ∂x(i) ∂x(i′) = ∂xi1 ∂xi′1 . . . ∂xip ∂xi′p , ∂φ(α) ∂φ(α′) = ∂φα1 ∂φα′ 1 . . . ∂φαs ∂φα′ s . Examples. 3.1. A simple example of mixed tensor provide variations ∇∇j1 . . . . . .∇∇jk ym(x, t). They form vector fields on index m in a coordinate chart ym(x, t) and covector on j1, . . . , jk in coordinate vicinity (x). 3.2. Another example of mixed tensor is given by product of tensors u (α) (β)(y x t )v(i) (j)(x) in vicinities (x) and (y). The change of coordinates at x does not influence u (α) (β)(y x t ) part, jacobians of coordinate changes arise only near v (i) (j)(x). However the different choice of coordinate vicinities for y evoke the tensorial transformation law for u (α) (β) multiple. Now let us suppose that (φm) coordinates of the mixed tensor depend in effective way on the coordinates (xk). An analogue of Definition 1.1 for mixed tensors is given by the following definition. Definition 3.2. ∇∇-derivative of a mixed tensor is defined by ∇∇x ku (i/α) (j/β) = ∂ ∂xk u (i/α) (j/β) + ∑ s∈(i) Γ s k h(x)u(i/α)\|s=h (j/β) − ∑ s∈(j) Γ h k s(x)u(i/α) (j/β)\|s=h + (3.3) + ∑ ρ∈(α) Γ ρ γ δ(φ(x))u(i/α)\|ρ=δ (j/β) ∂φδ ∂xk − ∑ ρ∈(β) Γ γ ρ δ(φ(x))u(i/α) (j/β)\|ρ=γ ∂φδ ∂xk . (3.4) Line (3.3) corresponds to the covariant derivative on (xk) coordinates, additional line (3.4) makes the resulting expression to be tensor with respect to the coordinates in im- ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 REGULARITY OF NONLINEAR FLOWS ON NONCOMPACT RIEMANNIAN MANIFOLDS ... 1019 age (φm). One may also note that the connection symbols above depend on different parameters and the additional jacobians ∂φ ∂x are required in line (3.4). Remarks. 3.1. The tensorial character of ∇∇-derivative is easily checked, like before: ∇∇-derivative defines a tensor of higher valence, i.e., the mixed tensor law holds ∇∇x ku (i/α) (j/β) = ∂xk′ ∂xk ∂x(i) ∂x(i′) ∂x(j′) ∂x(j) ∂φ(α) ∂φ(α′) ∂φ(β′) ∂φ(β) ∇∇x k′u (i′/α′) (j′/β′). The proof of this property is easy by application of the transformation of connection law [17]. 3.2. An important property of ∇∇-derivative is the superposition rule: let u (α) (β) be a tensor on manifold M , then ∇∇x ku (α) (β)(φ(x)) = ( ∇�u (α) (β) )( φ(x) ) ∂φ� ∂xk . (3.5) To check (3.5) we use Definition (3.3) (3.4) to obtain ∇∇x ku (α) (β)(φ(x)) = ∂x ku (0/α) (0/β)(φ(x))+ + ∑ ρ∈(α) Γ ρ γ δ(φ)u(0/α)\|s=γ (0/β) (φ) ∂φδ ∂xk − ∑ ρ∈(β) Γ γ ρ δ(φ)u(0/α) (0/β)\|ρ=γ ∂φδ ∂xk . By chain rule for ∂x k and definition of covariant derivative (2.7) one gets the statement. As we will soon see, property (3.5) simplifies the geometrically correct calculation of the higher-order variational equations. After the introduction of mixed tensor and its ∇∇-derivative, we can further transform equation on the first variation (3.1). By adding and subtracting the terms with Γ(y) to single out the ∇∇-derivative of vector fields A0(y), Aα(y) on image coordinates (y), we have δ ( ∂ym ∂xk ) = = ( ∇∇x kA m α (y) − Γ m p q(y)Ap α ∂yq ∂xk ) δWα + ( ∇∇x kA m 0 (y) − Γ m p q(y)Ap 0 ∂yq ∂xk ) dt. Noting that the terms near connection contain the differential of process y (2.1) and using the symmetry of connection Γ m p q = Γ m q p, we find another representation for the equation on first variation δ ( ∂ym ∂xk ) = −Γ m p q(y) ∂yp ∂xk δyq + ∇∇x k (Am α (y))δWα + ∇∇x k(A m 0 (y)) dt. (3.6) We obtain an additional argument in favor of new type variations and ∇∇-derivatives: up to the parallel transition term with Γ(y) the increments of first-order variation are determined by ∇∇-derivatives of coefficients. We take this observation as the recurrence base for the search of higher-order variational equations. Theorem 3.1. Suppose that the equation on ∇∇-variation ∇∇x γy m, \|γ\| ≥ 1, is written in form δ(∇∇x γy m) = −Γ m p q(∇∇x γy p)δyq + M m γ iδW i + Nm γ dt. (3.7) ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 1020 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK Then the next order variation ∇∇x k∇∇x γy m = ∇∇x γ∪{k}y m fulfills δ(∇∇x γ∪{k}y m) = −Γ m p q(∇∇x γ∪{k}y p)δyq + R m p �q(∇∇x γy p) ∂y� ∂xk δyq+ + (∇∇x kM m γ i )δW i + (∇∇x kN m γ )dt. (3.8) The coefficients of variational equations are recurrently related by M m γ∪{k} i = ∇∇x kM m γ i + R m p �q(∇∇x γy p) ∂y� ∂xk Aq i , Nm γ∪{k} = ∇∇x kN m γ + R m p �q(∇∇x γy p) ∂y� ∂xk Aq 0. (3.9) Here R forms curvature (1,3)-tensor with components R 2 1 34 = ∂Γ 2 1 3 ∂x4 − ∂Γ 2 1 4 ∂x3 + Γ j 1 3Γ 2 j 4 − Γ j 1 4Γ 2 j 3, (3.10) where for simplicity we only point the positions of corresponding indexes. Remark 3.3. The additional term with Γ(y) in the Definition 3.2 of ∇∇-derivative compactificates these noninvariant terms to the compact expressions with curvature. So it becomes possible to find the influence of curvature and nonlinearities of diffusion equa- tion on the any order regularity properties. The approaches to define the variation to be covariant Riemannian, derivative in the direction of vector field or stochastic derivative did not account the invariance on process yxt [4, 5, 7] and inevitably led to the growing number of noninvariant terms in the varia- tional equations. Therefore, it was principally hard to trace the influence of curvature in regular properties. Proof. For simplification we omit, where possible the dependence of connection Γ on variable y, however the dependence on x is always displayed precisely. Let us substitute the definition of ∇∇-derivative under Stratonovich integral∫ δ(∇∇x k∇∇x γy m) = = ∫ δ { ∂x k∇∇x γy m + Γ m p q(y) ∂yp ∂xk ∇∇x γy q − ∑ s∈γ Γ h k s(x)∇∇x γ\|s=h ym } . (3.11) For the first term in (3.11) we apply the inductive assumption (3.7) and, after differentia- tion of integral and application of property of Stratonovich integral∫ X δ (∫ Y δZ ) = ∫ XY δZ (3.12) obtain∫ δ { ∂x k∇∇x γy m } = ∫ δ ( ∂x k ∫ { −Γ m p q(∇∇x γy p)δyq + M m γ iδW i + Nm γ dt }) = = − ∫ ∂Γ m p q ∂y� ∂y� ∂xk (∇∇x γy p)δyq − ∫ Γ m p q(∇∇x γy p)δ ( ∂yq ∂xk ) − (3.13) ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 REGULARITY OF NONLINEAR FLOWS ON NONCOMPACT RIEMANNIAN MANIFOLDS ... 1021 − ∫ Γ m p q(∂ x k∇∇x γy p)δyq + ∫ {∂x kM m γ iδW i + ∂x kNm γ dt}. (3.14) The second term in brackets in (3.11) is first rewritten by Stratonovich – Itô formula for function f(x1, x2, x3) = x1x2x3: f ◦ Xt = f ◦ X0 + t∫ 0 Djf ◦ X δXj after that the inductive assumption (3.7) and properties of Stratonovich integrals are ap- plied ∫ δ { Γ m p q(y) ∂yp ∂xk ∇∇x γy q } = = ∫ Γ m p q ∂yp ∂xk δ(∇∇x γy q) + ∫ Γ m p q(∇∇x γy q)δ ( ∂yp ∂xk ) + ∫ ∂yp ∂xk (∇∇x γy q)δΓ m p q(y) = = ∫ Γ m p q ∂yp ∂xk { −Γ q � s(∇∇x γy �)δys + M m γ iδW i + Nq γdt } + (3.15) + ∫ Γ m p q(∇∇x γy q)δ ( ∂yp ∂xk ) + ∫ ∂yp ∂xk (∇∇x γy q) ∂Γ m p q ∂y� δy�. (3.16) For the last term in (3.11) we use again the inductive assumption (3.7) − ∫ δ {∑ s∈γ Γ h k s(x)∇∇x γ\|s=h ym } = = − ∑ s∈γ ∫ Γ h k s(x) { −Γ m p q(y)(∇∇x γ\|s=h yp)δyq + M m γ\|s=h iδW i + Nm γ\|s=h dt } . (3.17) Further we transform the first expression in (3.14) to the ∇∇-derivative − ∫ Γ m p q(∂ x k∇∇x γy p)δyq = − ∫ Γ m p q(∂ x k∇∇x γy p)δyq = = − ∫ Γ m p q(∇∇k∇∇x γy p)δyq + ∫ Γ m p qΓ p � n ∂y� ∂xk (∇∇x γy n)δyq− − ∑ s∈γ ∫ Γ m p q(y)Γ h k s(x)(∇∇x γ\|s=h yp)δyq. (3.18) Notice now that: a) the second expression in (3.13) contracts with the first expression in (3.16); b) the third expression in (3.18) contracts with the first expression in (3.17); c) the second and third terms in (3.14), (3.15) and (3.17) give the ∇∇-derivatives of M and N coefficients. We write the remaining terms, by redenoting indexes and gathering terms with deriva- tives ∂Γ and second powers Γ(y)Γ(y) of connection∫ δ { ∂x k∇∇x γy m + Γ m p q(y) ∂yp ∂xk ∇∇x γy q − ∑ s∈γ Γ h k s(x)∇∇x γ\|s=h ym } = ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 1022 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK = − ∫ Γ m p q(∇∇x k∇∇x γy p)δyq + ∫ { ∇∇x kM m γ iδW i + ∇∇x kN m γ dt } + + ∫ ∂y� ∂xk (∇∇x γy p)δyq× × { ∂Γ m � p(y) ∂yq − ∂Γ m p q(y) ∂y� + Γ m s q(y)Γ s � p(y) − Γ m � s(y)Γ s p q(y) } . (3.19) The terms in (3.19) appear correspondingly first one from the second term in (3.16); second one from the first term in (3.13); third from the first term in (3.18); last term from the first term in (3.15). But the expression in brackets {. . .} gives the curvature (3.10), so we conclude∫ δ(∇∇x k∇∇x γy m) = − ∫ { Γ m p q(∇∇x k∇∇x γy p)δyq + R m p �q(∇∇x γy p) ∂y� ∂xk δyq+ +∇∇x kM m γ iδW i + ∇∇x kN m γ dt } . The theorem is proved. 4. Symmetries of variational equations and nonlinear estimate on variations. Now we are going to use the symmetry of variational equations to find a set of nonlinear estimates on variations. First remark that by (3.6) the recurrence base for the definition of higher-order varia- tional systems (3.7) is given by M m k i = ∇∇x kA m i (yxt ), Nm k = ∇∇x kA m 0 (yxt ). Using recurrent properties (3.9) and (3.5) we can determine the nonlinear symmetries of variational equations. Because (∇∇x)nH(yxt ) = ∑ j1+...+js=n, s=1,...,n (∇y)sH(yxt ) · (∇∇x)j1y . . . (∇∇x)jsy we see that the nth order variation in the left-hand side of (3.7) is proportional to the nth power of first variation in the right-hand side, or n √ (∇∇x)nyxt ∼ ∇∇xyxt . (4.1) Introduce nonlinear expression that reflects this symmetry rn(y, t) = n∑ j=1 Epj(ρ2(yxt , z)) ∥∥(∇∇x)jyxt ∥∥q/j (4.2) and gives some nonlinear norm on the smoothness of process yxt with respect to the initial data. Here z ∈ M is some fixed point, ρ(x, y) is geodesic distance between points x, y, norm of variation is defined in (1.3). ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 REGULARITY OF NONLINEAR FLOWS ON NONCOMPACT RIEMANNIAN MANIFOLDS ... 1023 The following theorem provides necessary conditions for quasicontractive estimate on rn. Introduce notation Ã0 = A0 + 1 2 d∑ α=1 ∇AαAα. (4.3) Theorem 4.1. Suppose that the following conditions hold: coercitivity: there exists z ∈ M such that for any C ∈ R+ there exists KC ∈ R 1 such that for any x ∈ M 〈 Ã0(x),∇xρ2(x, z) 〉 + C d∑ α=1 ‖Aα(x)‖2 ≤ KC(1 + ρ2(x, z)); (4.4) dissipativity: for any C,C ′ ∈ R+ there exists KC ∈ R 1 such that for any x, y ∈ M 〈 ∇Ã0[h], h 〉 + C d∑ α=1 \|∇Aα[h]\|2 − C ′ d∑ α=1 〈R(Aα, h)Aα, h〉 ≤ KC‖h‖2; (4.5) notation ∇A[h] = h�∇�A means covariant directional derivative, 〈·, ·〉 and \| · \| cor- responding Riemannian scalar product and norm, and[ R(A,B)C ]m = R m i jkA jBkCi denotes curvature operator; nonlinear behaviour of coefficients and curvature: for any n there are constants k• such that for all j = 1, . . . , n and x ∈ M∥∥∥(∇)jÃ0(x) ∥∥∥ ≤ (1 + ρ(x, z))k0 , ∥∥(∇)jAα(x) ∥∥ ≤ (1 + ρ(x, z))kα , (4.6)∥∥(∇)jR(x) ∥∥ ≤ (1 + ρ(x, z))kR . Then there is some k = k(k0, kα, kR) such that if monotone polynomials pj ≥ 1 in (4.2) are hierarchied by ∀ j1 + js = i ≤ n : [ pi(·) ]i(1 + \| · \|2 )kq ≤ [ pj1(·) ]j1 . . . [ pjs(·) ]js (4.7) then the nonlinear estimate on variations holds ∃Kk ∀t ≥ 0 rn(y, t) ≤ e Kkt rn(y, 0). (4.8) Remarks. 4.1. For M = R d with the global Euclidean coordinate system (xi)di=1 both connection and curvature vanish. In this case ρ(x, y) = ‖x− y‖ and one can choose point z = 0 to reduce conditions (4.4), (4.5) to the classical conditions of coercitivity: 〈 Ã0(x), x 〉 + C d∑ α=1 ∥∥Aα(x) ∥∥2 ≤ KC(1 + ‖x‖2); dissipativity: ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 1024 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK 〈 ∇Ã0[h], h 〉 + C d∑ α=1 ∥∥∇Aα[h] ∥∥2 ≤ KC‖h‖2. They naturally arise in the proof of nonexplosion and uniqueness estimates on process yxt E‖yxt ‖2 ≤ eKt ( 1 + ‖x‖2 ) , E‖yxt − yzt ‖2 ≤ eK ′‖x − z‖2. (4.9) Indeed, by coercitivity condition and Itô formula ht ≡ E‖yxt ‖2 = ‖x‖2 + t∫ 0 E {〈 Ã0(yxs ), yxs 〉 + 1 2 d∑ α=1 ∥∥Aα(yxs ) ∥∥2 } ds ≤ ≤ ‖x‖2 + C t∫ 0 (1 + hs)ds. (4.10) Then Gronwall – Bellman inequality leads to the first estimate in (4.9). In a similar way dissipativity condition leads to the second estimate in (4.9), ensuring local uniqueness of solutions. In this sense the coercitivity and dissipativity conditions are natural for nonlinear dif- fusion equations. Their generalization for stochastic differential equations in infinite di- mensional linear spaces was found by Krylov, Pardoux, Rosovskii [18, 19]. 4.2. In fact, from coercitivity and dissipativity conditions follows that field Ã0 and, therefore, field A0 should be more than a square of fields Aα times curvature. For example, for C2 bounded diffusion coefficients supx∈M ( ‖Aα(x)‖, ‖∇Aα(x)‖, ‖∇∇Aα(x)‖ ) < ∞ and bounded geometry supx∈M ∥∥R(x) ∥∥ < ∞ only the monotonic- ity of field (−A0) is necessary for the validity of coercitivity and dissipativity conditions (C2 boundedness arises due to the structure of Ã0). In the domain of manifold, where the curvature form m(h, h) = ∑d α=1 〈 R(Aα, h)Aα, h 〉 is positive, the additonal terms with curvature improve the restrictions on diffu- sion and drift. However, the manifolds with strictly positive curvature forms are compact [20], so in this case process yxt actually lives on compact manifold and there are no non- linear complications. 4.3. Let us turn the attention of reader that the coercitivity and dissipativity assump- tions naturally arise in the proof of nonexplosion and uniqueness estimates on the process yxt . To check this fact one should proceed similar to (4.9), (4.10), with application of Itô formula to expressions Eρ2(yxt , o) and Eρ2(yxt , yzt ) and further use of estimates (4.31), (4.32). This is a subject of [21]. In Theorem 4.1 we actually state, that the coercitivity and dissipativity assumptions, combined with (4.6), are sufficient for any order regularity of process yxt with respect to the initial data. Moreover, as it will be clear from the proof, dissipativity assumption (4.5) represents the coercitivity condition for variational processes ∇∇yxt . 4.4. An example of manifold may be given by R d with conformally perturbed Euclidean metric tensor gij(x) = e2ψ(x)δij (such perturbations preserve angles between vectors). In this case [22] connection coefficients of metric g are nonvoid and equal to Γ h i j(x) = δhi ∂x j ψ(x) + δhj ∂x i ψ(x) − δijδ hk∂x kψ(x). The covariant derivative of vector field has representation ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 REGULARITY OF NONLINEAR FLOWS ON NONCOMPACT RIEMANNIAN MANIFOLDS ... 1025 ∇iV h(x) = ∂iV h(x) + Γ h i jV j = ∂iV h + δhi 〈V, ∂ψ〉 + ∂iψ · V h − ∂hψ · Vi and curvature tensor equals to R h i kj = δhkψij . . . δhj ψik + ψh kδij − ψh j δik + (δkkδij − δhj δik)‖∂ψ‖2, where we used notations ψij = ∂i∂jψ − ∂iψ · ∂jψ, ψh k = δhjψjk with Kronecker deltas δij , δhk, δhj . In this case the dissipativity condition (4.5) adopts form, which could be hard to guess by direct calculations in the global R d coordinate system of manifold (Rd, e2ψδij). Therefore, the use of geometric invariance arguments of Definition 1.1 actually per- mits to single out the pointwise conditions on the behaviour of coefficients and curvature, that guarantee the global regularity estimates on diffusion process even for more compli- cate manifolds. Proof. First let us note that Itô formula implies that for geodesic distance ρ2(yxt , z) = ρ2(x, y) + d∑ j=1 t∫ 0 (A1 αρ2)(yxs , z)dWα s + t∫ 0 (L1ρ2)(yxs , z)ds (4.11) with notation L1 for generator L of diffusion (2.3), acting on first coordinate of metric function. Therefore, writing the differential of one terms in nonlinear expression (4.2) we have by Itô formula (temporarily 2q = m/i, p = pi): h(t) = Ep(ρ2(yxt , z)) ∥∥(∇∇x)iyxt ∥∥2q = = h(0) + E t∫ 0 { p(ρ2(yxs , z)) d ∥∥(∇∇x)iyxs ∥∥2q + ∥∥(∇∇x)iyxs ∥∥2q dp ( ρ2(yxs , z) ) + + 1 2 d [ p(ρ2(yxs , z) ) , ∥∥(∇∇x)iyxs ∥∥2q ]} = = h(0) + t∫ 0 E { p(ρ2(yxs , z))(2q‖(∇∇x)iyxs ‖2(q−1)d ∥∥(∇∇x)iyxs ∥∥2+ +q(2q − 2) ∥∥(∇∇x)iyxs ∥∥2(q−2) d [∥∥(∇∇x)iyxs ∥∥2 , ∥∥(∇∇x)iyxs ∥∥2 ] + (4.12) + ∥∥(∇∇x)iyxs ∥∥2q ( p′ ( ρ2(yxs , z) ) dρ2(yxs , z)+ + 1 2 p′′ ( ρ2(yxs , z) ) d [ ρ2(yxs , z), ρ2(yxs , z) ]) + (4.13) + 1 2 p′ ( ρ2(yxs , z) )∥∥(∇∇x)iyxs ∥∥2(q−1) d [ ρ2(yxs , z), ∥∥(∇∇x)iyxs ∥∥2 ]} . (4.14) Next step is to find from recurrent relations of Theorem 3.1 the expressions for differ- entials of norms ∥∥(∇∇x)jyxt ∥∥2 (1.3). ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 1026 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK Recall, that by (3.7), (3.8) the general form of variational equations looks like δ(Xm γ ) = −Γ m p qX p γδyq + M m γ αδWα + Nm γ dt (4.15) with coefficients M m γ α, Nm γ , recurrently determined in (3.9). To simplify further notations, let us introduce an additional process, that formally corresponds to the index γ = ∅ δXm ∅ = −Γ m p qX p ∅ δyq + Am α δWα + Am 0 dt. Then the relations of coefficients M,N for the process Xm γ could be written in the fol- lowing form: 1) recurrent base: Mm ∅ α = Am α (yxt ), Nm ∅ = Am 0 (yxt ); (4.16) 2) recurrent step by (3.6) for γ = ∅ and (3.8) for γ �= ∅ M m γ∪{k} α = ∇∇x kM m ∅ α for γ = ∅, ∇∇x kM m γ α + R m p �qX p γ (∇∇x ky �)Aq α for γ �= ∅, (4.17) Nm γ∪{k} = ∇∇x kN m ∅ for γ = ∅, ∇∇x kN m γ + R m p �qX p γ (∇∇x ky �)Aq 0, for γ �= ∅. (4.18) Lemma 4.1. The differential of norm of process Xm γ (4.15) has form d‖X‖2 = gγε(x) { gmn(Xm γ M n ε α + Xn ε M m γ α)dWα+ +gmn(Xm γ Nn ε + Xn ε Nm γ + M m γ αM n ε α)dt + 1 2 gmn(Xm γ Pn ε + Xn ε Pm γ )dt } . (4.19) Expressions Pm γ are recurrently related by Pm k = ∇∇x k(∇AαAm α ) + R m p �qA p αAq α(∇∇x ky �), (4.20) Pm γ∪{k} = ∇∇x kP m γ + 2R m p �qM p γ α(∇∇x ky �)Aq α+ +(∇sR m p �q)X p γ (∇∇x ky �)Aq αAs α + R m p �qX p γ (∇∇x kA � α)Aq α+ +R m p �qX p γ (∇∇x ky �)(∇Aα Aα). (4.21) Proof. The detail and bookkeeping proof of this result will appear in [23]. Let us briefly present the main idea. First of all one verifies formula (4.19) and obtains expression for Pm γ Pm γ dt = d[M m γ α,Wα] + Γ m p qM p γ αAq αdt. (4.22) Then it is necessary to find the recurrent relations for Pm γ . From (4.17) follows Pm γ∪{k}dt = d[M m γ∪{k}α,Wα] + Γ m p qM p γ∪{k}αAq αdt = = d [ ∇∇x kM m γ α,Wα ] + Γ m p q(∇∇x kM p γ α)Aq αdt+ (4.23) ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 REGULARITY OF NONLINEAR FLOWS ON NONCOMPACT RIEMANNIAN MANIFOLDS ... 1027 +d [ R m p �qX p γ (∇∇x ky �)Aq α,Wα ] + Γ m p q(R p i �jX i γ(∇∇x ky �)Aj α)Aq αdt. (4.24) The last line (4.24) appears only for γ �= ∅. From arguments, analogous to the proof of Theorem 3.1, it follows representation (4.23) = { ∇∇x kP m γ + R m p �qM p γ α(∇∇x ky �)Aq α } dt. (4.25) To find the reccurence base it is sufficient to apply definitions (4.16), (4.22) and obtain Pm ∅ dt = d[Am α (y),Wα] + Γ m � hA � αAh αdt = = [ ∂Am α ∂y� + Γ m � hA h α ] A� αdt = (∇�A m α ) · A� αdt = (∇AαAm α )dt. (4.26) Therefore from (4.25) and (4.26) it follows (4.20). In a similar way, direct calculation of line (4.24) gives the remaining terms in (4.21). The lemma is proved. Now we use the result of Lemma 4.1 to single out the dissipativity condition (4.5) in the principal part in the right-hand side of (4.19). Let i = 1 and Xm k = ∇∇x l y m, then by (4.20) and (3.5) Pm k = ∇∇x k(∇Aα Am α (y)) + R m p �qA p αAq α∇∇x ky � = = ∇y �∇Aα Am α · ∇∇x ky � − R(Aα,∇∇x ky)Aα. Therefore, because in (4.21) Pm γ∪{k} = ∇∇x kP m γ + . . ., the higher-order coefficient permits representation Pm γ = ∇�∇AαAm α · ∇∇x γy � − R(Aα,∇∇x γy)Aα+ + ∑ β1∪...∪βs=γ, s≥2 Kβ1,...,βs(∇∇x β1 y, . . . ,∇∇x βs y) with coefficients Kβ1,...,βs , depending on A0, Aα, R and their covariant derivatives. Moreover, the dependence of Kβ1,...,βs(∇∇x β1 y, . . . ,∇∇x βs y) on lower order variations ∇∇x βy also manifests symmetries (4.1). In a similar way, due to (3.6) M m k α = ∇∇x kA m α (y) = ∇y �A m α (y) · ∇∇x ky �, N m k = ∇∇x kA m 0 (y) = ∇y �A m 0 (y) · ∇∇x ky � and relations (3.9), we have analogous asymptotic M m γ α = ∇y �A m α [ ∇∇x γy � ] + ∑ β1∪...∪βs=γ, s≥2 K ′ β1,...,βs (∇∇x β1 y, . . . ,∇∇x βs y), N m γ = ∇y �A 0 α [ ∇∇x γy � ] + ∑ β1∪...∪βs=γ, s≥2 K ′′ β1,...,βs (∇∇x β1 y, . . . ,∇∇x βs y) (4.27) with multilinear coefficients K ′, K ′′, depending on A0, Aα, R and their covariant deriva- tives. ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 1028 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK Therefore from (4.19) the principal part of differential is d ∥∥(∇∇x)iyxt ∥∥2 = 2 〈 (∇∇)iy,∇y �Aα [ (∇∇)iy� ]〉 dWα+ + { 2 〈 (∇∇)iy,∇y � Ã0 [ (∇∇)iy� ]〉 + d∑ α=1 ∥∥∥∇Aα [ (∇∇)iy ]∥∥∥2 − − d∑ α=1 〈 R(Aα, (∇∇)iy)Aα, (∇∇)iy 〉} dt+ + ∑ j1+...+js=i, s≥2 〈 (∇∇x)iy, { K1 j1,...,js,α ( (∇∇)j1y, . . . , (∇∇)jsy ) dWα+ + K2 j1,...,js ( (∇∇)j1y, . . . , (∇∇ )js y)dt }〉 , (4.28) i.e., the dissipativity condition arises in the principal part. Like before the coefficients K1, K2 depend on covariant derivatives of A0, Aα, R and display symmetry (4.1). Now we can turn to the estimation of (4.12) – (4.14). 1. Using asymptotic (4.28) we see that terms in (4.12) lead to the dissipativity con- dition (4.5) in principal part with some constants and additional terms with lower order variations h(0) + t∫ 0 E { p(ρ2(yxs , z))(2q‖(∇∇x)iyxs ‖2(q−1)d ∥∥(∇∇x)iyxs ∥∥2+ + q(2q − 2) ∥∥(∇∇x)iyxs ∥∥2(q−2) d [∥∥(∇∇x)iyxs ∥∥2 , ∥∥(∇∇x)iyxs ∥∥2 ] ≤ ≤ KE t∫ 0 p(ρ2(yxt , z)) ∥∥(∇∇)iyxt ∥∥2(q−1){dissipativity}C,C′((∇∇)iyxt , (∇∇)iyxt )dt+ + ∑ j1+...+js=i, s≥2 E t∫ 0 p ( ρ2(yxt , z) )∥∥(∇∇)iyxt ∥∥2(q−1)× × 〈 (∇∇)iy,Kj1,...,js((∇∇)j1y, . . . , (∇∇)jsy) 〉 dt (4.29) with coefficients K like before. 2. Term in (4.13) is transformed by monotonicity and polynomiality of p ( ∃C: p′′(u)u ≤ Cp′(u) ) 1∫ 0 E ∥∥(∇∇)iy ∥∥2q { p′(ρ2(y, z))dρ2(y, z) + 1 2 p′′(ρ2(y, z))d [ ρ2(y, z), ρ2(y, z) ]} = = 1∫ 0 E ∥∥(∇∇)iy ∥∥2q { p′(ρ2(y, z))L1ρ2(y, z)+ ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 REGULARITY OF NONLINEAR FLOWS ON NONCOMPACT RIEMANNIAN MANIFOLDS ... 1029 + 1 2 p′′(ρ2(y, z))ρ2(y, z) 1 ρ2(y, z) d∑ α=1 ( A1 αρ2(y, z) )2 } dt ≤ ≤ t∫ 0 E ∥∥(∇∇)iy ∥∥2q p′(ρ2(y, z)) { L1ρ2(y, z) + C ρ2(y, z) d∑ α=1 ( A1 αρ2(y, z) )2 } dt. (4.30) Then we apply results of [21] about upper estimates on second-order operators, acting on metric function. Theorem 4.2 [21]. Suppose that the generalized coercitivity and dissipativity con- ditions (4.4), (4.5) hold. Then there is constant K such that L1ρ2(x, y) ≤ K(1 + ρ2(x, y)). (4.31) Moreover, for any C there exists KC such that L1ρ2(x, y) + C d∑ α=1 (A1 αρ2(x, y))2 ρ2(x, y) ≤ KC(1 + ρ2(x, y)). (4.32) 3. Using representation (4.28) and (4.11) we find the principal asymptotic of (4.14). By (4.27) ∣∣∣p′(ρ2) ∥∥(∇∇)iy ∥∥2(q−1) d [ ρ2, ∥∥(∇∇)iy ∥∥2 ]∣∣∣ = = ∣∣∣∣p′(ρ2) ∥∥(∇∇)iy ∥∥2(q−1) d∑ α=1 A1 αρ2 · 2 〈 (∇∇)iy,∇Aα [ (∇∇)iy ] + + ∑ j1+...+js, s≥2 K ′ j1,...,js ( (∇∇)j1y, . . . , (∇∇)jsy )〉 \| ≤ ≤ p′(ρ2) ∥∥(∇∇)iy ∥∥2q d∑ α=1 (A1 αρ2))2 ρ2 + + p′(ρ2)ρ2 ∥∥(∇∇)iy ∥∥2(q−1)× × ∥∥∥∥∥∥∇Aα [ (∇∇)iy ] + ∑ j1+...+js, s≥2 K ′ j1,...,js ( (∇∇)j1y, . . . , (∇∇)jsy )∥∥∥∥∥∥ 2 . (4.33) The first term is added to (4.30), after that (4.32) is applied. The second term is combined with terms in (4.12), (4.29), leading to the dissipativity condition with modified constants. 4. Applying coercitivity and dissipativity (4.4), (4.5) we finally come to estimate h(t) = Ep(ρ2(yxt , z))‖(∇∇)iyxt ‖2q ≤ h(0) + C t∫ 0 h(t)dt+ + ∑ j1+...+js, s≥2 t∫ 0 Ep(ρ2(y, z)) ∥∥(∇∇)iy ∥∥2(q−1)× ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 1030 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK ×K ′ i;j1,...,js ( (∇∇)iy; (∇∇)j1y, . . . , (∇∇)jsy ) dt, where coefficient K ′ depends quadratically on lower order variations for the case of (4.33). It remains to apply estimates (4.6) and symmetry (4.1). By inequality \|xq−1y\| ≤ ≤ \|x\|q q + (q − 1) \|y\|q q we have for a = 1, 2 Ep(ρ2) ∥∥(∇∇)iy ∥∥2(q−1) Ki;j1,...,js ( (∇∇)iy; (∇∇)j1y, . . . , (∇∇)jsy ) ≤ ≤ Ep(ρ2)(1 + ρ2)k ∥∥(∇∇)iy ∥∥2q−a∥∥(∇∇)j1y ∥∥a . . . ∥∥(∇∇)jsy ∥∥a ≤ ≤ CEp ∥∥(∇∇)iy ∥∥2q + C ′Ep(ρ2)(1 + ρ2)2qk ∥∥(∇∇)j1y ∥∥2q . . . ∥∥(∇∇)jsy ∥∥2q . The first term is already of necessary form, to transform the last term we recall that 2q = m/i (4.2), so ‖xj1‖m/i . . . ‖xjs‖m/i = ( ‖xj1‖m/j1 )j1/i . . . ( ‖xjs‖m/js )js/i . Then the nonlinear hierarchies of polynomials (4.7) give pi(ρ2)(1 + ρ2)km/i ∥∥(∇∇)j1y ∥∥m/i . . . ∥∥(∇∇)jsy ∥∥m/i ≤ ≤ (pj1(ρ 2) ∥∥(∇∇)j1y ∥∥m/j1)j1/i . . . (pjs (ρ2) ∥∥(∇∇)jsy ∥∥m/js)js/i ≤ ≤ j1 i pj1(ρ 2) ∥∥(∇∇)j1y ∥∥m/j1 + . . . + js i pjs(ρ 2) ∥∥(∇∇)jsy ∥∥m/js , i.e., the differential of each term in (4.2) is estimated by terms of (4.2) hi(t) = Epi(ρ2) ∥∥(∇∇)iy ∥∥q/i ≤ hi(0) + const t∫ 0 rn(y, s)ds. The theorem is proved. 5. C∞ regular dependence of diffusion process yx t on initial data. Applica- tions to the regularity properties of semigroups. Turning to the questions of existence, uniqueness and differentiability of variational equations with respect to the initial data, one can show these properties under conditions (4.4) – (4.6). Theorem 5.1. Under conditions (4.4) – (4.6) process yxt is C∞ differentiable with respect to the initial data. Its variations (∇∇x)yxt represent strong solutions to variational systems (3.7), (3.8). Proof. First of all, by Theorem 3.1 and asymptotics (4.27), variational equation on process (∇∇x)iyxt represents nonautonomous and inhomogeneous equation on variable (∇∇x)iyxt , if all lower order variations (∇∇x)jyxt , j < i, are already constructed. The behaviour of nonautonomous part is controlled by coercitivity and dissipativity condition. In a similar way the nonlinear symmetries (4.1) and polynomial behaviour of coefficients (4.6) give a set of optimal estimates on inhomogeneous part, like (4.8). Therefore, like in [15, 17, 21, 24, 25], variational processes (∇∇x)iyxt , i ≥ 1, are constructed as strong solutions to systems (3.7), (3.8). To prove C∞ differentiability of process yxt on initial data, it only remains to show that the solutions of variational equations (3.7), (3.8) represent higher-order ∇∇-derivatives ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 REGULARITY OF NONLINEAR FLOWS ON NONCOMPACT RIEMANNIAN MANIFOLDS ... 1031 of process yxt , i.e., its differentiability. Like in [15, 17, 21, 24, 25] this can be obtained by application of nonlinear symmetries (4.1) in a recurrent on the order of differentiation way. Because we consider the finite-dimensional situation, we can also apply the stopping times techniques, e.g. [26]. They guarantee that derivatives of process with locally C∞ coefficients are represented as solutions of corresponding variational equations before exit times, i.e., give the required statement. The theorem is proved. Now we can turn to the applications of nonlinear estimate (4.8) to the regular prop- erties of semigroup. We construct the spaces of continuously differentiable functions, that are preserved under the action of diffusion semigroup (Ptf)(x) = Ef(yxt ). Because of globally non-Lipschitz coefficients, such semigroups fail the strong continuity in time property even in space of continuous bounded functions Cb(M). Therefore the appli- cation of operator techniques of semigroups theory and corresponding constructions of functional spaces, does not seem to be adapted to this case. To solve this problem we use pure stochastic representations (2.13). Because in for- mula (2.13) the derivatives of semigroup are related with derivatives of function via ker- nels, represented by higher-order ∇∇-derivatives of process yxt , the nonlinear estimates (4.8) help to solve the situation. Moreover, the structure of topologies in these spaces is influenced by nonlinearity parameters of initial diffusion equation and geometry of man- ifold, see also [27]. Definition 5.1. Let q0, q1, . . . , qn ≥ 1 be a family of monotone functions of polyno- mial behaviour, that fulfill ∀i ≥ 1 qi(b)(1 + b)k ≤ qi+1(b) ∀b ≥ 0. (5.1) Function f ∈ Cn $q (M) iff it is n-times continuously covariantly differentiable and the norm is finite ‖f‖Cn �q (M) = max i=0,...,n sup x∈M ‖(∇x)if(x)‖ qi(ρ2(x, z)) . (5.2) Due to the triangle inequalities for metric and properties of functions qi the choice of some point z above does not influence on the topology of space Cn $q (M), but only the choice of norm. Next theorem is an application of nonlinear estimate (4.8) to the smooth properties of diffusion semigroups. Theorem 5.2. Suppose conditions (4.4) – (4.6) hold. Then there is parameter k such that for weights (q0, . . . , qn) (5.1) the space Cn $q (M) is preserved under the action of semigroup ∀t ≥ 0 Pt : Cn $q (M) → Cn $q (M) and the quasicontractive estimate holds: there are constants K, M such that ‖Ptf‖Cn �q (M) ≤ KeMt‖f‖Cn �q (M) ∀f ∈ Cn $q (M). (5.3) Proof. First recall that by Theorem 2.1 the covariant derivatives of semigroup are related with the covariant derivatives of function via the kernels, given by variational processes (∇∇x)iyxt ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 1032 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK (∇x)iPtf(x) = ∑ j1+...+js=i E 〈 (∇y)sf(yxt ), (∇∇x)j1yxt ⊗ . . . ⊗ (∇∇x)jsyxt 〉 . (5.4) To estimate these kernels we use nonlinear estimate (4.8). Let us first note that by Definition 1.1 (1.2) the initial data for variations are: ∇∇x k(y x t )m ∣∣∣ t=0 = ∂xm ∂xk = δmk , ∇∇x kj(y x t )m ∣∣∣ t=0 = ∇∇x k(∇∇x j y m) ∣∣∣ t=0 = ∂k(δmj ) − Γ h k j(x)δmh + Γ m p q(y x 0 )δpj δ q k = 0 and (∇∇x)iyxt ∣∣∣ t=0 = 0 ∀i ≥ 1. Choose a system of weights ∀j = 1, . . . , n p̃j(x) = P (x)(1 + x)kq(1/j−1/n) that fulfills hierarchy (4.7). For this choice p̃n = P and nonlinear estimate (4.8) has form EP ( ρ2(yxt , z) )∥∥(∇∇)nyxt ∥∥q/n ≤ eMtρn(y, 0) = = eMtP (ρ2(x, z)) ( 1 + ρ2(x, z) )kq(n−1)/n . (5.5) Now we estimate derivatives (5.4) in topologies Cn $q (M): ‖(∇x)iPtf(x)‖ T (0,i) x qi(ρ2(x, z)) ≤ ≤ ∑ j1+...+js, s≥1 ‖E < (∇y)sf(yxt ) , (∇∇x)j1yxt ⊗ . . . ⊗ (∇∇x)jsyxt > T (0,i) y ‖ T (0,i) x qi(ρ2(x, z)) ≤ ≤ ∑ j1+...+js, s≥1 ( sup yx t ∈M ‖(∇y)sf(yxt )‖ T (0,s) y qs(ρ2(yxs , z)) ) × ×Eqs(ρ2(yxs , z))‖(∇∇x)j1yxt ‖ . . . ‖(∇∇x)jsyxt ‖ qi(ρ2(x, z)) ≤ ≤ ∑ j1+...+js, s≥1 ‖f‖Cn �q ∏s �=1 (Eqs(ρ2(yxs , z))‖(∇∇x)j�yxt ‖i/j�)j�/i qi(ρ2(x, z)) . Here we substituted an intermediate weight qs(y) and at last step applied Hölder in- equality. The last fraction is estimated by (5.5) 1 qi(ρ2(x, z)) s∏ �=1 (Eqs(ρ2(yxs , z))‖(∇∇x)j�yxt ‖i/j�)j�/i ≤ ≤ 1 qi(ρ2(x, z)) s∏ �=1 (eMtqs(ρ2(x, z))(1 + ρ2(x, z))ki(j�−1)/j�)j�/i = ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8 REGULARITY OF NONLINEAR FLOWS ON NONCOMPACT RIEMANNIAN MANIFOLDS ... 1033 = const eMt qs(ρ 2(x, z))(1 + ρ2(x, z))k(i−s) qi(ρ2(x, z)) ≤ const eMt, where we applied hierarchy (5.1). Above we used that q = i, n = j� in notations of (4.2) and that j1 + . . . + js = i. The last inequality follows from hierarchy (5.1). This gives statement (5.3). To obtain the continuous differentiability of semigroup on x it is sufficient, similar to [15, 17, 21, 24, 25], to use representations (2.13). The C∞-differentiability of process yxt implies pathwise weak relation between solution of initial equation and first variation process f(yxt ) − f(yyt ) = x∫ y 〈 ∇yz t f(yzt ), ∂yzt ∂z 〉 dz and similar relations for higher-order variations, e.g. [25] (Theorem 3.4). After that one takes the expectation with respect to the random parameter to obtain Ptf(x) − Ptf(y) = Ef(yxt ) − Ef(yyt ) = = E x∫ y 〈 ∇yz t f(yzt ), ∂yzt ∂z 〉 dz = ∫ x y ( E 〈 ∇yz t f(yzt ), ∂yzt ∂z 〉) dz and similar relations for the higher-order derivatives. Therefore the increments of semigroup are represented as integrals ∫ x y of aggregates in the right-hand side of (2.13) and these expressions form the derivatives of semigroup. Final conclusion about continuous differentiability of semigroup follows from estimates on the continuity in mean of variational processes with respect to the initial data. This fact can be proved in a similar to nonlinear estimate (4.8) way, by application of symme- tries (4.1), e.g. [15, 17, 21, 24, 25]. The theorem is proved. 1. Itô K. Stochastic differential equations in a differentiable manifold // Nagoya Math. J. – 1950. – 1. – P. 35 – 47. 2. Itô K. The Brownian motion and tensor fields on Riemannian manifold // Proc. Int. Cong. Math. – Stock- holm, 1962. – P. 536 – 539. 3. Ikeda N., Watanabe S. Stochastic differential equations and diffusion processes. – Dordrecht: North- Holland Publ., 1981. 4. Elworthy K. D. 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Anal. – 1996. – 135. – P. 488 – 518. Received 15.12.2004, after revision — 10.06.2005 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8

Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural?

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