Search

Let M be a d-dimensional Riemannian manifold of class C∞ with Riemannian metric and let D be a connected domain in M having a non-empty smooth boundary and a compact closure . Suppose that are given and that converges uniformly to on D′ as for some neighborhood D′ of D. Consider the diffusion process on D′ with a small parameter generated by(1.1)

Let G be a compact connected Lie group with identity element e , and let PeG denote the space of continuous maps y: [0, 1] —> G such that y(0) = e . When equipped with the natural group structure and sup metric, PeG becomes an interesting example of an infinite dimensional nonlinear topological group. The purpose of this paper is to consider certain aspects of analysis on PeG. Stimulated by a theorem of M. Malliavin and P. Malliavin, we prove the existence of a natural Brownian motion on PeG which depends only on a choice of bi-invariant metric for G. Our main results, however, concern the heat semigroup associated to the Brownian motion on PeG. We identify the action of the generator of this semigroup when applied to certain highly regular functions, with a result similar to that obtained earlier by L. Gross in the (linear) abstract Wiener space context.

Basic derivative formulas are presented for hypoelliptic heat semigroups and harmonic functions extending earlier work in the elliptic case. According to our approach, special emphasis is placed on integration by parts formulas at the level of local martingales. Combined with the optional sampling theorem, this turns out to be an efficient way of dealing with boundary conditions, as well as with difficulties related to finite lifetime of the underlying diffusion. Our formulas require hypoellipticity of the diffusion in the sense of Malliavin calculus (integrability of the inverse Malliavin covariance) and are formulated in terms of the derivative flow, the Malliavin covariance and its inverse. Finally, some extensions to the nonlinear setting of harmonic mappings are discussed.

By using Hsu's multiplicative functional for the Neumann heat equation, a natural damped gradient operator is defined for the reflecting Brownian motion on compact manifolds with boundary. This operator is linked to quasi-invariant flows in terms of a integration by parts formula, which leads to the standard log-Sobolev inequality for the associated Dirichlet form on the path space., Comment: 14 pages

Let $P_t$ be the diffusion semigroup generated by $L:=\Delta +\nabla V$ on a complete connected Riemannian manifold with $\operatorname {Ric}\ge-(\sigma ^2\rho_o^2+c)$ for some constants $\sigma, c>0$ and $\rho_o$ the Riemannian distance to a fixed point. It is shown that $P_t$ is hypercontractive, or the log-Sobolev inequality holds for the associated Dirichlet form, provided $-\operatorname {Hess}_V\ge\delta$ holds outside of a compact set for some constant $\delta >(1+\sqrt{2})\sigma \sqrt{d-1}.$ This indicates, at least in finite dimensions, that $\operatorname {Ric}$ and $-\operatorname {Hess}_V$ play quite different roles for the log-Sobolev inequality to hold. The supercontractivity and the ultracontractivity are also studied., Comment: Published in at http://dx.doi.org/10.1214/08-AOP444 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We introduce a class of non-commutative Heisenberg like infinite dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the corresponding heat kernel measures, $\{��_t\}_{t>0},$ are also studied. We show that these heat kernel measures admit: 1) Gaussian like upper bounds, 2) Cameron-Martin type quasi-invariance results, 3) good $L^p$ -- bounds on the corresponding Radon-Nykodim derivatives, 4) integration by parts formulas, and 5) logarithmic Sobolev inequalities. The last three results heavily rely on the boundedness of the Ricci tensor.

We derive weighted log-Sobolev inequalities from a class of super Poincar\'e inequalities. As an application, the Talagrand inequality with larger distances are obtained. In particular, on a complete connected Riemannian manifold, we prove that the $\log^\dd$-Sobolev inequality with $\dd\in (1,2)$ implies the $L^{2/(2-\dd)}$-transportation cost inequality $$W^\rr_{2/(2-\dd)}(f\mu,\mu)^{2/(2-\dd)}\le C\mu(f\log f), \mu(f)=1, f\ge 0$$ for some constant $C>0$, and they are equivalent if the curvature of the corresponding generator is bounded below. Weighted log-Sobolev and entropy-cost inequalities are also derived for a large class of probability measures on $\R^d$.

By establishing the intrinsic super-Poincar\'e inequality, some explicit conditions are presented for diffusion semigroups on a non-compact complete Riemannian manifold to be intrinsically ultracontractive. These conditions, as well as the resulting uniform upper bounds on the intrinsic heat kernels, are sharp for some concrete examples.

We derive weighted log-Sobolev inequalities from a class of super Poincar�� inequalities. As an application, the Talagrand inequality with larger distances are obtained. In particular, on a complete connected Riemannian manifold, we prove that the $\log^\dd$-Sobolev inequality with $\dd\in (1,2)$ implies the $L^{2/(2-\dd)}$-transportation cost inequality $$W^\rr_{2/(2-\dd)}(f��,��)^{2/(2-\dd)}\le C��(f\log f), ��(f)=1, f\ge 0$$ for some constant $C>0$, and they are equivalent if the curvature of the corresponding generator is bounded below. Weighted log-Sobolev and entropy-cost inequalities are also derived for a large class of probability measures on $\R^d$.

We first prove the L^p-convergence (p\geq 1) and a Fernique-type exponential integrability of divergence functionals for all Cameron-Martin vector fields with respect to the pinned Wiener measure on loop spaces over a compact Riemannian manifold. We then prove that the Driver flow is a smooth transform on path spaces in the sense of the Malliavin calculus and has an \infty-quasi-continuous modification which can be quasi-surely well defined on path spaces. This leads us to construct the Driver flow on loop spaces through the corresponding flow on path spaces. Combining these two results with the Cruzeiro lemma [J. Funct. Anal. 54 (1983) 206-227] we give an alternative proof of the quasi-invariance of the pinned Wiener measure under Driver's flow on loop spaces which was established earlier by Driver [Trans. Amer. Math. Soc. 342 (1994) 375-394] and Enchev and Stroock [Adv. Math. 119 (1996) 127-154] by Doob's h-processes approach together with the short time estimates of the gradient and the Hessian of the logarithmic heat kernel on compact Riemannian manifolds. We also establish the L^p-convergence (p\geq 1) and a Fernique-type exponential integrability theorem for the stochastic anti-development of pinned Brownian motions on compact Riemannian manifold with an explicit exponential exponent. Our results generalize and sharpen some earlier results due to Gross [J. Funct. Anal. 102 (1991) 268-313] and Hsu [Math. Ann. 309 (1997) 331-339]. Our method does not need any heat kernel estimate and is based on quasi-sure analysis and Sobolev estimates on path spaces., Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000045

We extend to Riemannian manifolds the theory of conditioned stochastic differential equations. We also provide some enlargement formulas for the Brownian filtration in this nonflat setting., Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000126

The infinite Brownian loop $\{B_t^0,t\ge 0\}$ on a Riemannian manifold $\mathbb M$ is the limit in distribution of the Brownian bridge of length $T$ around a fixed origin $0$, when $T\to+\infty$. It has no spectral gap. When $\mathbb M$ has nonnegative Ricci curvature, $B^0$ is the Brownian motion itself. When $\mathbb M=G/K$ is a noncompact symmetric space, $B^0$ is the relativized $\Phi_0$--process of the Brownian motion, where $\Phi_0$ denotes the basic spherical function of Harish--Chandra, i.e. the $K$--invariant ground state of the Laplacian. In this case, we consider the polar decomposition $B_t^0=(K_t,X_t)$, where $K_t\in K/M$ and $X_t\in\overline{\mathfrak a_+}$, the positive Weyl chamber. Then, as $t\to+\infty$, $K_t$ converges and $d(0,X_t)/t\to 0$ almost surely. Moreover the processes $\{X_{tT}/\sqrt{T},t\ge 0\}$ converge in distribution, as $T\to+\infty$, to the intrinsic Brownian motion of the Weyl chamber. This implies in particular that $d(0,X_{tT})/\sqrt{T}$ converges to a Bessel process of dimension $D=\operatorname{rank}\mathbb M+2j$, where $j$ denotes the number of positive indivisible roots. An ingredient of the proof is a new estimate on $\Phi_0$.

The remarkable link between the structure of the white noise and that of the infinite dimensional rotation group has been exemplified by various approaches in probability theory and harmonic analysis. Such a link naturally becomes more intricate as the dimension of the time-parameter space of the white noise increases. One of the powerful method to illustrate this situation is to observe the structure of certain subgroups of the infinite dimensional rotation group that come from the diffeomorphisms of the time-parameter space, that is the time change. Indeed, those subgroups would shed light on the probabilistic meanings hidden behind the usual formal observations. Moreover, the subgroups often describe the way of dependency for Gaussian random fields formed from the white noise as the time-parameter runs over the basic parameter space.The main purpose of this note is to introduce finite dimensional subgroups of the infinite dimensional rotation group that have important probabilistic meanings and to discuss their roles in probability theory. In particular, we shall see that the conformal invariance of white noise can be described in terms of the conformal group which is a finite dimensional Lie subgroup of the infinite dimensional rotation group.

Harmonic maps are viewed as maps sending a fixed diffusion to manifold-valued martingales. Under a convexity condition, we prove that the continuity of real-valued harmonic functions implies the continuity of harmonic maps. Then we prove with a probabilistic method that continuous harmonic maps are smooth under Hormander's condition; the proof relies on the study of martingales with values in the tangent bundle.

Under a natural invariance assumption on the Levy measure we construct compound Poisson processes and more general isotropic Levy processes on Riemannian manifolds by projection of a suitable horizontal process in the bundle of orthonormal frames.We characterize such Levy processes through their infinitesimal generators and we show that they can be realized as the limit of a sequence of Brownian motions which are interlaced with jumps along geodesic segments.

The infinitesimal reduced variation of the stochastic parallel displacement on the commutative principal bundle is martingale if and only if the connections satisfy the Maxwell equations.

We show that there are no stochastically complete minimal submanifolds immersed in Cartan-Hadamard manifolds which lie in a non-degenerate cone type domain.

By using the maximum principle and analysis of heat semigroups, Harnack inequalities are studied for log-Sobolev functions. From this, some lower bound estimates of the log-Sobolev constant are presented by using the spectral gap inequality and the coupling method. The resulting inequalities either recover or improve the corresponding ones proved by Chung and Yau. Especially, Harnack inequalities and estimates of log-Sobolev constants can be dimension-free.

The geometric stochastic analysis on the Riemannian path space developed recently gives rise to the concept of tangent processes. Roughly speaking, it is the infinitesimal version of the Girsanov theorem. Using this concept, we shall establish a formula of integration by parts on the path space over a loop group. Following the martingale method developed in Capitaine, Hsu and Ledoux, we shall prove that the logarithmic Sobolev inequality holds on the full paths. As a particular case of our result, we obtain the Driver–Lohrenz’s heat kernel logarithmic Sobolev inequalities over loop groups. The stochastic parallel transport introduced by Driver will play a crucial role.

We consider the class of stationary, zero-mean Gaussian processes, indexed by the circle, satisfying a two-point Markov property and taking values in a vector bundle over the circle with given holonomy. We establish, subject to certain additional symmetry properties, a classification of all such processes. We then propose a construction of a Brownian motion of loops, in which these processes provide the infinitesimal increments.

For nonlinear functions f of a random vector Y, E[f(Y)] and f(E[Y]) usually differ. Consequently the mathematical expectation of Y is not intrinsic: when we change coordinate systems, it is not invariant.This article is about a fundamental and hitherto neglected property of random vectors of the form Y = f(X(t)), where X(t) is the value at time t of a diffusion process X: namely that there exists a measure of location, called the "intrinsic location parameter" (ILP), which coincides with mathematical expectation only in special cases, and which is invariant under change of coordinate systems. The construction uses martingales with respect to the intrinsic geometry of diffusion processes, and the heat flow of harmonic mappings. We compute formulas which could be useful to statisticians, engineers, and others who use diffusion process models; these have immediate application, discussed in a separate article, to the construction of an intrinsic nonlinear analog to the Kalman Filter. We present here a numerical simulation of a nonlinear SDE, showing how well the ILP formula tracks the mean of the SDE for a Euclidean geometry., 25 pages, 1 figure

We investigate the escape rate of the Brownian motion $W_x (t)$ on a complete noncompact Riemannian manifold. Assuming that the manifold has at most polynomial volume growth and that its Ricci curvature is bounded below, we prove that $$\dist (W_x (t), x) \leq \sqrt{Ct \log t}$$ for all large $t$ with probability 1. On the other hand, if the Ricci curvature is nonnegative and the volume growth is at least polynomial of the order $n > 2$ then $$\dist (W_x (t), x) \geq \frac{\sqrt{Ct}}{\log^{1/(n-2)} t \log \log^{(2+\varepsilon)/(n-2)} t} again for all large $t$ with probability 1 (where $\varepsilon > 0$).

This note is concerned with the differentiation of heat semigroups on Riemannian manifolds. In particular, the relation $dP_tf=P_tdf$ is investigated for the semigroup generated by the Laplacian with Dirichlet boundary conditions. By means of elementary martingale arguments it is shown that well-known properties which hold on complete Riemannian manifolds fail if the manifold is only BM-complete. In general, even if $M$ is flat and $f$ smooth of compact support, $\Vert dP_tf\Vert_\infty$ cannot be estimated on compact time intervals in terms of $f$ or $df$.

We present a general method to construct $m$-symmetric diffusion processes $(X_t, \mathbf{P}_x)$ on any given locally compact metric space $(X, d)$ equipped with a Radon measure $m$. These processes are associated with local regular Dirichlet forms which are obtained as $\Gamma$-limits of approximating nonlocal Dirichlet forms. This general method works without any restrictions on $(X, d, m)$ and yields processes which are well defined for quasi every starting point. ¶ The second main topic of this paper is to formulate and exploit the so-called Measure Contraction Property. This is a condition on the original data $(X, d, m)$ which can be regarded as a generalization of curvature bounds on the metric space $(X, d)$. It is a bound for distortions of the measure $m$ under contractions of the state space $X$ along suitable geodesics or quasi geodesics w.r.t. the metric $d$. In the case of Riemannian manifolds, this condition is always satisfied. Several other examples will be discussed, including uniformly elliptic operators, operators with weights, certain subelliptic operators, manifolds with boundaries or corners and glueing together of manifolds. ¶ The Measure Contraction Property implies upper and lower Gaussian estimates for the heat kernel and a Harnack inequality for the associated harmonic functions. Therefore, the above-mentioned diffusion processes are strong Feller processes and are well defined for every starting point.

We obtain a formula for Liapounov exponents of stochastic flows generated by stochastic differential equations on compact manifolds. As an application, the exponents of a class of stochastic flows on spheres are determined explicitly.

This paper is concerned with the intrinsic metrics of the two main classes of superprocesses. For the Fleming-Viot process, we identify it as the Bhattacharya distance, and for Dawson-Watanabe processes, we find the Kakutani-Hellinger metric. The corresponding geometries are studied in some detail. In particular, representation formulas for geodesics and arc length functionals are obtained. The relations between the two metrics yield a geometric interpretation of the identification of the Fleming-Viot process as a Dawson-Watanabe superprocess conditioned to have total mass 1. As an application, a functional limit theorem for super-Brownian motion conditioned on local extinction is proved.

We show how the Clark-Ocone-Haussmann formula for Brownian motion on a compact Riemannian manifold put forward by S. Fang in his proof of the spectral gap inequality for the Ornstein-Uhlenbeck operator on the path space can yield in a very simple way the logarithmic Sobolev inequality on the same space. By an appropriate integration by parts formula the proof also yields in the same way a logarithmic Sobolev inequality for the path space equipped with a general diffusion measure as long as the torsion of the corresponding Riemannian connection satisfies Driver's total antisymmetry condition.

Consider the first exit time and position from small geodesic balls for Brownian motion on Riemannian manifolds. We establish a smooth Besselization technique and calculate the asymptotic expansion for the joint distributions by a purely probabilistic approach.

It is known that any smooth, nondegenerate, second-order elliptic operator on a manifold (dimension $\not= 2$) has the form $\Delta +B$, where B is a vector field and $\Delta$ is the Laplace-Beltrami operator under some Riemannian metric on the manifold. In this paper we give several conditions on the "Ricci curvature" Ric $-\nabla_B^s$ associated with the operator $\Delta + B$ to ensure that the diffusion semigroup generated by $\Delta + B$ conserves probability and possesses the Feller property.

We consider conditional propagation of chaos and use it to solve a class of quasilinear equations of parabolic type. In addition, we construct a class of continuous stochastic processes associated with the above nonlinear equations. Our method imposes fewer smoothness conditions on the coefficients and allows a degenerate nonlinear weight before a divergence form operator. We hope this probabilistic approach will introduce a better microscopic picture for understanding some Stefan type problems.

Let $V_W$ and $V_Y$ be Euclidean vector spaces and let $V_Z \equiv L(V_W \rightarrow V_Y)$. Given a Wiener process $W$ on $V_W$, with natural filtration $\{\mathscr{I}_t\}$, and a $\mathscr{I}_T$-measurable random variable $U$ in $V_Y$, we seek adapted processes $(Y, Z)$ in $V_Y \times V_Z$ satisfying the SDE $U = Y(t) + \int_{(t,T\rbrack}ZdW - \int_{(t, T\rbrack} \Gamma(Y, ZZ^\ast) ds/2, 0 \leq t \leq T,$ under local Lipschitz and convexity conditions on the map $(y, A) \rightarrow \Gamma(y, A)$. These conditions apply in particular in the case $\Gamma(y, A) = \sum\Gamma^i_{jk}(y)A^{jk}$, where $\Gamma$ is a linear connection on $V_Y$ whose Christoffel symbols $\Gamma^i_{jk}$ are bounded and Lipschitz, and $\Gamma$ has certain convexity properties. In that case the solution $Y$ above is known as a $\Gamma$-martingale with terminal value $U$. The solution $(Y, Z)$ is constructed explicitly using the Pardoux-Peng theory of backwards SDE's. Applications include the Dirichlet problem and the heat equation for harmonic mappings, and other PDE's.

We prove a central limit theorem for the distance of the Brownian point on the universal cover of a compact negatively curved Riemannian manifold. The technical point is a contraction property for the leafwise Brownian motion along the stable foliation.

An upper bound is given for the behaviour of the radial part of a $\Gamma$-martingale, generalizing previous work of the author on the radial part of Riemannian Brownian motion. This upper bound is applied to establish an integral curvature condition to determine when $\Gamma$-martingales cannot "implode" in finite intrinsic time, answering a question of Emery and generalizing work of Hsu on the $C_0$-diffusion property of Brownian motion.

Let $\rho_t$ be the radial part of a Brownian motion in an $n$-dimensional Riemannian manifold $M$ starting at $x$ and let $T = T_\varepsilon$ be the first time $t$ when $\rho_t = \varepsilon$. We show that $E\lbrack \rho^2_{t\wedge T} \rbrack = nt - (1/6)S(x)t^2 + \sigma(t^2)$, as $t \downarrow 0$, where $S(x)$ is the scalar curvature. The same formula holds for $E\lbrack\rho^2_t\rbrack$ under some boundedness condition on $M$.

We present short and elementary proofs of three recent results on exponential integrability of Lipschitz functions and quantitative bounds on the diameter under logarithmic Sobolev inequalities due respectively to S. Aida, T. Masuda, I. Shigekawa [A-M-S], D. Bakry, D. Michel [B-M] and L. Saloff-Coste [SC]. Although the first two results we aim to simplify deal with abstract Markov generators on probability spaces, we would like to briefly present the purpose of this note in the setting of the Laplace-Beltrami operator ∆ on a complete connected Riemannian manifold M of finite volume V . We will consider the normalized measure dμ = 1 V dv where dv denote the Riemannian measure and let ∇ be the Riemannian gradient on M . For a nonnegative bounded (say) real valued function f on M , let E(f) denote the entropy of f with respect to μ defined by E(f) = ∫ f log fdμ− ∫ fdμ log (∫ fdμ ) . We will say that ∆ satisfies a logarithmic Sobolev inequality if there exists ρ > 0 such that for all C∞, compactly supported or bounded, functions f on M , ρE(f) ≤ 2 ∫ f(−∆f)dμ = 2 ∫ |∇f |dμ. The largest possible value ρ0 for ρ is called the logarithmic Sobolev constant of the Laplacian ∆ on M , or simply of M . More generaly, one may consider, following [B], inequalities between entropy and energy of the type E(f) ≤ Φ (∥∥|∇f |∥∥2 2 ) for all C∞ bounded functions f with ‖f‖2 = 1 where Φ is a nonnegative function on [0,∞). With these notations, S. Aida, T. Masuda and I. Shigekawa [A-M-S] recently showed that, when ρ0 > 0, whenever f is a function on M such that ‖|∇f |‖∞ ≤ 1 (that is f is Lipschitz with Lipschitz norm less than or equal to 1), then, for every 0 < α < ρ0/2

In this paper we study multidimensional stochastic reaction-diffusion equations (SRDE's) with white noise boundary data. More precisely, we consider a general SRDE with Robin data to be a white noise field. Because this boundary data is very irregular, we formulate a set of conditions that a random field must satisfy to solve the SRDE. We show that a unique solution exists, and we study the boundary-layer behavior of the solution. This boundary-layer analysis reveals some natural restrictions on the reaction term of the SRDE that ensure that the reaction term does not qualitatively affect the boundary layer. The boundary-layer analysis also leads to the definition of some functional Banach spaces into which are encoded the boundary-layer degeneracies and that would be the natural settings for other analyses of the SRDE of this paper (e.g., large deviations and central limit theorems, approximation theorems).

There is a lack of appropriate replication of the asymptotical behaviour of stationary stochastic differential equations solved by numerical methods. The paper illustrates this fact with the stationary Ornstein-Uhlenbeck process with real-diagonalizable drift and the familiy of implicit Euler methods. For the description of the occuring bias the notions of asymptotical p-th mean, mean, variance and equilibrium preservation are introduced. The main result can be useful for the implementation of numerical algorithms requiring more precise long-term runs, such as in discrete parametric estimation or in numerical computation of top Lyapunov exponents.

A number of asymptotic problems for "classical" stochastic processes leads to diffusion processes on graphs. In this paper we study several such examples and develop a general technique for these problems. Diffusion in narrow tubes, processes with fast transmutations and small random perturbations of Hamiltonian systems are studied.

The joint density of the total winding and the radius of a planar Brownian motion is calculated, by solving the associated backward equation. An explicit expression for the distribution of the hitting time of an angular barrier is shown; in particular, the behavior of the tail of the distribution is determined.