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Inspired by Y. Ollivier's coarse Ricci curvature, we introduce a novel concept of coarse extrinsic curvature on Riemannian submanifolds. This is defined through Wasserstein distances between test probability measures supported in the tubular neighbourhood of the submanifold. This framework provides an understanding of the geometric properties of embeddings, offering valuable insights into their curvature dynamics and intrinsic structures. Additionally, this coarse curvature can also be extracted from empirical measures supported on random point clouds generated by a Poisson point process, and has the potential to extend to metric embeddings., Comment: 52 pages, 6 figures

The initial and boundary conditions of the Galactic star formation in molecular clouds are not well understood. In an effort to shed new light on this long-standing problem, we measured properties of dense cores and filamentary structures in the Vela C molecular cloud, observed with Herschel. We applied the getsf extraction method to separate the components of sources and filaments from each other and their backgrounds, before detecting, measuring, and cataloging the structures. The cores and filamentary structures constitute 40% of the total mass of Vela C, most of the material is in the low-density molecular background cloud. We selected 570 reliable cores, of which 149 are the protostellar cores and 421 are the starless cores. Almost 78% of the starless cores were identified with the gravitationally bound prestellar cores. The exponent of the CMF (alpha = 1.35) is identical to that of the Salpeter IMF. We selected 68 filaments with at least one side that appeared not blended with adjacent structures. The filament widths are in the range of 0.15 pc to 0.63 pc, and have a median value of W = 0.3(0.11) pc. The surface densities of filaments are well correlated with their contrasts and linear densities. Within uncertainties of the filament instability criterion, many filaments may well be both supercritical and subcritical. A large fraction of filaments may definitely be considered supercritical, in which are found 94 prestellar cores, 83 protostellar cores, and only 1 unbound starless core. Taking into account the uncertainties, the supercritical filaments contain only prestellar and protostellar cores. Our findings support the idea that there exists a direct relationship between the CMF and IMF and that filaments play a key role in the formation of prestellar cores, which is consistent with the previous Herschel results., Comment: 15 pages, 12 figures, 3 tables, accepted for publication in A&A

We study the large scale fluctuations of the solution of a nonlinear stochastic heat equation (SHE) in dimensions $d \geq 3$ driven by a multiplicative Gaussian noise, white in time and coloured in space with non-integrable spatial covariance decaying at the rate of $|x|^{-\kappa}$, with $\kappa \in (2, d)$. Motivated by recent works on SHE and KPZ driven by noise with compactly supported spatial correlation, we show that the fluctuations converge to the solution of a stochastic heat equation with an explicit additive spatially correlated noise that has a Riesz kernel of degree $-\kappa$ as its spatial covariance. The convergence of the fluctuations is shown to take place in the optimal H\"older topologies.

We show that the generalized Ricci tensor of a weighted complete Riemannian manifold can be retrieved asymptotically from a scaled metric derivative of Wasserstein 1-distances between normalized weighted local volume measures. As an application, we demonstrate that the limiting coarse curvature of random geometric graphs sampled from Poisson point process with non-uniform intensity converges to the generalized Ricci tensor., Comment: 34 pages, 5 figures

We investigate the stationary measure $\pi$ of SDEs driven by additive fractional noise with any Hurst parameter and establish that $\pi$ admits a smooth Lebesgue density obeying both Gaussian-type lower and upper bounds. The proofs are based on a novel representation of the stationary density in terms of a Wiener-Liouville bridge, which proves to be of independent interest: We show that it also allows to obtain Gaussian bounds on the non-stationary density, which extend previously known results in the additive setting. In addition, we study a parameter-dependent version of the SDE and prove smoothness of the stationary density, jointly in the parameter and the spatial coordinate. With this we revisit the fractional averaging principle of Li and Sieber [Ann. Appl. Probab. 32(5) (2022)] and remove an ad-hoc assumption on the limiting coefficients. Avoiding any use of Malliavin calculus in our arguments, we can prove our results under minimal regularity requirements., Comment: Accepted version, 31 pages

Cores and filamentary structures are the prime birthplaces of stars, and play key roles in the process of star formation. Latest advances in the methods of multi-scale source and filament extraction, and in making high-resolution column density map from $Herschel$ multi-wavelength observations enable us to detect the filamentary network structures in highly complex molecular cloud environments. The statistics for physical parameters shows that core mass strongly correlates with core dust temperature, and $M/L$ strongly correlates with $M/T$, which is in line with the prediction of the blackbody radiation, and can be used to trace evolutionary sequence from unbound starless cores to robust prestellar cores. Crest column densities of the filamentary structures are clearly related with mass per unit length ($M_{\rm line}$), but are uncorrelated by three orders ranging from $\sim 10^{20}$ to $\sim 10^{22}$ $ \rm cm^{-2}$ with widths. Full width at half maximum (FWHM) have a median value of 0.15 pc, which is consistent with the 0.1 pc typical inner width of the filamentary structures reported by previous research. We find $\sim $70\% of robust prestellar cores (135/199) embedded in supercritical filaments with $M_{\rm line}>16~M_{\odot}/{\rm pc}$, which implies that the gravitationally bound cores come from fragmentation of supercritical filaments. And on the basis of observational evidences that probability distribution function (PDF) with power-law distribution in the Perseus south is flatter than north, YSO number is significantly less than that in the north, and dust temperature difference. We infer that south region is more gravitationally bound than north region., Comment: 15 pages, 9 figures, accepted for publication in RAA

We investigate the effective behaviour of a small transversal perturbation of order $\epsilon$ to a completely integrable stochastic Hamiltonian system, by which we mean a stochastic differential equation whose diffusion vector fields are formed from a completely integrable family of Hamiltonian functions $H_i, i=1,\dots n$. An averaging principle is shown to hold and the action component of the solution converges, as $\epsilon \to 0$, to the solution of a deterministic system of differential equations when the time is rescaled at $1/\epsilon$. An estimate for the rate of the convergence is given. In the case when the perturbation is a Hamiltonian vector field, the limiting deterministic system is constant in which case we show that the action component of the solution scaled at $1/\epsilon^2$ converges to that of a limiting stochastic differentiable equation.

We study fast / slow systems driven by a fractional Brownian motion $B$ with Hurst parameter $H\in (\frac 13, 1]$. Surprisingly, the slow dynamic converges on suitable timescales to a limiting Markov process and we describe its generator. More precisely, if $Y^\varepsilon$ denotes a Markov process with sufficiently good mixing properties evolving on a fast timescale $\varepsilon \ll 1$, the solutions of the equation $$ dX^\varepsilon = \varepsilon^{\frac 12-H} F(X^\varepsilon,Y^\varepsilon)\,dB+F_0(X^\varepsilon,Y^\varepsilon)\,dt\; $$ converge to a regular diffusion without having to assume that $F$ averages to $0$, provided that $H< \frac 12$. For $H > \frac 12$, a similar result holds, but this time it does require $F$ to average to $0$. We also prove that the $n$-point motions converge to those of a Kunita type SDE. One nice interpretation of this result is that it provides a continuous interpolation between the homogenisation theorem for random ODEs with rapidly oscillating right-hand sides ($H=1$) and the averaging of diffusion processes ($H= \frac 12$).

Our first result is a stochastic sewing lemma with quantitative estimates for mild incremental processes, with which we study SPDEs driven by fractional Brownian motions in a random environment. We obtain uniform $L^p$-bounds. Our second result is a fractional averaging principle admitting non-stationary fast environments. As an application, we prove a fractional averaging principle for SPDEs., Comment: To appear in Stochastics and Dynamics; 38 pages

The main results of the article are short time estimates and asymptotic estimates for the first two order derivatives of the logarithmic heat kernel of a complete Riemannian manifold. We remove all curvature restrictions and also develop several techniques. A basic tool developed here is intrinsic stochastic variations with prescribed second order covariant differentials, allowing to obtain a path integration representation for the second order derivatives of the heat semigroup $P_t$ on a complete Riemannian manifold, again without any assumptions on the curvature. The novelty is the introduction of an $\epsilon^2$ term in the variation allowing greater control. We also construct a family of cut-off stochastic processes adapted to an exhaustion by compact subsets with smooth boundaries, each process is constructed path by path and differentiable in time, furthermore the differentials have locally uniformly bounded moments with respect to the Brownian motion measures, allowing to by-pass the lack of continuity of the exit time of the Brownian motions on its initial position.

I was asked to make my, by now quite old PhD thesis, available on the arxiv, for parts of it was never submitted for publication. The thesis offers a systematic study of stochastic differential equations (SDEs) on non-compact spaces. In particular we solve the open problem on strong completeness. An SDE is strongly complete if its solution can be chosen to depend continuously in space and in time for all time. The question is whether non-explosion, with possibly additional assumptions, implies strong completeness. Strong completeness of an SDE implies that its solution depends continuously on the initial condition, opening up possibility for numerical solutions, and the existence of a perfect Cocycle (a basic assumption on random dynamical systems). This was known only for compact manifolds and for linear state spaces, methods for either are not applicable to a general space. We also obtain existence of the global smooth solution flow of SDEs on $R^n$ (sometimes allowing substantial growth of the derivative of the coefficients, removing the global Lipschitz continuity condition). Non-explosion, the $C_0$-property, and the derivative flow are studied. We showed Bismut-Witten Laplacians are essentially self-adjojnt, paving the way for studying theirs semigroups acting on functions and on differential forms. We relate the Markovian semi-group on differential one forms with the semi-group $P_t$ on functions (inter-twining), find a method for proving path integration formulas for $dP_tf$, path integration formula for semi-group on differential forms, moment bounds for the derivative flows. Relation are obtained between intrinsic topological and geometrical properties of the manifold and that of SDEs. Information on the homotopy and cohomology of the manifolds are obtained from moment stability of the stochastic flows., Comment: PhD thesis, 1992

We prove an enhanced limit theorem for additive functionals of a multi-dimensional Volterra process $(y_t)_{t\geq 0}$ in the rough path topology. As an application, we establish weak convergence as $\varepsilon\to 0$ of the solution of the random ordinary differential equation (ODE) $\frac{d}{dt}x^\varepsilon_t=\frac{1}{\sqrt \varepsilon} f(x_t^\varepsilon,y_{\frac{t}{\varepsilon}})$ and show that its limit solves a rough differential equation driven by a Gaussian field with a drift coming from the L\'evy area correction of the limiting rough driver. Furthermore, we prove that the stochastic flows of the random ODE converge to those of the Kunita type It\^o SDE $dx_t=G(x_t,dt)$, where $G(x,t)$ is a semi-martingale with spatial parameters., Comment: Published version with minor typos corrected; 32 pages

We prove a fractional averaging principle for interacting slow-fast systems. The mode of convergence is in H\"older norm in probability. The main technical result is a quenched ergodic theorem on the conditioned fractional dynamics. We also establish geometric ergodicity for a class of fractional-driven stochastic differential equations, improving a recent result of Panloup and Richard., Comment: 37 pages. Accepted version

We review recent developments of slow/fast stochastic differential equations, and also present a new result on Diffusion Homogenisation Theory with fractional and non-strong-mixing noise and providing new examples. The emphasise of the review will be on the recently developed effective dynamic theory for two scale random systems with fractional noise: Stochastic Averaging and `Rough Diffusion Homogenisation Theory'. We also study the geometric models with perturbations to symmetries.

With recently developed tools, we prove a homogenisation theorem for a random ODE with short and long-range dependent fractional noise. The effective dynamics are not necessarily diffusions, they are given by stochastic differential equations driven simultaneously by stochastic processes from both the Gaussian and the non-Gaussian self-similarity universality classes. A key lemma for this is the `lifted' joint functional central and non-central limit theorem in the rough path topology., Comment: This is the revised second part of arXiv:1911.12600

We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein-Uhlenbeck process, providing the foundation for the fluctuation theory of slow/fast systems driven by such a noise. Our main contribution is on the joint convergence to a limit with both Gaussian and non-Gaussian components. This is valid for any $L^2$ functions, whereas for functions with stronger integrability properties the convergence is shown to hold in the H\"older topology. As an application we prove a `rough creation' result, i.e. the weak convergence of a family of random smooth curves to a non-Markovian random process with rough sample paths. This includes the second order problem and the kinetic fractional Brownian motion model., Comment: To appear in the Journal of Theoretical Probability. arXiv admin note: text overlap with arXiv:1911.12600

We consider a system of differential equations in a fast long range dependent random environment and prove a homogenization theorem involving multiple scaling constants. The effective dynamics solves a rough differential equation, which is `equivalent' to a stochastic equation driven by mixed It\^o integrals and Young integrals with respect to Wiener processes and Hermite processes. Lacking other tools we use the rough path theory for proving the convergence, our main technical endeavour is on obtaining an enhanced scaling limit theorem for path integrals (Functional CLT and non-CLT's) in a strong topology, the rough path topology, which is given by a H\"older distance for stochastic processes and their lifts. In dimension one we also include the negatively correlated case, for the second order / kinetic fractional BM model we also bound the error., Comment: 70 pages

We prove a Kodaira-Hodge decomposition on differential 1-forms on the space of non-smooth paths over a Riemannian manifold, allowing us to define the corresponding first cohomology group. This uses the It\^o map of a Brownian system and damped stochastic parallel translation.

We use a basic martingale method to show a differentiation formula for the derivatives $$d(P_tf)(x_0)(v_0)={1\over t} E f(x_t) \int_0^t \langle Y(x_s)(v_s),dB_t\rangle_{R^m}.$$ These are proved first on $R^n$, then on manifolds. Afterwards for solutions of heat equations on differential forms, and a second order formula.

There are two open problem on the analysis of continuous paths on a Riemannian manifold, the Markov uniqueness and the independence of the closure of the differential operator $d$ on its initial domain. The operator $d$ acts naturally on $BC^1$ functions, one is concerned with its extensions to the $L^2$ spaces. With a suitable choice of an initial domain we denote by $D^{2,1}$ its closure under the graph norm. For the Wiener space, the domain of $d$ can be classified, as a consequence its extension is unique whether the initial domain is smooth cylindrical or in $BC^1$ etc. This has not shown to be the same when the measure is the probability distribution of any smooth elliptic diffusion. In an earlier paper, we have shown that the closure of $BC^\infty$ functions agree with that of smooth cylindrical functions, leaving an undesirable gap. The Markov uniqueness is essentially concerned with the problem whether there exists a unique Markov process on the path space whose Markov generator agrees with the infinite-dimensional Laplacian on $C^\infty$ cylindrical functions. Here we reduce Markov uniqueness to whether the pull back of $D^{2,1}$ by the ito map is $ D^{2,1}$ (i.e. a surjection). We also propose a possible approach for tackle this problem.

An approach to analysis on path spaces of Riemannian manifolds is described. The spaces are furnished with `Brownian motion' measure which lies on continuous paths, though differentiation is restricted to directions given by tangent paths of finite energy. An introduction describes the background for paths on ${\mathbb R}^m$ and Malliavin calculus. For manifold valued paths the approach is to use `It\^o' maps of suitable stochastic differential equations as charts . `Suitability' involves the connection determined by the stochastic differential equation. Some fundamental open problems concerning the calculus and the resulting `Laplacian' are described. A theory for more general diffusion measures is also briefly indicated. The same method is applied as an approach to getting over the fundamental difficulty of defining exterior differentiation as a closed operator, with success for one \& two forms leading to a Hodge -Kodaira operator and decomposition for such forms. Finally there is a brief description of some related results for loop spaces.

An integration by parts formula is the foundation for stochastic analysis on path spaces over a (finite dimensional) Riemannian manifold or over $R^n$, from which we may deduce the operator $d$ is closable and define the Laplacian operator on path spaces. A useful formula on the Riemannian manifold is $$dP_tf(v)=(1/t)E f(x_t) \int_0^t \langle d\{x_s\}, v_s\rangle ,$$ for $P_t$ the heat semi-group, $x_t$ the BM, $v_t$ the derivative flow or its conditional expectation (which is a damped parallel translation), $d\{x_s\}$ is the martingale part of $x_t$. As a meta theorem, this leads to the Clark-Ocone formula (martingale representation theorem with specific integrand) and Logrithmic Sobolev inequalities. Interpreted appropriately, the latter formula is obviously a special case of the integration by parts formula. Here we show by the Markov property and by induction that the latter formula implies the integration by parts formula. WE also use Bismut's original approach to prove an integration by parts formula, using a connection with torsion and one on the free path space.

Integration by parts formulae are given for a class of measures on the space of paths of a smooth manifold $M$ determined by the laws of degenerate diffusions. The mother of such formulae, on the path space of diffeomorphism group of $M$ is shown to arise from a quasi-invariance property of measures determined by stochastic flows. From this the other formulae are derived by filtering out redundant noise using an associated LeJan-Watanabe connection.

Formulae are given for $dP_t \phi$, $d^*P_t\phi$ and $\Delta P_t\phi$ for $P_t$ the heat semigroup acting on a q-form $\phi$. The formulae are Brownian motion expectations of $\phi$ composed with random translations determined by Weitzenbock curvarure terms. Derivatives of the curvature are not involved.

A great open problem is: can one learn the topology of the non-smooth path spaces with an L2 Hodge-deRham theory This one hopes to establish through a suitable complex of differential forms. Since the space is a Banach manifolds, and the Hodge theory is based on Hilbert spaces, it is trick to find such a complex. The relatively simpler Bismut tangent spaces and their tensor products, whose dual spaces are natural candidates for the complex, cannot be used, because these spaces are not necessarily closed under the Lie bracket operation if there is the effect of curvature. In this article we seek out a class of nice vector fields whose brackets behaves are calculable and behave nicely the damped tensor vector fields, using an It\^o mp, which is also used by the authors in `Special It\^o maps and an L2 Hodge theory for one forms on path spaces. Stochastic processes, physics and geometry: new interplays, I (Leipzig, 1999), 145-162, CMS Conf. Proc., 28, Amer. Math. Soc., Providence, RI, 2000').

Given a pair of second order diffusion operators, one on the total space of a principle bundle $N$ and the other on the base space $M$, intertwined by the projection $\pi:N\to M$, if the operator ${\mathcal A}$ on the base manifold has constant rank, we define a semi-connection on the principal bundle which allows to split the diffusion operator ${\mathcal B}$ on the total space into the sum of the horizontal lift of ${\mathcal A}$ and the other vertical. This allow to conclude a disintegration theorem for the law of ${\mathcal B}$. As an application, a decomposition of stochastic flow is given.

Following Le Jan and Watanabe we define a connection associated with a non-degenrrate diffusion operators. This connection is characterized here and shown to be the Levi-Civita connection for gradient systems. This both explains why such systems have useful properties and allows us to extend these properties to more general systems. Topics described here include: moment estimates for $T\xi_t$, a Weitzenb\"ock formula for the generator of the semigroup on p-forms induced by the flow, a Bismut type formula for $d\log p_t$ in terms of an arbitrary metric connection, and a generalized Bochner vanishing theorem. A comprehensive theory on this, its generalization to semi-elliptic case and applications is published in the book `On the geometry of diffusion operators and stochastic flows'. Related to this is also the book `The geometry of filtering'. This article is easier to read.

In this paper we discuss the stability of stochastic differential equations and the interplay between the moment stability of a SDE and the topology of the underlying manifold. Sufficient and necessary conditions are given for the moment stability of a SDE in terms of the coefficients. Finally we prove a vanishing result for the fundamental group of a complete Riemannian manifold in terms of purely geometrical quantities.

A unified treatment is given of some results of H. Donnelly-P. Li and L. Schwartz concerning the behaviour of heat semigroups on open manifolds with given compactifications, on one hand, and the relationship with the behaviour at infinity of solutions of related stochastic differential equations on the other. A principal tool is the use of certain covers of the manifold: which also gives a non-explosion test. As a corollary we obtain known results about the behaviour of Brownian motions on a complete Riemannian manifold with Ricci curvature decaying at most quadratically in the distance function.

We introduce strong p-completeness and use them for studying the continuous dependence of solutions of SDE's on non-compact manifolds. We obtain conditions for the existence of global smooth solution flow, and prove their diffeomorphism properties. The criterion is in terms of the growth of the solutions of the linearized equations (i.e. the derivative flows). It is also given in terms of the driving vector fields and their derivatives. For the Euclidean spaces there are three principle classes of examples: (1) the driving vector fields grow at most $|x|^{1-2\alpha}$ and the derivatives of the driving vector are allowed to be the order $|x|^\alpha$; (2) the coefficients of the SDE grows linearly while its derivative allowed to grow logarithmically, (3) The coefficients are bounded while the derivatives are allowed to grow linearly. We also given test on non-explosions.

Let $M$ be a compact Riemannian manifold and $h$ a smooth function on $M$. Let $\rho^h(x)=\inf_{|v|=1}\left(Ric_x(v,v)-2Hess(h)_x(v,v) \right)$. Here $Ric_x$ denotes the Ricci curvature at $x$ and $Hess(h)$ is the Hessian of $h$. Then $M$ has finite fundamental group if $\Delta ^h-\rho^h<0$. Here $\Delta ^h=: \Delta +2L_{\nabla h}$ is the Bismut-Witten Laplacian. This leads to a quick proof of recent results on extension of Myers' theorem to manifolds with mostly positive curvature. There is also a similar result for noncompact manifolds.

It is well known that some important Markov semi-groups have a "regularization effect" -- as for example the hypercontractivity property of the noise operator on the Boolean hypercube or the Ornstein-Uhlenbeck semi-group on the real line, which applies to functions in $L^p$ for $p>1$. Talagrand had conjectured in 1989 that the noise operator on the Boolean hypercube has a further subtle regularization property for functions that are just integrable, but this conjecture remains open. Nonetheless, the Gaussian analogue of this conjecture was proven in recent years by Eldan-Lee and Lehec, by combining an inequality for the log-Hessian of the Ornstein-Uhlenbeck semi-group with a new deviation inequality for log-semi-convex functions under Gaussian measure. In this work, we explore the question of how much more general this phenomenon is. Specifically, our first goal is to explore the validity of both these ingredients for some diffusion semi-groups in $\mathbb{R}^n$, as well as for the $M/M/\infty$ queue on the non-negative integers and the Laguerre semi-groups on the positive real line. Our second goal is to prove a one-dimensional regularization effect for these settings, even in those cases where these ingredients are not valid., Comment: 33 pages

We consider slow / fast systems where the slow system is driven by fractional Brownian motion with Hurst parameter $H>{1\over 2}$. We show that unlike in the case $H={1\over 2}$, convergence to the averaged solution takes place in probability and the limiting process solves the 'na\"ively' averaged equation. Our proof strongly relies on the recently obtained stochastic sewing lemma.