Xue-Mei Li’s work lies at the vibrant intersection of probability theory, stochastic analysis, differential geometry (both finite and infinite dimensional), and multiscale dynamics, including fractional and non-Markovian systems. Combining deep geometric insights with rigorous analysis, she explores how noise “feels” curvature and topology, revealing how stochastic flows encode geometric and topological features and deepening our understanding of how randomness interacts with geometry within complex structures. She has forged remarkable collaborations, advancing the field through innovative applications of rough path theory to extend classical results to multiscale SDEs driven by rough or long-range dependent noise such as fractional Brownian motion and Volterra processes. By combining rough path theory with Malliavin calculus, she has extended large-scale dynamics and fluctuation theory for SPDEs to spatially long-range systems.
Among her many contributions are solving the longstanding open problem of strong completeness for SDEs by establishing fundamental criteria guaranteeing global smooth flows on non-compact manifolds; proving the derivative (BEL) formula for diffusion semigroups using an innovative martingale method; and coining and pioneering the study of strict local martingales. Her work in infinite-dimensional analysis and Malliavin calculus on manifolds includes profound contributions to the analysis of degenerate diffusion measures, proving Poincaré inequalities for loop measures, and obtaining sharp logarithmic estimates for heat kernels.
While on the topic of rough path, a recent progress is:A well-posed SDE is called complete (or conservative) if, for every initial point \( x \), its solution \( F_t(x, \omega) \) exists for almost every \( \omega \).
An SDE is said to be strongly complete (i.e. it has a smooth global solution flow) if there exists a set of full measure in \( \omega \) on which \( F_t(x, \omega) \) is jointly continuous in \( x \) and \( t \). In this case, the regularity of the solution is automatically determined by the regularity of its vector fields. Furthermore, if the adjoint SDE (that is, the SDE with its drift vector field reversed) is also strongly complete, then \( F_t(\cdot, \omega) \) defines diffeomorphisms.
When the driving vector fields of an SDE are smooth, solutions are unique. However, uniqueness and existence do not imply that solutions depend continuously on their initial data. In general, the random solution flow must be allowed to start from a random initial condition, for example by restarting from a random variable. The technical difficulty lies in the fact that lifetimes and exit times from relatively compact open sets may not be continuous with respect to initial values. For instance, the lifetime of the ODE
\[ \frac{d}{dt} z(t) = z(t)^2 \] on the complex plane is not continuous on any open set containing a segment of the real line. The solution to this equation is global if and only if the initial point is not on the real axis.
On compact smooth manifolds, strong completeness is automatic (this can be proved by lifting the SDE to the diffeomorphism group). For Lipschitz continuous vector fields in Euclidean space, strong completeness can be proved by the fixed point method. However, neither method extends easily to non-compact non-linear manifolds.
The question of under what conditions a complete SDE driven by smooth vector fields is strongly complete was solved by Xue-Mei Li. This problem had remained open since the 1970s after the first counterexample was discovered. Their results also yield non-trivial improvements to the known results in \( \mathbb{R}^n \).
The strong completeness problem is roughly equivalent to asking whether, for a set of full measure in \( \omega \), a connected relatively compact open set pushed forward by the solution remains connected for all time.
The existence of a derivative of \( F_x(\cdot, -) \) in probability is a simpler problem; this leads to solving an SDE known as the derivative flow. Li proved that the order of integration
\[ \int f(F_t(x, \omega)) \, dP(\omega), \] where \( f \) is a smooth compactly supported function, and differentiation (in probability with respect to the initial point) can be exchanged, provided the following holds: the solution starting from a smooth compact curve remains a connected smooth curve. This property is called strong 1-completeness.
Li also introduced the concept of strong p-completeness, and provided examples of SDEs which are strong p-complete but not strongly (p-1)-complete.
The main existence theorem, Theorem 4.1, requires control on the moments of the derivative flow. It was found that both the growth of the driving vector fields and the growth of their derivatives are required, and they compensate each other. Applications to SDEs in Euclidean space are given in Section 6, and applications to differentiation of the heat semigroup are in Section 9.
This results was extended for non-smooth coefficients, for SDEs in Euclidean space in
Strong completeness for a class of stochastic differential equations with irregular coefficients, X. Chen and Xue-Mei Li, Electronic Journal of Probability, 19 (2014), no. 91, 1–34. [article link], also on arXiv:1402.5079.
The merit of the article is to make clear that the difficulties by the lack of regularity in the driving force and those arising from unbounded derivatives of the vector fields play distinct roles, allowing also the ellipticity of the Markov generator to decay at infinity.
This is original research, not published elsewhere, focusing on diffusion processes and their associated Riemannian and sub-Riemannian geometry. The main theorem is presented at the beginning of the book (page 8), with the remainder devoted to its applications.
The main theorem can be summarised as follows. Consider a sub-elliptic operator of the form:
\[ \sum X_i^2 + X_0, \]
where \( X_1, \ldots, X_p \) are smooth vector fields. This defines a sub-bundle \( E \) of the tangent bundle. The theorem states that a semi-elliptic second-order differential operator without a zero-order term (i.e. a diffusion operator) in Hörmander form determines a metric linear connection on the sub-bundle with respect to its intrinsic metric. Moreover, every metric connection on such a sub-bundle arises in this way. This connection is characterised by the following property: if \( X \) denotes the bundle map from the trivial bundle over the manifold to the tangent bundle, then the covariant derivative of the vector field \( X(e) \) vanishes in directions where the vector fields are trivial.
In general, this connection has torsion. For example, the vector fields induced by an isometric embedding of a Riemannian manifold yield the Levi-Civita connection. The book provides a detailed computation and analysis of this connection: its torsion, curvature, Weitzenböck formulas, and the symmetry properties of its Ricci curvature in relation to torsion. It also studies the associated semigroups. A Hörmander form diffusion operator of constant rank leads to a semi-elliptic stochastic differential equation (SDE), providing probabilistic representations for various problems in analysis, particularly analysis on path spaces. In particular, the work derives Bismut-type formulas, logarithmic Sobolev inequalities, Clark-Ocone formulas, and decompositions of noise.
In The Geometry of Filtering (Birkhäuser, 2010), co-authored with K. D. Elworthy and Yves Le Jan, Xue-Mei Li presents original research, not published elsewhere, on intertwined diffusion and differential operators on principal bundles. In this book, she studies second-order differential operators of constant rank, their connections, and associated stochastic flows.
This book is also original research, studying intertwined diffusion and differential operators on principal bundles. In it, they study second-order differential operators of constant rank, denoted A and B, without zero-order terms, that are intertwined by a map from one manifold to another. The intertwining property determines a splitting of the operator B into the sum of two Hörmander-type operators, referred to as the horizontal and vertical operators. The vertical operator can be viewed as being independent of operator A, leading to applications in stochastic filtering and analysis on path spaces for probability measures determined by degenerate diffusion operators. This method is also useful even when the state spaces of the operators are Euclidean.
To provide context for the following topics, it is useful to note that one of the most common and topologically non-trivial infinite-dimensional spaces is the space of loops on a Riemannian manifold. The idea is to studying the topology by finite dimensional analogue, establishing De Rham cohomologies. Forcompact smooth Riemannian manifold, the de Rham cohomology coincides with the singular cohomology. While singular cohomology is a topological concept, de Rham cohomology belongs to differential geometry. For non-compact complete Riemannian manifolds, additional considerations are required.
The first step towards such a theorem involves an \( L^2 \) Hodge decomposition theorem for differential forms, where the \( L^2 \) space is defined using the volume measure. In the infinite-dimensional loop space, a natural replacement for the volume measure is the Brownian bridge measure. However, loop measures are nowadays replaced in favour of other desired properties in the current study of loop soups.
In another direction, Leonard Gross proved the logarithmic Sobolev inequality on the path space (Wiener space) over \( \mathbb{R}^n \), which has become a fundamental tool. There has been substantial effort to extend such inequalities to manifolds, with considerable success on path spaces using the Brownian motion measure. However, on loop spaces, the inequality was shown to hold only for asymptotically Euclidean manifolds. Otherwise, counterexamples arise; for example, the Poincaré inequality fails on the loop space of a compact Riemannian manifold when equipped with the Brownian bridge measure.
Developing a Sobolev calculus, including Malliavin derivatives, on path spaces has been necessary, with integration by parts formulas providing a crucial foundation.
Specifically, interest focuses on the differential operator \( d^*d \), where the adjoint is taken with respect to the probability distribution of continuous paths of a Brownian motion or Brownian bridge on a finite-dimensional manifold. These inequalities for measures generated by semi-elliptic (degenerate) diffusion operators are discussed in the above mentioned books.
Li et al. showed that the differentiation formula (BEL formula) is equivalent to the integration by parts formula; see A Class of Integration by Parts Formulae in Stochastic Analysis I. Therefore, it is desirable to obtain a range of such formulas and explore their applications. These results are presented in the following articles. See also “Generalised Brownian Bridges: Examples” and “On Hypoelliptic Bridges”.
They discuss semilinear parabolic equations (reaction-diffusion equations) on \\( \\mathbb{R}^n \\) with a small parameter. It is well known that, under suitable conditions, there exists a function \\( V \\) such that solutions converge to 0 or 1 depending on whether \\( V(t,x) \\) is negative or positive. They study the convergence of the first spatial derivatives of the solutions and prove that these derivatives also converge on the trough, where \\( V(t,x) < 0 \\).
The focus is on the Brownian bridge measure on the space of continuous loops over finite-dimensional manifolds.
The well-known de Rham theorem relates de Rham cohomology, a differential geometric concept, with singular cohomology, a topological concept, yielding far-reaching consequences. As a step towards such a theory on infinite-dimensional spaces, the existence of an L² Hodge decomposition theorem is studied on path spaces, which are manifolds modelled on infinite-dimensional Banach spaces. A key difficulty is the lack of natural L² tensor spaces for the tangent spaces.
The aim is to establish a Sobolev calculus in infinite-dimensional spaces, where derivatives are understood in the sense of Malliavin calculus. The idea is to construct charts that pull the corresponding calculus from Wiener space. One technique involves using stochastic flows from SDEs whose measures provide the basis for the analysis and whose vector fields induce a suitable covariant derivative on the finite-dimensional manifold, combined with stochastic filtering techniques to keep all concepts intrinsic.
Here, they attempt to prove Markov uniqueness for the operator d*d. They manage to show that the closure of BC² functions coincides with that of smooth compactly supported functions. However, the gap remains in proving the same for BC¹ functions.