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We develop the structure theory for transformations of weakly geometric rough paths of bounded $1 < p$-variation and their controlled paths. Our approach differs from existing approaches as it does not rely on smooth approximations. We derive an explicit combinatorial expression for the rough path lift of a controlled path, and use it to obtain fundamental identities such as the associativity of the rough integral, the adjunction between pushforwards and pullbacks, and a change of variables formula for rough differential equations (RDEs). As applications we define rough paths, rough integration and RDEs on manifolds, extending the results of [CDL15] to the case of arbitrary $p$.

We prove the Makeenko-Migdal equation for two-dimensional Euclidean Yang-Mills theory on an arbitrary compact surface, possibly with boundary. In particular, we show that two of the proofs given by the first, third, and fourth authors for the plane case extend essentially without change to compact surfaces., Comment: Final version, minor typographical corrections. To appear in Comm. Math. Phys

We show well-posedness for the parabolic Anderson model on $2$-dimensional closed Riemannian manifolds. To this end we extend the notion of regularity structures to curved space, and explicitly construct the minimal structure required for this equation. A central ingredient is the appropriate re-interpretation of the polynomial model, which we build up to any order.

The free multiplicative Brownian motion $b_{t}$ is the large-$N$ limit of the Brownian motion on $\mathsf{GL}(N;\mathbb{C}),$ in the sense of $\ast $-distributions. The natural candidate for the large-$N$ limit of the empirical distribution of eigenvalues is thus the Brown measure of $b_{t}$. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region $\Sigma_{t}$ that appeared work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density $W_{t}$ on $\bar{\Sigma}_{t},$ which is strictly positive and real analytic on $\Sigma_{t}$. This density has a simple form in polar coordinates: \[ W_{t}(r,\theta)=\frac{1}{r^{2}}w_{t}(\theta), \] where $w_{t}$ is an analytic function determined by the geometry of the region $\Sigma_{t}$. We show also that the spectral measure of free unitary Brownian motion $u_{t}$ is a "shadow" of the Brown measure of $b_{t}$, precisely mirroring the relationship between Wigner's semicircle law and Ginibre's circular law. We develop several new methods, based on stochastic differential equations and PDE, to prove these results., Comment: Added references to subsequent works building on these results. Made a notational change, replacing the regularization parameter x with epsilon

We consider two unitary representations of the infinite-dimensional groups of smooth paths with values in a compact Lie group. The first representation is induced by quasi-invariance of the Wiener measure, and the second representation is the energy representation. We define these representations and their basic properties, and then we prove that these representations are unitarily equivalent., This an article to be published in "Zapiski Seminarov POMI" dedicated to the memory of Mi.I.Gordin

We introduce a new form of the Segal–Bargmann transform for a connected Lie group K of compact type. We show that the heat kernel ( ρ t ( x ) ) t > 0 , x ∈ K has a space-time analytic continuation to a holomorphic function ( ρ C ( τ , z ) ) Re τ > 0 , z ∈ K C , where K C is the complexification of K. The new transform is defined by the integral ( B τ f ) ( z ) = ∫ K ρ C ( τ , z k − 1 ) f ( k ) d k , z ∈ K C . If s > 0 and τ ∈ D ( s , s ) (the disk of radius s centered at s), this integral defines a holomorphic function on K C for each f ∈ L 2 ( K , ρ s ) . We construct a heat kernel density μ s , τ on K C such that, for all s , τ as above, B s , τ : = B τ | L 2 ( K , ρ s ) is an isometric isomorphism from L 2 ( K , ρ s ) onto the space of holomorphic functions in L 2 ( K C , μ s , τ ) . When τ = t = s , the transform B t , t coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type. When τ = t ∈ ( 0 , 2 s ) , the transform B s , t coincides with the one introduced by the first two authors.

We study the (two-parameter) Segal‐Bargmann transform B N on the unitary group UN , for large N . Acting on matrix valued functions that are equivariant under the adjoint action of the group, the transform has a meaningful limit Gs,t as N → ∞, which can be identified as an operator on the space of complex Laurent polynomials. We introduce the space of trace polynomials, and use it to give effective computational methods to determine the action of the heat operator, and thus the Segal‐Bargmann transform. We prove several concentration of measure and limit theorems, giving a direct connection from the finite-dimensional transform B N to its limit Gs,t. We characterize the operator Gs,t through its inverse action on the standard polynomial basis. Finally, we show that, in the case s = t, the limit transform Gt,t is the “free Hall transform” G t introduced by Biane.

We give three short proofs of the Makeenko-Migdal equation for the Yang-Mills measure on the plane, two using the edge variables and one using the loop or lasso variables. Our proofs are significantly simpler than the earlier pioneering rigorous proofs given by T. L\'evy and by A. Dahlqvist. In particular, our proofs are "local" in nature, in that they involve only derivatives with respect to variables adjacent to the crossing in question. In an accompanying paper with F. Gabriel, we will show that two of our proofs can be adapted to the case of Yang-Mills theory on a compact surface., Comment: Final version. One of the three proofs was rewritten for greater clarity. To appear in Communications in Mathematical Physics

Given a non-negative Hermitian form on the dual of the Lie algebra of a complex Lie group, one can associate to it a (possibly degenerate) Laplacian on the Lie group. Under Hormander's condition on the Laplacian there exists a smooth time-dependent measure, convolution by which gives the semigroup generated by the Laplacian. Fixing a positive time, we may form the Hilbert space of holomorphic functions on the group which are square integrable with respect to this heat kernel measure. At the same time, under Homander's condition, the given Hermitian form extends to a time dependent norm on the dual of the universal enveloping algebra. In previous work we have shown that, for each positive time, the Taylor map, which sends a holomorphic function to its set of Taylor coefficients at the identity element, is a unitary map from the previous Hilbert space of square integrable holomorphic functions onto a Hilbert space contained in the dual of the universal enveloping algebra. The present paper is concerned with the behavior of these two families of Hilbert spaces when the Lie group is replaced by a product of complex Lie groups or by a quotient by a not necessarily normal subgroup. We obtain thereby the first example of unitarity of the Taylor map for a complex manifold which is not a Lie group. In addition, we determine the behavior of these spaces as the given Hermitian form varies.

We introduce a class of non-commutative, complex, infinite-dimensional Heisenberg like Lie groups based on an abstract Wiener space. The holomorphic functions which are also square integrable with respect to a heat kernel measure μ on these groups are studied. In particular, we establish a unitary equivalence between the square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the “Lie algebra” of this class of groups. Using quasi-invariance of the heat kernel measure, we also construct a skeleton map which characterizes globally defined functions from the L 2(ν)-closure of holomorphic polynomials by their values on the Cameron–Martin subgroup.

A Hermitian formq on the dual space, g , of the Lie algebra, g; of a Lie group,G; de- termines a sub-Laplacian,1; onG: It will be shown that Hcondition for hypoellipticity of the sub-Laplacian holds if and only if the associated Hermitian form, induced by q on the dual of the universal enveloping algebra, U 0 , is non-degenerate. The subelliptic heat semigroup,e t1=4 ; is given by convolution by aC 1 probability density t: WhenG is complex andu : G! C is a holomorphic function, the collection of derivatives ofu at the identity inG gives rise to an element, O u.e/2 U 0 . We will show that ifG is complex, connected, and simply connected then the "Taylor" map,u7! O u.e/, defines a unitary map from the space of holomorphic functions inL 2 .G; t/ onto a natural Hilbert space lying in U 0 .

A Hermitian formqon the dual space,*, of the Lie algebra,, of a simply connected complex Lie group,G, determines a sub-Laplacian, Δ, onG. Assuming Hörmander's condition for hypoellipticity, there is a smooth heat kernel measure, ρt, onGassociated toetΔ/4. In a companion paper [6], we proved the existence of a unitary “Taylor” map from the space of holomorphic functions inL2(G, ρt) ontoJt0(a subspace of) the dual of the universal enveloping algebra of. Here we give a very different proof of the surjectivity of the Taylor map under the assumption thatGis nilpotent. This proof provides further insight into the structure of the Taylor map. In particular we show that the finite rank tensors are dense inJt0when the Lie algebra is graded and the Laplacian is adapted to the gradation. We also show how the Fourier–Wigner transform produces a natural family of holomorphic functions inL2(G, ρt), for appropriatet, whenGis the complex Heisenberg group.

Diffusions on the pinned loop and pathgroup of a general (non-compact) Lie group are constructed. The laws of the endpoint processes satisfy heat equations related to second order differential operators associated to subspaces of the Cameron–Martin space.

We prove Cheng–Yau type inequalities for positive harmonic functions on Riemannian manifolds by using methods of Stochastic Analysis. Rather than evaluating an exact Bismut formula for the differential of a harmonic function, our method relies on a Bismut type inequality which is derived by an elementary integration by parts argument from an underlying submartingale. It is the monotonicity inherited in this submartingale which allows us to establish the pointwise estimates.

This paper is concerned with the characterization of spaces of square integrable holomorphic functions on a complex manifold, $G$, in terms of the derivatives of the function at a fixed point $o\in G$. The reproducing kernel properties of square integrable holomorphic functions are reviewed and a number of examples are given. These examples include square integrable holomorphic functions relative to Gaussian measures on complex Euclidean spaces along with their generalizations to heat kernel measures on complex Lie groups. These results are intimately related to the Itô's chaos expansion in stochastic analysis and to the Fock space description of free quantum fields in physics.

We introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural. On the way we develop a notion of rough integration and an efficient and intrinsic theory of rough differential equations (RDEs) on manifolds. The theory of RDEs is then used to construct parallel translation along manifold valued rough paths. Finally, this framework is used to show there is a one to one correspondence between rough paths on a d-dimensional manifold and rough paths on d-dimensional Euclidean space. This last result is a rough path analogue of Cartan's development map and its stochastic version which was developed by Eeels and Elworthy and Malliavin.

We study the existence of “Lp-type” gradient estimates for the heat kernel of the natural hypoelliptic “Laplacian” on the real three-dimensional Heisenberg Lie group. Using Malliavin calculus methods, we verify that these estimates hold in the case p>1. The gradient estimate for p=2 implies a corresponding Poincaré inequality for the heat kernel. The gradient estimate for p=1 is still open; if proved, this estimate would imply a logarithmic Sobolev inequality for the heat kernel.

We show that heat kernel measure and pinned Wiener measure on loop groups over simply connected compact Lie groups are equivalent.

Let W.M/ be the based (at o2 M/ path space of a compact Riemannian manifold M equipped with Wiener measure": This paper is devoted to considering vector fields on W.M/ of the form X h.ae/ D Ps.ae/hs.ae/ where Ps.ae/ denotes stochastic parallel translation up to time s along a Wiener path ae 2 W.M/ and fhsgs2(0;1) is an adapted To M-valued process on W.M/: It is shown that there is a large class of processes h (called adapted vector fields) for which we may view X h as first-order differential operators acting on functions on W.M/. Moreover, if h and k are two such processes, then the commutator of X h with X k is again a vector

It is asserted in Definition 4.2 in [1] that the random operators U ( t ) defined there are unitary. As was pointed out to the author by Shizan Fang, it is clear that U ( t ) is an isometry but it is not obvious that U ( t ) is surjective. The purpose of this note is to fill this gap.

We study the law of a hypoelliptic Brownian motion on an infinite-dimensional Heisenberg group based on an abstract Wiener space. We show that the endpoint distribution, which can be seen as a heat kernel measure, is absolutely continuous with respect to a certain product of Gaussian and Lebesgue measures, that the heat kernel is quasi-invariant under translation by the Cameron-Martin subgroup, and that the Radon-Nikodym derivative is Malliavin smooth., 34 pages

LetGbe a connected compact type Lie group equipped with anAdG-invariant inner product on the Lie algebra g ofG. Given this data there is a well known left invariant “H1-Riemannian structure” on L = L (G)—the infinite dimensional group of continuous based loops inG. Using this Riemannian structure, we define and construct a “heat kernel”νT(g0, ·) associated to the Laplace–Beltrami operator on L (G). HereT>0,g0∈ L (G), andνT(g0,·) is a certain probability measure on L (G). For fixedg0∈ L (G) andT>0, we use the measureνT(g0,·) and the Riemannian structure on L (G) to construct a “classical” pre-Dirichlet form. The main theorem of this paper asserts that this pre-Dirichlet form admits a logarithmic Sobolev inequality.

The results in Driver [13] for quasi-invariance of Wiener measure on the path space of a compact Riemannian manifold (M) are extended to the case of pinned Wiener measure. To be more explicit, let h : [ 0 , 1 ] → T 0 M h:[0,1] \to {T_0}M be a C 1 {C^1} function where M is a compact Riemannian manifold, o ∈ M o \in M is a base point, and T o M {T_o}M is the tangent space to M at o ∈ M o \in M . Let W ( M ) W(M) be the space of continuous paths from [0,1] into M, ν \nu be Wiener measure on W ( M ) W(M) concentrated on paths starting at o ∈ M o \in M , and H s ( ω ) {H_s}(\omega ) denote the stochastic-parallel translation operator along a path ω ∈ W ( M ) \omega \in W(M) up to "time" s. (Note: H s ( ω ) {H_s}(\omega ) is only well defined up to ν \nu -equivalence.) For ω ∈ W ( M ) \omega \in W(M) let X h ( ω ) {X^h}(\omega ) denote the vector field along ω \omega given by X s h ( ω ) ≡ H s ( ω ) h ( s ) X_s^h(\omega ) \equiv {H_s}(\omega )h(s) for each s ∈ [ 0 , 1 ] s \in [0,1] . One should interpret X h {X^h} as a vector field on W ( M ) W(M) . The vector field X h {X^h} induces a flow S h ( t , ∙ ) : W ( M ) → W ( M ) {S^h}(t, \bullet ):W(M) \to W(M) which leaves Wiener measure ( ν ) (\nu ) quasi-invariant, see Driver [13]. It is shown in this paper that the same result is valid if h ( 1 ) = 0 h(1) = 0 and the Wiener measure ( ν ) (\nu ) is replaced by a pinned Wiener measure ( ν e ) ({\nu _e}) . (The measure ν e {\nu _e} is proportional to the measure ν \nu conditioned on the set of paths which start at o ∈ M o \in M and end at a fixed end point e ∈ M e \in M .) Also as in [13], one gets an integration by parts formula for the vector-fields X h {X^h} defined above.

Given a non-negative Hermitian form on the dual of the Lie algebra of a complex Lie group, one can associate to it a (possibly degenerate) Laplacian on the Lie group. Under Hörmander's condition on the Laplacian there exists a smooth time-dependent measure, convolution by which gives the semigroup generated by the Laplacian. Fixing a positive time, we may form the Hilbert space of holomorphic functions on the group which are square integrable with respect to this “heat kernel” measure. At the same time, under Hörmander's condition, the given Hermitian form extends to a time dependent norm on the dual of the universal enveloping algebra. ¶ In previous work we have shown that, for each positive time, the Taylor map, which sends a holomorphic function to its set of Taylor coefficients at the identity element, is a unitary map from the previous Hilbert space of square integrable holomorphic functions onto a Hilbert space contained in the dual of the universal enveloping algebra. ¶ The present paper is concerned with the behavior of these two families of Hilbert spaces when the Lie group is replaced by a product of complex Lie groups or by a quotient by a not necessarily normal subgroup. We obtain thereby the first example of unitarity of the Taylor map for a complex manifold which is not a Lie group. In addition, we determine the behavior of these spaces as the given Hermitian form varies.

We show that the logarithmic derivatives of the convolution heat kernels on a uni-modular Lie group are exponentially integrable. This result is then used to prove an "integrated" Harnack inequality for these heat kernels. It is shown that this integrated Harnack inequality is equivalent to a version of Wang's Harnack inequality. (A key feature of all of these inequalities is that they are dimension independent.) Finally, we show these inequalities imply quasi-invariance properties of heat kernel measures for two classes of infinite dimensional "Lie" groups., Comment: 41 pages A section added where we show that this integrated Harnack inequality is equivalent to a version of Wang's Harnack inequality. New abstract

It is well known that the solution of x′(t) = bx(t − 1) for t ⩾ 0 with x(t) = 1 on [−1, 0] (or other initial function) can be approximated for “large t” by a linear combination of one or two exponentials eλt, where λ = be−λ. (In case b = −1 e the solution is approximated by cte−t.) This paper gives estimates for the error in that approximation when −5π 2 ⩽ b ⩽ 3π 2 . The smallness of the error even for modest values of t > 0 may be surprising.

The purpose of these notes is to first provide some basic background to Riemannian geometry and stochastic calculus on manifolds and then to cover some of the more recent developments pertaining to analysis on “curved Wiener spaces.” Essentially no differential geometry is assumed, however, it is assumed that the reader is comfortable with stochastic calculus and differential equations on Euclidean spaces. Here is a brief description of what will be covered in the text below.

Let $t > 0, K$ be a connected compact Lie group equipped with an $Ad_K$- invariant inner product on the Lie Algebra of $K$. Associated to this data are two measures $\mu^0_t$ and $\nu^0_t$ on $\mathscr{L}(K)$ – the space of continuous loops based at $e \in K$. The measure $\mu^0_t$ is pinned Wiener measure with “variance $t$” while the measure $\nu^0_t$ is a “heat kernel measure” on $\mathscr{L}(K)$. The measure $\mu^0_t$ is constructed using a $K$-valued Brownian motion while the measure $\nu^0_t$ is constructed using a $\mathscr{L}(K)$-valued Brownian motion. In this paper we show that $\nu^0_t$ is absolutely continuous with respect to $\mu^0_t$ and the Radon-Nikodym derivative $d\nu^0_t /d\mu^0_t$ is bounded.

We use a variant of the classical Segal-Bargmann transform to understand the canonical quantization of Yang-Mills theory on a space-time cylinder. This transform gives a rigorous way to make sense of the Hamiltonian on the gauge-invariant subspace. Our results are a rigorous version of the widely accepted notion that on the gauge-invariant subspace the Hamiltonian should reduce to the Laplacian on the compact structure group. We show that the infinite-dimensional classical Segal-Bargmann transform for the space of connections, when restricted to the gauge-invariant subspace, becomes the generalized Segal-Bargmann transform for the the structure group.

It is shown that in four space-time dimensions the compactU(1) lattice gauge theory with general energy function converges to a renormalized free electromagnetic field on the current sector as the lattice spacing approaches zero, provided the coupling constant is sufficiently large. For the Wilson energy function, it is possible, by judicious choice of the Gibbs state, to get convergence for arbitrary coupling strengths. Furthermore, for all but a countable number of values of the coupling constant, the limit exists and is independent of the particular state chosen to define the lattice model.

Let M be a connected manifold and G be a closed Lie subgroup of GL(V)—the general linear group on a finite dimensional vector space V. Denote by ( Ω m ) the space of H1-loops in M starting at a fixed point (m). Let M be the set of P ϵ C ∞ (Ω m , G) , modulo conjugation by an element of G, such that P(στ) = P(σ) P(τ) for σ, τ ϵ Ω m (στ is the concatenation of the loops σ and τ), P(σ′) = P(σ) if σ′ is reparameterization of σ, and the differential of P satisfies a “locality” condition. It is shown that the bundle connection pairs (E, ▽) (up to equivalence), with structure group G and fiber V, are in one to one correspondence with M —a similar result has been announced by Kobayashi. The correspondence is induced by the parallel translation operators of connections. Furthermore, if the manifold (M) is simply connected, then the space of bundle connection pairs can be classified by a collection of Lie algebra valued 1-forms on the manifold Ω m (called integrated lassos). These 1-forms are related to the differentials of elements of M . This last result generalizes Weil's characterization of U(1)—line bundle connection pairs by the curvature 2-form. It is also a generalization of Gross' results to base manifolds (M) other than R n.

The two dimensional Yang-Mills theory (YM2) is analyzed in both the continuum and the lattice. In the complete axial gauge the continuum theory may be defined in terms of a Lie algebra valued white noise, and parallel translation may be defined by stochastic differential equations. This machinery is used to compute the expectations of gauge invariant functions of the parallel translation operators along a collection of curvesC. The expectation values are expressed as finite dimensional integrals with densities that are products of the heat kernel on the structure group. The time parameters of the heat kernels are determined by the areas enclosed by the collectionC, and the arguments are determined by the crossing topologies of the curves inC. The expectations for the Wilson lattice models have a similar structure, and from this it follows that in the limit of small lattice spacing the lattice expectations converge to the continuum expectations. It is also shown that the lasso variables advocated by L. Gross [36] exist and are sufficient to generate all the measurable functions on the YM2-measure space.

Integration by parts formulas are established both for Wiener measure on the path space of a loop group and for the heat kernel measures on the loop group. The Wiener measure is defined to be the law of a certain loop group valued “Brownian motion” and the heat kernel measures are time t , t >0, distributions of this Brownian motion. A corollary of either of these integrations by parts formulas is the closability of the pre-Dirichlet form considered by B. K. Driver and T. Lohrenz [1996, J. Functional Anal. 140 , 381–448]. We also show that the heat kernel measures are quasi-invariant under right under right and left translations by finite energy loops.

Certain natural geometric approximation schemes are developed for Wiener measure on a compact Riemannian manifold. These approximations closely mimic the informal path integral formulas used in the physics literature for representing the heat semi-group on Riemannian manifolds. The path space is approximated by finite dimensional manifolds consisting of piecewise geodesic paths adapted to partitions $P$ of $[0,1]$. The finite dimensional manifolds of piecewise geodesics carry both an $H^{1}$ and a $L^{2}$ type Riemannian structures $G^i_P$. It is proved that as the mesh of the partition tends to $0$, $$ 1/Z_P^i e^{- 1/2 E(\sigma)} Vol_{G^i_P}(\sigma) \to \rho_i(\sigma)\nu(\sigma) $$ where $E(\sigma )$ is the energy of the piecewise geodesic path $\sigma$, and for $i=0$ and $1$, $Z_P^i$ is a ``normalization'' constant, $Vol_{G^i_P}$ is the Riemannian volume form relative $G^i_P$, and $\nu$ is Wiener measure on paths on $M$. Here $\rho_1 = 1$ and $$ \rho_0 (\sigma) = \exp( -1/6 \int_0^1 Scal(\sigma(s))ds ) $$ where $Scal$ is the scalar curvature of $M$. These results are also shown to imply the well know integration by parts formula for the Wiener measure., Comment: 48 pages, latex2e using amsart and amssymb

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, { ν 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–Nikodym derivatives, (4) integration by parts formulas, and (5) logarithmic Sobolev inequalities. The last three results heavily rely on the boundedness of the Ricci tensor.

Stochastic calculus proofs of the integration by parts formula for cylinder functions of parallel translation on the Wiener space of a compact Riemannian manifold (M) are given. These formulas are used to prove a new probabilistic formula for the logarithmic derivative of the heart kernel on M. This new formula is well suited for generalizations to infinite dimensional manifolds.