# Research interests

Optimal transport, stochastic partial differential equations, stochastic analysis.

# Research papers

hide all abstracts.

or**On the geometry of geodesics in discrete optimal transport**

with M. Erbar and M. Wirth*Abstract*: We consider the space of probability measures on a discrete set $\mathcal{X}$, endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset $\mathcal{Y} \subseteq \mathcal{X}$, it is natural to ask whether they can be connected by a constant speed geodesic with support in $\mathcal{Y}$ at all times. Our main result answers this question affirmatively, under a suitable geometric condition on $\mathcal{Y}$ introduced in this paper. The proof relies on an extension result for subsolutions to discrete Hamilton-Jacobi equations, which is of independent interest.

18 pages, submitted for publication.**Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance**

with E. Carlen*Abstract*: We study a class of ergodic quantum Markov semigroups on finite-dimensional unital $C^*$-algebras. These semigroups have a unique stationary state $\sigma$, and we are concerned with those that satisfy a quantum detailed balance condition with respect to $\sigma$. We show that the evolution on the set of states that is given by such a quantum Markov semigroup is gradient flow for the relative entropy with respect to $\sigma$ in a particular Riemannian metric on the set of states. This metric is a non-commutative analog of the $2$-Wasserstein metric, and in several interesting cases we are able to show, in analogy with work of Otto on gradient flows with respect to the classical $2$-Wasserstein metric, that the relative entropy is strictly and uniformly convex with respect to the Riemannian metric introduced here. As a consequence, we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum Markov semigroups with detailed balance.

J. Funct. Anal.**273**(5) (2017), 1810-1869.**Transport based image morphing with intensity modulation**

with M. Rumpf and S. Simon*Abstract*: We present a generalized optimal transport model in which the mass-preserving constraint for the $L^2$-Wasserstein distance is relaxed by introducing a source term in the continuity equation. The source term is also incorporated in the path energy by means of its squared $L^2$-norm in time of a functional with linear growth in space. This extension of the original transport model enables local density modulation, which is a desirable feature in applications such as image warping and blending. A key advantage of the use of a functional with linear growth in space is that it allows for singular sources and sinks, which can be supported on points or lines. On a technical level, the $L^2$-norm in time ensures a disintegration of the source in time, which we use to obtain the well-posedness of the model and the existence of geodesic paths. The numerical discretization is based on the proximal splitting approach and selected numerical test cases show the potential of the proposed approach. Furthermore, the approach is applied to the warping and blending of textures.

Scale Space and Variational Methods in Computer Vision. SSVM 2017. Lecture Notes in Computer Science, vol 10302. Springer, Cham**From large deviations to Wasserstein gradient flows in multiple dimensions**

with M. Erbar and M. Renger*Abstract*: We study the large deviation rate functional for the empirical distribution of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peletier and Zimmer that this functional is asymptotically equivalent (in the sense of $\Gamma$-convergence) to the Jordan–Kinderlehrer–Otto functional arising in the Wasserstein gradient flow structure of the Fokker–Planck equation. In higher dimensions, part of this statement (the lower bound) has been recently proved by Duong, Laschos and Renger, but the upper bound remained open, since the proof in \cite{DLR2013} relies on regularity properties of optimal transport maps that are restricted to one dimension. In this note we present a new proof of the upper bound, thereby generalising the result of \cite{ADPZ2011} to arbitrary dimensions.

Electron. Comm. Probab.**20**(2015), no 89, 1-12.**Long-time behavior of a finite volume discretization for a fourth order diffusion equation**

with D. Matthes*Abstract*: We consider a non-standard finite-volume discretization of a strongly non-linear fourth order diffusion equation on the $d$-dimensional cube, for arbitrary $d \geq 1$. The scheme preserves two important structural properties of the equation: the first is the interpretation as a gradient flow in a mass transportation metric, and the second is an intimate relation to a linear Fokker-Planck equation. Thanks to these structural properties, the scheme possesses two discrete Lyapunov functionals. These functionals approximate the entropy and the Fisher information, respectively, and their dissipation rates converge to the optimal ones in the discrete-to-continuous limit. Using the dissipation, we derive estimates on the long-time asymptotics of the discrete solutions. Finally, we present results from numerical experiments which indicate that our discretization is able to capture significant features of the complex original dynamics, even with a rather coarse spatial resolution.

Nonlinearity**29**(7) (2016), 1992-2023.**Entropic Ricci curvature bounds for discrete interacting systems**

with M. Fathi*Abstract*: We develop a new and systematic method for proving entropic Ricci curvature lower bounds for Markov chains on discrete sets. Using different methods, such bounds have recently been obtained in several examples (e.g., 1-dimensional birth and death chains, product chains, Bernoulli–Laplace models, and random transposition models). However, a general method to obtain discrete Ricci bounds had been lacking. Our method covers all of the examples above. In addition we obtain new Ricci curvature bounds for zero-range processes on the complete graph. The method is inspired by recent work of Caputo, Dai Pra and Posta on discrete functional inequalities.

Ann. Appl. Probab.**26**(3) (2016), 1774-1806.**Discrete Ricci curvature bounds for Bernoulli-Laplace and random transposition models**

with M. Erbar and P. Tetali*Abstract*: We calculate a Ricci curvature lower bound for some classical examples of random walks, namely, a chain on a slice of the $n$-dimensional discrete cube (the so-called Bernoulli–Laplace model) and the random transposition shuffle of the symmetric group of permutations on $n$ letters.

Ann. Fac. Sci. Toulouse Math**24**(4) (2015), 781-800.**A generalized model for optimal transport of images including dissipation and density modulation**

with M. Rumpf, C. Schönlieb and S. Simon*Abstract*: In this paper the optimal transport and the metamorphosis perspectives are combined. For a pair of given input images geodesic paths in the space of images are defined as minimizers of a resulting path energy. To this end, the underlying Riemannian metric measures the rate of transport cost and the rate of viscous dissipation. Furthermore, the model is capable to deal with strongly varying image contrast and explicitly allows for sources and sinks in the transport equations which are incorporated in the metric related to the metamorphosis approach by Trouvé and Younes. In the non-viscous case with source term existence of geodesic paths is proven in the space of measures. The proposed model is explored on the range from merely optimal transport to strongly dissipative dynamics. For this model a robust and effective variational time discretization of geodesic paths is proposed. This requires to minimize a discrete path energy consisting of a sum of consecutive image matching functionals. These functionals are defined on corresponding pairs of intensity functions and on associated pairwise matching deformations. Existence of time discrete geodesics is demonstrated. Furthermore, a finite element implementation is proposed and applied to instructive test cases and to real images. In the non-viscous case this is compared to the algorithm proposed by Benamou and Brenier including a discretization of the source term. Finally, the model is generalized to define discrete weighted barycentres with applications to textures and objects.

ESAIM Math. Model. Numer. Anal.**49**(6) (2015), 1745-1769.**Poisson stochastic integration in Banach spaces**

with S. Dirksen and J. van Neerven*Abstract*: We prove new upper and lower bounds for Banach space-valued stochastic integrals with respect to a compensated Poisson random measure. Our estimates apply to Banach spaces with non-trivial martingale (co)type and extend various results in the literature. We also develop a Malliavin framework to interpret Poisson stochastic integrals as vector-valued Skorohod integrals, and prove a Clark-Ocone representation formula.

Electron. J. Probab.**18**(2013), no. 100, 1-28.**Gradient flow structures for discrete porous medium equations**

with M. Erbar*Abstract*: We consider discrete porous medium equations of the form $\partial_t\rho_t = \Delta \varphi(\rho_t)$, where $\Delta$ is the generator of a reversible continuous time Markov chain on a finite set $\mathcal{X}$, and $\varphi$ is an increasing function. We show that these equations arise as gradient flows of certain entropy functionals with respect to suitable non-local transportation metrics. This may be seen as a discrete analogue of the Wasserstein gradient flow structure for porous medium equations in $\mathbf{R}^n$ discovered by Otto. We present a one-dimensional counterexample to geodesic convexity and discuss Gromov-Hausdorff convergence to the Wasserstein metric.

Discrete Contin. Dyn. Syst.**34**(4) (2014), 1355-1374.**Gromov-Hausdorff convergence of discrete transportation metrics**

with N. Gigli*Abstract*: This paper continues the investigation of `Wasserstein-like’ transportation distances for probability measures on discrete sets. We prove that the discrete transportation metrics $\mathcal{W}_N$ on the $d$-dimensional discrete torus ${\mathbf{T}_N^d}$ with mesh size $\frac1N$ converge, when $N\to\infty$, to the standard 2-Wasserstein distance $W_2$ on the continuous torus in the sense of Gromov-Hausdorff. This is the first convergence result for the recently developed discrete transportation metrics $\mathcal{W}$. The result shows the compatibility between these metrics and the well-established 2-Wasserstein metric.

SIAM J. Math. Anal.**45**(2) (2013), 879-899.**An analog of the 2-Wasserstein metric in non-commutative probability under which the fermionic Fokker-Planck equation is gradient flow for the entropy**

with E. Carlen*Abstract*: Let $\mathfrak{C}$ denote the Clifford algebra over $\mathbb{R}^n$, which is the von Neumann algebra generated by $n$ self-adjoint operators $Q_j$, $j=1,\dots,n$ satisfying the canonical anticommutation relations, $Q_iQ_j+Q_jQ_i = 2\delta_{ij}I$, and let $\tau$ denote the normalized trace on $\mathfrak{C}$. This algebra arises in quantum mechanics as the algebra of observables generated by $n$ Fermionic degrees of freedom. Let ${\mathfrak P}$ denote the set of all positive operators $\rho\in\mathfrak{C}$ such that $\tau(\rho) =1$; these are the non-commutative analogs of probability densities in the non-commutative probability space $(\mathfrak{C},\tau)$. The Fermionic Fokker-Planck equation is a quantum-mechanical analog of the classical Fokker-Planck equation with which it has much in common, such as the same optimal hypercontractivity properties. In this paper we construct a Riemannian metric on ${\mathfrak P}$ that we show to be a natural analog of the classical $2$-Wasserstein metric, and we show that, in analogy with the classical case, the Fermionic Fokker-Planck equation is gradient flow in this metric for the relative entropy with respect to the ground state. We derive a number of consequences of this, such as a sharp Talagrand inequality for this metric, and we prove a number of results pertaining to this metric. Several open problems are raised.

Comm. Math. Phys.**331**(3) (2014), 887-926.**Approximating rough stochastic PDEs**

with M. Hairer and H. Weber*Abstract*: We study approximations to a class of vector-valued equations of Burgers type driven by a multiplicative space-time white noise. A solution theory for this class of equations has been developed recently in [Hairer, Weber,*Probab. Theory Related Fields*, 2013]. The key idea was to use the theory of*controlled rough paths*to give definitions of weak/mild solutions and to set up a Picard iteration argument. In this article the limiting behaviour of a rather large class of (spatial) approximations to these equations is studied. These approximations are shown to converge and convergence rates are given, but the limit may depend on the particular choice of approximation. This effect is a spatial analogue to the Itô-Stratonovich correction in the theory of stochastic ordinary differential equations, where it is well known that different approximation schemes may converge to different solutions.

Comm. Pure Appl. Math.**67**(5) (2014), 776-870.**Ricci curvature of finite Markov chains via convexity of the entropy**

with M. Erbar*Abstract*: We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy.

Using this notion of Ricci curvature we prove discrete analogues of fundamental results by Bakry-Émery and Otto-Villani. Furthermore we show that Ricci curvature bounds are preserved under tensorisation. As a special case we obtain the sharp Ricci curvature lower bound for the discrete hypercube.

Arch. Ration. Mech. Anal.**206**(3) (2012), 997-1038.**Gradient flows of the entropy for finite Markov chains**

J. Funct. Anal.*Abstract*: Let $K$ be an irreducible and reversible Markov kernel on a finite set $\mathcal{X}$. We construct a metric $\mathcal{W}$ on the set of probability measures on $\mathcal{X}$ and show that with respect to this metric, the law of the continuous time Markov chain evolves as the gradient flow of the entropy. This result is a discrete counterpart of the Wasserstein gradient flow interpretation of the heat flow in $\mathbb{R}^n$ by Jordan, Kinderlehrer, and Otto (1998).

The metric $\mathcal{W}$ is similar to, but different from, the $L^2$-Wasserstein metric, and is defined via a discrete variant of the Benamou-Brenier formula.**261**(8) (2011), 2250-2292.**A spatial version of the Itô-Stratonovich correction**

with M. Hairer*Abstract*: We consider a class of stochastic PDEs of Burgers type in spatial dimension 1,

driven by space-time white noise. Even though it is well-known that these equations are well-posed, it turns out that if one performs a spatial discretisation of the nonlinearity in the “wrong” way, then the sequence approximate equations does converge to a limit, but this limit exhibits an additional correction term.

This correction term is proportional to the local quadratic cross-variation (in space!) of the gradient of the conserved quantity with the solution itself. This can be understood as a consequence of the fact that for any fixed time, the law of the solution is locally equivalent to Wiener measure, where space plays the role of time. In this sense, the correction term is similar to the usual Itô-Stratonovich correction term that arises when one considers different temporal discretisations of stochastic ODEs.

Ann. Probab.**40**(4) (2012), 1675–1714.**A Trotter product formula for gradient flows in metric spaces**

with Ph. Clément*Abstract*: We prove a Trotter product formula for gradient flows in metric spaces. This result is applied to establish convergence in the $L^2$-Wasserstein metric of the splitting method for some Fokker-Planck equations and porous medium type equations perturbed by a potential.

*The published version of the article contains a typo in Theorem 1.1. The factor $\frac12$ on the left hand side of (1.4) should be removed. This has been corrected in an erratum.*

J. Evol. Equ.**11**(2) (2011), 405-427.**Conical square functions and non-tangential maximal functions with respect to the Gaussian measure**

with J. van Neerven and P. Portal*Abstract*: We study, in $L^{1}(\mathbb{R}^n;\gamma)$ with respect to the Gaussian measure, non-tangential maximal functions and conical square functions associated with the Ornstein-Uhlenbeck operator by developing a set of techniques which allow us, to some extent, to compensate for the non-doubling character of the Gaussian measure. The main result asserts that conical square functions can be controlled in $L^1$-norm by non-tangential maximal functions. Along the way we prove a change of aperture result for the latter. This complements recent results on Gaussian Hardy spaces due to Mauceri and Meda.

Publ. Mat.**55**(2) (2011), 313-341.**Whitney coverings and the tent spaces T**^{1,q}(γ) for the Gaussian measure

with J. van Neerven and P. Portal*Abstract*: We introduce a technique for handling Whitney decompositions in Gaussian harmonic analysis and apply it to the study of Gaussian analogues of the classical tent spaces $T^{1,q}(\gamma)$ of Coifman-Meyer-Stein.

Ark. Mat.**50**(2) (2012), 379-395.**Gradient estimates and domain identification for analytic Ornstein-Uhlenbeck operators**

with J. van Neerven*Abstract*: Let $P$ be the Ornstein-Uhlenbeck semigroup associated with the stochastic Cauchy problem \[ dU(t) = AU(t) dt + dWH (t), \] where $A$ is the generator of a $C_0$-semigroup $S$ on a Banach space $E$, $H$ is a Hilbert subspace of $E$, and $W_H$ is an $H$-cylindrical Brownian motion. Assuming that $S$ restricts to a $C_0$-semigroup on $H$, we obtain $L^p$-bounds for $D_H P(t)$. We show that if $P$ is analytic, then the invariance assumption is fulfilled. As an application we determine the $L^p$-domain of the generator of $P$ explicitly in the case where $S$ restricts to a $C_0$-semigroup on $H$ which is similar to an analytic contraction semigroup.

Parabolic Problems: The Herbert Amann Festschrift, Birkhäuser (2011), 463-477.**Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces**

with J. van Neerven*Abstract*: Let $(E,H,\mu)$ be an abstract Wiener space and let $D_V := VD$, where $D$ denotes the Malliavin derivative and $V$ is a closed and densely defined operator from $H$ into another Hilbert space ${\underline{H}}$. Given a bounded operator $B$ on ${\underline{H}},$ coercive on the range $\overline{\mathsf{R}(V)}$, we consider the operators $A:= V^* BV$ in $H$ and $\underline{A}:= VV^* B$ in ${\underline{H}}$, as well as the realisations of the operators $L: = D_V^* BD_V$ and $\underline{L} := D_VD_V^* B$ in $L^p(E,\mu)$ and $L^p(E,\mu;{\underline{H}})$ respectively, where $1 < p < \infty$.

Our main result asserts that the following four assertions are equivalent:- ${\mathsf D}(\sqrt{L}) = {\mathsf D}(D_V)$ with $\| \sqrt{L}f\|_{p} \eqsim \| D_V f\|_{p}$ for $f\in {\mathsf D}(\sqrt{L})$;
- $\underline{L}$ admits a bounded $H^\infty$-functional calculus on $\overline{\mathsf{R}(D_V)}$;
- ${\mathsf D}(\sqrt{A}) = {\mathsf D}(V)$ with $\| \sqrt{A}h\| \eqsim \| Vh \|$ for $h\in {\mathsf D}(\sqrt{A})$;
- $\underline{A}$ admits a bounded $H^\infty$-functional calculus on $\overline{\mathsf{R}(V)}$.

This is a nonsymmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where ${\underline{H}}=H$, $V = I$, $B = \frac12 I$). A one-sided version of the main result, giving $L^p$-boundedness of the Riesz transform $D_V/\sqrt{L}$ in terms of a square function estimate, is also obtained.

As an application let $-A$ generate an analytic $C_0$-contraction semigroup on a Hilbert space $H$ and let $-L$ be the $L^p$-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of $L$ are equivalent with the Kato square root property for $A$.

J. Funct. Anal.**257**(8) (2009), 2410-2475.**Malliavin calculus and decoupling inequalities in Banach spaces**

J. Math. Anal. Appl.*Abstract*: We develop a theory of Malliavin calculus for Banach space-valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Itô isometry to Banach spaces. In the white noise case we obtain two sided $L^p$-estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-Itô chaoses. Our main tools are decoupling inequalities for vector-valued random variables. In the opposite direction we use Meyer’s inequalities to give a new proof of a decoupling result for Gaussian chaoses in UMD Banach spaces.**363**(2) (2010), 383-398.**A Clark-Ocone formula in UMD Banach spaces**

with J. van Neerven*Abstract*: Let $H$ be a separable real Hilbert space and let $\mathbb{F}=(\mathscr{F}_t)_{t\in [0,T]}$ be the augmented filtration generated by an $H$-cylindrical Brownian motion $(W_H(t))_{t\in [0,T]}$ on a probability space $(\Omega,\mathscr{F},\mathbb{P})$. We prove that if $E$ is a UMD Banach space, $1\leq p<\infty$, and $F\in \mathbb{D}^{1,p}(\Omega;E)$ is $\mathscr{F}_T$-measurable, then $$ F = \mathbb{E} (F) + \int_0^T P_{\mathbb{F}} (DF)\,dW_H,$$ where $D$ is the Malliavin derivative of $F$ and $P_{\mathbb{F}}$ is the projection onto the ${\mathbb{F}}$-adapted elements in a suitable Banach space of $L^p$-stochastically integrable $\mathcal{L}(H,E)$-valued processes.

Electron. Comm. Probab.**13**(2008), 151-164.**On the domain of non-symmetric Ornstein-Uhlenbeck operators in infinite dimensions**

with J. van Neerven*Abstract*: We consider the linear stochastic Cauchy problem \[ dX(t) = AX(t)\,dt + B\,dW_H(t),\qquad t\geq 0, \] where $A$ generates a $C_0$-semigroup on a Banach space $E$, $W_H$ is a cylindrical Brownian motion over a Hilbert space $H$, and $B:H\to E$ is a bounded operator. Assuming the existence of a unique minimal invariant measure $\mu_\infty$, let $L_p$ denote the realization of the Ornstein-Uhlenbeck operator associated with this problem in $L^p(E,\mu_\infty)$. Under suitable assumptions concerning the invariance of $\mathrm{Ran}(B)$ under the semigroup generated by $A$, we prove the following domain inclusions, valid for $1 < p \leq 2$: \begin{equation}\begin{aligned}\label{eq:} \mathscr{D}((-L_p)^{1/2}) & \hookrightarrow W_H^{1,p}(E,\mu_\infty), \\ \mathscr{D}(L_p) &\hookrightarrow W_H^{2,p}(E,\mu_\infty). \end{aligned}\end{equation} Here $W_H^{k,p}(E,\mu_\infty)$ denotes the $k$-th order Sobolev space of functions with Fréchet derivatives up to order $k$ in the direction of $H$. No symmetry assumptions are made on $L_p$.

Infin. Dimens. Anal. Quantum Probab. Relat. Topics**11**(4) (2008), 603-626.**On analytic Ornstein-Uhlenbeck semigroups in infinite dimensions**

with J. van Neerven*Abstract*: We extend to infinite dimensions an explicit formula of Chill, Fašangová, Metafune, and Pallara for the optimal angle of analyticity of analytic Ornstein-Uhlenbeck semigroups. The main ingredient is an abstract representation of the Ornstein-Uhlenbeck operator in divergence form.

Arch. Math. (Basel)**89**(3) (2007), 226-236.

# Short notes

**Entropic Ricci curvature for discrete spaces***Abstract*: We give a short overview on a recently developed notion of Ricci curva- ture for discrete spaces. This notion relies on geodesic convexity properties of the relative entropy along geodesics in the space of probability densities, for a metric which is similar to (but different from) the 2-Wasserstein metric. The theory can be considered as a discrete counterpart to the theory of Ricci curvature for geodesic measure spaces developed by Lott–Sturm–Villani.

In: Modern Approaches to Discrete Curvature, Springer Lecture Notes in Mathematics**2184**(2017), 159-174.

# PhD thesis

I defended my thesis at 21 April 2009 at TU Delft. Please send me an email if you would like to receive a printed copy.

# Coauthors

- Eric Carlen (Rutgers)
- Philippe Clément (Delft)
- Sjoerd Dirksen (RWTH Aachen)
- Matthias Erbar (Bonn)
- Max Fathi (Toulouse)
- Nicola Gigli (SISSA)
- Martin Hairer (Warwick)
- Daniel Matthes (TU Munich)
- Jan van Neerven (Delft)
- Pierre Portal (ANU Canberra)
- Michiel Renger (WIAS Berlin)
- Martin Rumpf (Bonn)
- Carola Schönlieb (Cambridge)
- Stefan Simon (Bonn)
- Prasad Tetali (Georgia Tech)
- Hendrik Weber (Warwick)
- Melchior Wirth (Jena)