**Workshop in Probability and its Applications**

**17-20 March 2014 in Mathematical Institute, University of Oxford.**

- Dominique BAKRY (Toulouse, France)
- Vincent BEFFARA (ENS, Lyons, France)
- Horatio BOEDIHARDJO (OMI, Oxford University)
- Rainer BUCKDAHN (Univ Brest, France)
- Tomas CASS (Imperial College, London)
- Zengjing CHEN (Shandong University, China)
- Sam COHEN(University of Oxford)
- David CROYDON (Warwick, UK)
- Zhao DONG (Chinese Academy of Science, China)
- Shizan FANG (Université de Bourgogne, France)
- Fuzhou GONG (Chinese Academy of Science, China)
- Alexander GRIGORIAN
- Elton P. HSU (Northwestern University, Evanston, USA)
- Jamie INGLIS (INRIA Sophia Antipolis, France)
- Juan LI (Shandong University, China)
- Zenghu LI (Beijing Normal University, China)
- Wei LIU (Jiangsu Normal University, China)
- Shunlong LUO (Chinese Academy of Science, Beijing)
- James MARTIN (University of Oxford)
- Michela OTTOBRE (Imperial College, London)
- Shige PENG (Shandong University, China)
- Zhen WU (Shandong University, China)
- Hao WU (MIT, USA)
- Bo WU (Fudan University, Shanghai)
- Danyu YANG (OMI, Oxford University)
- B. ZEGARLINSKI (Imperial College London)
- Weian ZHENG (East China Normal University, Shanghai)

The idea is to bring together people from around Oxford who are interested in probability and related areas for a one day meeting. We hope to make it a regular event with broad coverage of the topic. The first meeting was on 9th December 2013. The second meeting will be 17-20 March 2014. The speakers and will be:

- Dominique BAKRY (Toulouse, France)
- Vincent BEFFARA (ENS, Lyons, France)
- Horatio BOEDIHARDJO (OMI, Oxford University)
- Rainer BUCKDAHN (Univ Brest, France)
- Tomas CASS (Imperial College, London)
- Zengjing CHEN (Shandong University, China)
- Sam COHEN(University of Oxford)
- David CROYDON (Warwick, UK)
- Zhao DONG (Chinese Academy of Science, China)
- Shizan FANG (Université de Bourgogne, France)
- Fuzhou GONG (Chinese Academy of Science, China)
- Alexander GRIGORIAN
- Elton P. HSU (Northwestern University, Evanston, USA)
- Jamie INGLIS (INRIA Sophia Antipolis, France)
- Juan LI (Shandong University, China)
- Zenghu LI (Beijing Normal University, China)
- Wei LIU (Jiangsu Normal University, China)
- Shunlong LUO (Chinese Academy of Science, Beijing)
- James MARTIN (University of Oxford)
- Michela OTTOBRE (Imperial College, London)
- Shige PENG (Shandong University, China)
- Zhen WU (Shandong University, China)
- Hao WU (MIT, USA)
- Bo WU (Fudan University, Shanghai)
- Danyu YANG (OMI, Oxford University)
- B. ZEGARLINSKI (Imperial College London)
- Weian ZHENG (East China Normal University, Shanghai)

**Speakers: **

**Participants: **

All talks will be in Mathematical Institute, lecture rooms L2 and L4. The seminar room C4 is reserved for discussions. You can download the program in PDF.

** Monday 17 March, lecture room 2**

**Joint Session with East Midlands Stochastic Analysis Seminar**

9:30 | D. Bakry (Toulouse, France) | Brownian motion on symmetric matrices constructed over Clifford algebras |

14: 30 | B. Zegarlinski (Imperial College London) | Linear and nonlinear semigroups: smoothing and ergodicity |

Tea and Coffee | ||

12:00 | Tom Cass (Imperial College London) | Rough path analysis: a survey and some illustrative applications |

13:00 | Buffet Lunch | |

14:30 | Vincent Beffara | Percolation on mesoscopic lattices |

15:30 | Rainer Buckdahn (Brest, France) | Mean-Field SDEs and associated PDE |

16:30 | Tea and Coffee | |

17:00 | Shige Peng (Shandong University, Jinan) | TBA |

**East Midlands Stochastic Analysis seminars are partially
supported by the London Mathematical Society**

**Tuesday 18 March, lecture room 4**

9:30 | Zheng (ECNU, Shanghai) | TBA |

10:30 | Shizan Fang (Bourgogne, France) | On stochastic differential equations |

11:10 | Tea and Coffee | |

11:40 | Michela Ottobre (Imperial College, London) | How to use infinite dimensional SDEs to sample from a given probability distribution |

12:20 | Victor Fedyashov (Oxford) | Ergodic BSDEs with jumps and time dependence |

13:00 | Buffet Lunch | |

10:30 | Alexander Grigorian | TBA |

15:30 | James Inglis (INRIA, France) | A nonlinear mean-field renewal process and approximating particle system describing neuronal activity |

16:10 | Tea and Coffee | |

16:40 | Horatio Boedihardjo (Oxford-Man Institute, Oxford) | Iterated integrals and winding number |

17:20 | Xi Geng (Oxford) | The Uniqueness of Signature Problem for Gaussian Processes |

**Wednesday 19 March, lecture room 4**

9:00 | Elton P. Hsu (Northwestern University, USA) | Geometric Deviation From Levy's Occupation Time Arcsine Law |

9:40 | Shunlong LUO (Chinese Academy of Sciences, Beijing) | Quantifying Quantum Non-Markovianity |

10:20 | Tea and Coffee | |

10:50 | Juan Li (Shandong University, Weihai) | Reflected mean-field BSDEs, and the related control problems |

11:30 | Hao Wu (MIT, USA) | Intersections of SLE paths |

12:10 | Zengjing Chen (Shandong University, Jinan) | Necessary and sufficient conditions for the law of large numbers for capacities |

12:50 | Buffet Lunch |

**Thursday 20 March, lecture room 2**

9:30 | James Martin | Random graph processes with forest fires |

10:10 | Zenghu Li (Beijing Normal University) | Forward and backward stochastic equations of super-Levy processes |

10:50 | Tea and Coffee | |

11:20 | Sam Cohen (Oxford) | Uniformly Uniformly-Ergodic Markov Chains and related BSDEs |

12:00 | Bo Wu (Fudan University, Shanghai) | |

12:40 | Buffet Lunch | |

14:00 | Zhen Wu (Shandong University, China) | BSDEs with two-time-scale Markov chains and applications |

14:40 | David Croydon (Warwick, UK) | Quenched invariance principles for random walks and random divergence forms in random media with a boundary |

15:20 | Tea and Coffee | |

16:00 | Danyu Yang (Oxford-Man Institute, Oxford) | Rough differential equation in Banach space and partial sum process of orthogonal series |

16:40 | Fuzhou Gong (Chinese Academy of Science, Beijing ) | Comparison Theorems of Spectral Gaps of Schrodinger Operators and Diffusion Operators on Abstract Wiener Spaces |

17:20 | Zhao Dong (Chinese Academy of Science, Beijing) | Malliavin Matrix of Degenerate PDE and Gradient Estimates |

**D. Bakry (Toulouse, France): Brownian motion on symmetric matrices constructed over Clifford algebras**

Abstract:Clifford algebras are associative algebraic structures living in spaces with dimension . With the help of the algebra structure, one may construct real symmetric matrices (with dimension on which live natural diffusion processes such as Brownian motion or Ornstein-Uhlenbeck operator. It turns out that the spectral measures of these matrices are again diffusion processes, and that the structure of these processes reflects the algebraic structure of the algebra. We thus recover the well-known Bott's periodicity property of the Clifford algebra. We shall also consider briefly octonionic structures where the situation is much more delicate.

**Vincent Beffara: Percolation on mesoscopic lattices**

In a celebrated paper, Smirnov proved that critical site-percolation on the regular triangular lattice has a non-trivial, conformally invariant scaling limit and that this can be used to derive for instance the value of critical exponents. The argument is unfortunately very specific to this particular lattice, and so far has not been generalized to any other natural case - in particular, percolation on is much beyond reach of current methods. I will present one direction in which the proof can be extended into a non-trivial class of models that somehow interpolate between the triangular lattice and general planar cases.

**Horatio Boedihardjo (Oxford-Man Institute, Oxford): Iterated integrals and winding number**

Abstract:The formal series of iterated path integrals has been an object of significant interest in topology, algebra and more recently, rough path theory. The non-commutative multiplication makes it difficult to prove precise facts about the series. We shall discuss two of these open problems. The first is "surjectiveness" problem, that is, which tensor elements can be realised as the series of iterated integrals of some paths. The second is "injectiveness", that is what information about a path can one recovers from its iterated integrals. It turns out that some results on these two problems can be obtained using the concept of winding number.

** Rainer Buckdahn (Brest, France): Mean-Field SDEs and associated PDE.**

The talk presents a common work with Juan Li (SDU, Weihai), Shige Peng (SDU, Jinan, PRC) and Catherine Rainer (UBO, Brest, France). In our work, after a general discussion of derivatives of functions of probability measures and the second order Taylor expansion for such functions, we study a flow associated with stochastic differential equations (SDEs) of mean-field type, i.e., SDEs driven by a Brownian motion, which coefficients depend at time on the solution process but also on its probability law , as well as the derivatives of this flow with respect to the space variable and the probability law. This discussion allows to derive a non local PDE which unique solution is described with the help of the mean-field SDE.

**Tom Cass (Imperial College): Rough path analysis: a survey and some illustrative applications**

Abstract: We review the theory of rough path analysis and rough differential equation. We then illustrate the usefulness of this approach by analysing in detail some recent applications in stochastic analysis.

**Zengjing Chen (Shandong University, Jinan): Necessary and sufficient conditions for the law of large numbers for capacities**

Abstract: In this paper, we investigate strong laws of large numbers for capacities under weaken conditions. We obtain two results: One is a sufficient and almost necessary condition under which any cluster point of empirical average lies, with probability (capacity) one, between upper and lower Choquet integrations; The other is a sufficient and almost necessary condition under which the interval between upper and lower Choquet integrations is the unique smallest interval in which any cluster point of empirical average lies with probability (capacity) one. Furthermore, we study some examples to explain the application about the strong laws of large numbers for capacities. Keywords: Capacity, Choquet expectation, independent, law of large numbers, sub-linear expectation.

**David Croydon (Warwick, UK): Quenched invariance principles for random walks and random divergence forms in random media with a boundary**

Abstract: I will discuss recent joint work with Zhen-Qing Chen (University of Washington) and Takashi Kumagai (Kyoto University) that establishes, via a Dirichlet form extension theorem and making full use of two-sided heat kernel estimates, quenched invariance principles for random walks and random divergence forms in random media with a boundary. In particular, our results demonstrate that the random walk on a supercritical percolation cluster or amongst random conductances bounded uniformly from below in a half-space, quarter-space, etc., converges when rescaled diffusively to a reflecting Brownian motion, which has been one of the important open problems in this area. I will also discuss a similar result for the random conductance model in a box, which allows improvements to be made to existing asymptotic estimates for the relevant mixing time. Furthermore, in the uniformly elliptic case, I will present quenched invariance principles for domains with more general boundaries.

**Sam Cohen (Oxford): Uniformly Uniformly-Ergodic Markov Chains and related BSDEs**

Abstract: If one starts with a uniformly ergodic Markov chain on countable states, what sort of perturbation can one make to the transition rates and still retain uniform ergodicity? In this talk, we will consider a class of perturbations that can be simply described, where a uniform estimate on convergence to an ergodic distribution can be obtained. We shall see how this is related to Ergodic BSDEs and BSDEs up to stopping times in this setting and outline some novel applications of this approach.

**Zhou Dong (Chinese Academy of Sciences, Beijing):**

Abstract: In this talk, we present the boundedness of the inverse of Malliavin Matrix for Degenerate stochastic differential equations with some new conditions, which is equivalent to the Hoermander conditions as the coefficients are smooth. Also, the gradient estimates for the semigroup is given.

**Shizan Fang (Bourgogne, France): On stochastic differential equations. **

Abstract:We will give a survey on recent development of stochastic differential equations, as well as its connection with hydrodynamics.

**Victor Fedyashov (Oxford): Ergodic BSDEs with jumps and time dependence **

Abstract:We look at ergodic BSDEs in the case where the forward dynamics are given by a solution to a non-autonomous (time-periodic coefficients) Ornstein-Uhlenbeck SDE with Lévy noise, taking values in a separable Hilbert space. We establish the existence of a unique bounded solution to an infinite horizon discounted BSDE. We then use the vanishing discount approach together with coupling techniques to obtain a Markovian solution to the EBSDE. We also prove uniqueness under certain growth conditions. Applications are then given, in particular to risk-averse ergodic optimal control and power plant evaluation under uncertainty.

**Xi Geng (Oxford): The Uniqueness of Signature Problem for Gaussian Processes**

Abstract: In this talk we are going to show that, for a certain class of Gaussian processes including fractional Brownian motion with Hurst parameter H>1/4, with probability one, the signature over [0,1] (the collection of iterated integrals over [0,1] of all orders) determines the sample paths uniquely up to reparametrization. The main strategy is a strengthened Le Jan-Qian approximation scheme and the Malliavin calculus. In the end I will also give some remarks on possible ways to attack the deterministic uniqueness of signature problem for geometric p-rough paths.

**Fuzhou Gong (Chinese Academy of Sciences, Beijing):**

**Elton P. Hsu (Northwestern University, USA): Geometric Deviation From Levy's Occupation Time Arcsine Law**

Abstract: Levy showed that the total time that a standard Brownian motion stays positive up to time 1 obeys the arcsine law. We will discuss the deviation of this law for a Brownian motion on a Riemannian manifold near a smooth hypersurface. The deviation has the order of the square root of the total time and is proportional to the mean curvature of the hypersurface. Its explicit form depends on the local time of the transversal Brownian motion properly scaled. This is a joint work with Cheng Ouyang at University of Illinois at Chicago.

**James Inglis (INRIA, France): A nonlinear mean-field renewal process and approximating particle system describing neuronal activity.**

Abstract: We will introduce the noisy integrate-and-fire model used to describe the evolution of the electrical potential across a single neuron, before considering a network of such neurons that interact through an instantaneous mean-field threshold dynamic (when the potential of a single neuron reaches a threshold, it is reset while all the others receive an instantaneous kick). We show that in the limit when the size of the network becomes infinite, the resulting equation may exhibit a blow-up phenomenon under certain conditions, when a large proportion of neurons all emit a spike at the same time (which has been linked with epilepsy). We moreover show that the particle system does indeed exhibit propagation of chaos, and propose a new way to give sense to a solution after a blow-up. This is based on joint research with F. Delarue (Nice), E. Tanré (INRIA) and S. Rubenthaler (Nice).

**Juan Li (Shandong University, Weihai, China): Reflected mean-field BSDEs, and the related control problems**

Abstract: In this talk we will study a new special mean-field problem in a purely probabilistic method, to characterize its limit which is the solution of mean-field backward stochastic differential equations (BSDEs) with reflections, and we will give the probabilistic interpretation of the nonlinear and nonlocal partial differential equations with the obstacles by the solutions of reflected mean-field BSDEs. Furthermore, we will study the optimal control problems of such reflected mean-field BSDEs. Finally, we will show that the value function which our reflected MFBSDE is coupled with is the unique viscosity solution of the related nonlocal parabolic partial differential equation with obstacle. Based on a joint work with Wenqiang Li (Shandong University, Weihai).

**Zenghu Li (Beijing Normal University): Forward and backward stochastic equations of super-Levy processes**

Abstract: The process of distribution functions of a one-dimensional super-Levy process is characterized as the pathwise unique solution of a stochastic integral equation driven by Gaussian and Poisson time-space noises, which generalizes the recent work of Xiong (AOP, 2013) on super-Brownian motion. To prove the pathwise uniqueness of the solution we establish a connection of the stochastic integral equation with some backward doubly stochastic equation with jumps. This is based on a joint work with Hui He and Xu Yang.

**Shunlong LUO (Chinese Academy of Sciences, Beijing): Quantifying Quantum Non-Markovianity**

Abstract: In the study of open quantum systems, memory effects are usually ignored, and this leads to dynamical semi-groups and Markovian dynamics. However, in practice, non-Markovian dynamics is the rule rather than exception. With the recent emergence of quantum information theory, there is a flurry of investigations of quantum non-Markovian dynamics. In this talk, we first review several significant measures for non-Markovianity, such as deviation from divisibility, information exchange between a system and its environment, or entanglement with the environment. Then by exploiting the correlations flow between a system and an arbitrary ancillary, we study a considerably intuitive measure for quantum non-Markovianity by use of correlations as quantified by quantum mutual information rather than entanglement. The measure captures quite directly and deeply the characteristics of quantum non-Markovianity from the perspective of information. A simplified version based on Jamiolkowski-Choi isomorphism which encodes operations via bipartite states and does not involve any optimization is also investigated.

**James Martin (Oxford): **

Abstract: Consider the following extension of the Erdos-Renyi random graph process; in a graph on vertices, each edge arrives at rate 1, but also each vertex is struck by lightning at rate , in which case all the edges in its connected component are removed. Such a "mean-field forest fire" model was introduced by Rath and Toth. For appropriate ranges of , the model exhibits "self-organised criticality". We investigate scaling limits, involving a multiplicative coalescent with an added "deletion" mechanism. I'll mention a few other related models, including epidemic models and "frozen percolation" processes. Joint work with Balazs Rath.

**Michela Ottobre (Imperial College London): How to use infinite dimensional SDEs
to sample from a given probability distribution
**

Abstract: MCMC methods are popular algorithms to sample from a given probability measure. Roughly, these algorithms are based on constructing an ergodic Markov Chain which has the target measure as unique invariant measure. We describe a new MCMC method optimized for the sampling of probability measures defined on infinite dimensional Hilbert spaces and having a density with respect to a Gaussian; such measures arise in the Bayesian approach to inverse problems, and in conditioned diffusions. Our algorithm is based on two key design principles: (i) algorithms which are well-defined in infinite dimensions result in methods that do not suffer from the curse of dimensionality when they are applied to approximations of the infinite dimensional target measure on R^N (i.e. the cost of the algorithm does not increase with dimension); (ii) non-reversible chains can have better convergence properties compared to their reversible counterparts. The method we introduce is based on Hamiltonian mechanics as well, and it is tailored to incorporate the two described design principles. The sequence of Markov Chains that we construct according to these design principles converges weakly to a second order Langevin diffusion on Hilbert space; as a consequence the algorithm explores the approximate target measures on R^N in a number of steps which is independent of N. This is a joint work with N. Pillai, F. Pinski and A. Stuart

**Shige Peng (Shandong University, China):**

Abstract:

**Bo Wu (Fudan University, Shanghai): Functional inequality on path space over a non-compact Riemannian
manifold**

**Hao Wu (MIT, USA): Intersections of SLE paths.**

Abstract: SLE curves are introduced by Oded Schramm as the candidate of the scaling limit of discrete models. In this talk, we first describe basic properties of SLE curves and their relation with discrete models. Then we summarize the Hausdorff dimension results related to SLE curves, in particular the new results about the dimension of cut points and double points. Third we introduce Imaginary Geometry, and from there give the idea of the proof of the dimension results.

**Zhen Wu (Shandong University, China): BSDEs with two-time-scale Markov chains and applications**

Abstract: This talk is concerned with backward stochastic differential equations (BSDEs) coupled by a finite-state Markov chains. This kind of BSDEs has wide applications in optimal control theory and mathematical finance. The underlying Markov chain is assumed to have a two-time scale structure. Namely, the states of the Markov chain can be divided into a number of groups so that the chain jumps rapidly within a group and slowly between the groups. In this talk, we illustrate two convergence results as the fast jump rate goes to infinity, which can be used to reduce the complexity of the original problem. This method is also referred to as singular perturbation. The first one is the weak convergence of the BSDEs with two-time-scale BSDEs. It is proved that the solution of the original BSDE system converges weakly under the Meyer-Zheng topology. The limit process is a solution of aggregated BSDEs. The results are applied to a set of partial differential equations and used to validate their convergence to the corresponding limit system. The second one is the optimal switching problem for regime-switching model with two-time-scale Markov chains. We use a probabilistic approach to solve the problem. To be specific, we obtain the optimal switching strategy by virtue of the oblique reflected BSDEs with Markov chains. Under the two-time-scale structure, we prove the convergence of the variational inequalities. Numerical examples are given for both of the problems to demonstrate the approximation results.

**Danyu Yang (Oxford-Man Institute, Oxford): Rough differential equation in Banach space and partial sum process of orthogonal series**

Abstract: We extend the universal limit theorem to rough differential equation in Banach space driven by weak geometric rough path, and give the quantitative dependence of solution in term of the initial value, vector field and driving rough path. As an infinite dimensional rough path, we prove that the partial sum process of general orthogonal series is a geometric 2-rough process under the same condition as in Menshov-Rademacher Theorem.

**B. Zegarlinski (Imperial College London): Linear and nonlinear semigroups: smoothing and ergodicity.**

Abstract: This will include diffusions with Hoermander type generators in infinite dimensions, as well as Reaction-Diffusion problems.

**Weian Zheng (East China Normal University, Shanghai)**

Abstract:

All talks will be in Mathematical Institute, lecture rooms L2 and L4. Directions to the Institute can be found here.