Titles and Abstracts
**Title: **Optimal investment under information driven default contagion

**Abstract: **We introduce a novel dynamic optimization framework to analyze optimal portfolio allocations within an information driven default
contagion model. The investor can allocate his wealth across several defaultable stocks whose growth rates and default intensities are driven by a hidden
Markov chain. We provide a rigorous analysis of default contagion arising through recursive dependence of the optimal strategies on the gradient of value functions. We establish uniform bounds for solutions to a sequence of approximation problems, show their convergence to the unique Sobolev solution of the recursive HJB system of PDEs in terms of default states, and prove the corresponding verification theorem.

(Joint work with Agostino Capponi.)

**Title: **Gaussian bounds on locally irregular graphs

**Abstract: **A well known theorem of Delmotte is that Gaussian bounds,
parabolic Harnack inequality, and the combination of volume doubling and
Poincaré inequality are equivalent for graphs. In this talk,
we consider graphs for which these conditions hold, but only for sufficiently large balls,
and show a similar equivalence. We also show more precise sufficient conditions on
the range of balls for which 'good behaviour' is required in order to obtain heat kernel
bounds in a fixed ball.

(Joint work with Martin T. Barlow.)

**Title: **Two-state stochastic models of central dogma at the single-cell level

**Abstract: **Chemical master equation (Markovian jumping process) incorporating multiple gene states are used to model the kinetics of central dogma at the single-cell level. (1) We utilized this model without feedback to help uncover the origin of the ubiquitous mechanism of transcriptional burst in bacteria, which is a mystery in biology for about ten years. (2) Under the condition that the gene state switching can be neither extremely slow nor exceedingly rapid as many previous theoretical treatments assumed, the model with positive feedback can be simplified to a fluctuating-rate model, which is indeed stochastic coupled Ordinary Differential Equations. The simplified kinetics yields a nonequilibrium landscape function, which, similar to the energy function for equilibrium fluctuation, provides the leading orders of fluctuations around each phenotypic state, as well as the transition rates between the two phenotypic states. This rate formula is analogous to Kramers' theory for chemical reactions. The rigorous proof needs to integrate the well-known Donsker-Varadhan theory and Freidlin-Wentzell theory of large deviation principle in such an averaging case.

**Title: **Feynman-Kac formulas for solutions to degenerate elliptic boundary value problems with Dirichlet boundary conditions

**Abstract: **We prove stochastic representation formulas for solutions to the elliptic boundary value and obstacle problems associated with a degenerate Markov diffusion process. The degeneracy in the diffusion
coefficient is proportional to the $\alpha$-power of the distance to the boundary of the half-space, where
$\alpha\in (0,1)$. This generalizes the well-known Heston stochastic volatility process, which is widely used
as an asset price model in mathematical finance and a paradigm for a degenerate diffusion process.
The generator of this degenerate diffusion process with killing, is a second-order, degenerate-elliptic
partial differential operator where the degeneracy in the operator symbol is proportional to the
2$\alpha$-power of the distance to the boundary of the half-plane. Our stochastic representation formulas
provide the unique solutions to the elliptic boundary value and obstacle problems, when we seek
solutions which are suitably smooth up to the boundary portion
$\Gamma_0$ contained in the boundary of
the half-plane. In the case when the full Dirichlet condition is given, our stochastic representation
formulas provide the unique solution which are not guaranteed to be any more than continuous up
to the boundary portion
$\Gamma_0$.

(Joint work with Paul Feehan and Jian Song.)

**Title: **Limit theory for pruning processes of Galton-Watson trees

**Abstract: **Pruning processes have been studied separately for Galton-Watson trees and Levy trees. We establish here a limit theory that strongly connects the two studies.

(Joint work with Matthias Winkel.)

**Title: **Random motion along co-adjoint orbits

**Abstract: **We study Lagrangian motion generated by a random Eulerian motion on
the co-adjoint orbit of a (finite dimensional) group $G$. Our choice of random
Eulerian motion preserves the energy. We discuss long--time behavior of the
Lagrangian motion. Examples are shown in the case of both compact and non-compact
groups. Our attempt could be viewed as an effort (in finite dimensions) towards the
understanding of (inviscid) turbulence for ideal incompressible fluids.

**Title: **Cycle symmetry and circulation fluctuations of some Markov processes

**Abstract: **Markov chains are widely used to model various stochastic systems in physics, chemistry, biology, etc. The trajectories of a recurrent Markov chain constantly form various cycles. For a family of cycles passing through the same set of states, we prove that the distributions of the forming times of these cycles, respectively conditioned on that the corresponding cycle is formed earlier than the others, are exactly the same. This cycle symmetry can be regarded as a generalization of the Haldane relation for reversible enzyme kinetics. We then prove that this cycle symmetry leads to the large deviation principle for the sample circulations along these cycles, in which the rate function has a non-obvious symmetry. This symmetry implies the Gallavotti-Cohen type fluctuation theorem for the sample net circulations. We also obtain the transient fluctuation theorem and the integral fluctuation theorem in non-equilibrium statistical physics for sample circulations. Similar results hold for diffusion processes on the circle

**Title: **A lower bound for disconnection by simple random walk

**Abstract: **In this talk the speaker will mainly talk about results in the speaker's recent paper of the same title (arXiv:1412:3959): we consider simple random walk on $Z^d$, $d\geq 3$. We investigate the asymptotic behaviour of the probability that a large body gets disconnected from infinity by the set of points visited by a simple random walk, motivated by the recent work of A.-S. Sznitman and the speaker in ''Large deviations for occupation time profiles of random interlacements'' (Probab. Theory Relat. Fields, 161:309-350, 2015) and ''A lower bound for disconnection by random interlacement'' (Electron. J. Probab., 19(17):1-26, 2014). We derive asymptotic lower bounds that bring into play random interlacements. Although open at the moment, some of the lower bounds we obtain possibly match the asymptotic upper bounds recently obtained by A.-S. Sznitman in ''Disconnection, random walks, and random interlacements'' (arXiv:1412:3960). This potentially yields special significance to the tilted walks that we use in this work, and to the strategy that we employ to implement disconnection.

**Title: **Self-avoiding walk and connective constant

**Abstract: **A self-avoiding walk (SAW) is a path on a graph that revisits no vertex. The connective constant of a graph is defined to be the exponential growth rate of the number of n-step SAWs with respect to n. We prove that sqrt{d-1} is a universal lower bound for connective constants of any infinite, connected, transitive, simple, d-regular graph. We also prove that the connective constant of a Cayley graph decreases strictly when a new relator is added to the group and increases strictly when a non-trivial word is declared to be a generator. I will also present a locality result regarding to the connective constants proved by defining a linearly increasing harmonic function on Cayley graphs. In particular, the connective constant is local for all solvable groups.

(Joint work with Geoffrey Grimmett.)

**Title: **Transport inequalities and the Lyapunov condition

**Abstract: **In this talk, we will give a review of some recent work on functional inequalities such as transport inequality and log-Sobolev inequality, derived from the Lyapunov condition.

**Title: **Singular stochastic PDEs

**Abstract: **I will discuss some recent development of stochastic PDEs. These equations have very singular solutions, and one needs renormalization techniques to interpret these solutions. I will focus on the theory of regularity structures recently developed by Martin Hairer which provides solutions to a wide class of such equations.

**Title: **Stationarity as a path property

**Abstract: **Stationarity is the shift invariance of the distribution for a stochastic process. In this work we rediscover
stationarity as a path property rather than a distributional property. More precisely, we characterize a set of paths denoted
as A, which corresponds to the notion of stationarity in the following sense: on one hand, the set A is shown to be large
enough, so that for any stationary process, almost all of its paths are in A; on the other hand, we prove that
any path in A will behave in the optimal way under any stationarity test satisfying some mild conditions. The results
justify our intuition about how a typical stationary process should look like, and potentially lead to new
families of stationarity tests.

(Joint work with Tony Wirjanto.)

**Title: **On a class of backward doubly stochastic differential equations

**Abstract: **We study a class of backward doubly stochastic differential equations (BDSDEs) involving a standard Brownian motion and a martingale with spacial parameter, and its connection with a class of nonlinear stochastic partial differential equations (SPDEs) driven by a (generalised) random field.

(Joint work with Xiaoming Song.)

**Title: **Scaling limits of disorder relevant systems

**Abstract: **We discuss the notion of disorder relevance vs irrelevance, and our
recent results on the scaling limit of disorder relevant systems, which
include the pinning model, the long-range directed polymer model,
and the two-dimensional random field Ising model. The key
ingredient is a result on convergence of polynomial chaos expansions
to Wiener chaos expansions.

(Joint work with F. Caravenna and N. Zygouras.)

**Title: **Almost sure multifractal spectrum of SLE

**Abstract: **15 years ago B. Duplantier predicted the multifractal spectrum of Schramm Loewner Evolution (SLE), which encodes the fine structure of the harmonic measure of SLE curves. In this talk, I will report our recent rigorous derivation of this prediction. As a byproduct, we also confirm a conjecture of Beliaev and Smirnov for the a.s. bulk integral means spectrum of SLE. The proof uses various couplings of SLE and Gaussian free field, which are developed in the theory of imaginary geometry and Liouville quantum gravity.

(Joint work with E. Gwynne and J. Miller.)

**Title: **A Harnack inequality for stochastic partial differential equations

**Abstract: **Under general conditions we show an \textit{a priori} probabilistic Harnack inequality for the non-negative solution of a stochastic partial differential equation of the following form
$$\partial_t u=div g(A\nabla u)+f(t,x,u;\omega)+g_i(t,x,u;\omega)\dot{w}_t^i.$$
We will also show that the solution of the above equation will be almost surely strictly positive if the initial condition is non-negative and not identically vanishing.

**Title: **Correlation kernels for sums and products of random matrices

**Abstract: **In this talk I report new results on the correlation kernels for the distribution of eigenvalues of (1) H+M, where H is a GUE random matrix, (2) (GM)*(GM), where G is a Ginibre random matrix, and (3) (TM)*(TM), where T is a truncated random unitary matrix. We only assume that M is a random matrix whose eigenvalues are in a polynomial ensemble, and our results are expressed in double contour integral formulas, versatile in asymptotic analysis.

(Joint work with Tom Claeys and Arno Kuijlaars.)

**Title: **Laws of the iterated logarithm for symmetric jump processes

**Abstract: **Based on two-sided heat kernel estimates for a class of symmetric jump processes on metric measure spaces, the laws of the iterated logarithm (LIL) for sample paths, local times and ranges are established. In particular, the LILs are obtained for stable-like processes on $\alpha$-sets.

(Joint work with Panki Kim and Takashi Kumagai.)

**Title: **Dependence uncertainty and Frechet problems

**Abstract: **Modeling inter-dependence among multiple risks often faces statistical as well as modeling challenges, with considerable uncertainty arising naturally. In this talk, we present the framework of risk aggregation with dependence uncertainty, its currently expanding research developments, and open questions. We discuss selected mathematical problems which may include: mixability, asymptotic shape of the aggregation, model uncertainty of risk measures, extreme risk scenarios, and a connection with extreme-value theory.

**Title: **Some properties of stochastic partial differential equations

**Abstract: **In this talk, based on the history of the research on stochastic heat equations and combined with my research interests, I will mention some fundamental theories of stochastic partial differential equations and their applications. I will mainly consider the topics on the invariant measures and the applications of stochastic partial differential equations in the interface model.

**Title: **Asymptotics of entropy production rate of OU processes

**Abstract: **In the context of non-equilibrium statistical physics, the entropy production rate is an important concept to describe how far a specific state of a system is from its equilibrium state. In this talk, we establish a central limit theorem and a moderate deviation principle for the entropy production rate of $d$-dimensional Ornstein-Uhlenbeck processes, by the techniques of functional inequalities such as Poincaré inequality and log-Sobolev inequality. As an application, we obtain a law of iterated logarithm for the entropy production rate.

(Joint work with Ran Wang.)

**Title: **Contact Processes on Some Inhomogeneous Graphs and Random Graphs

**Abstract: **The study of stochastic processes in inhomogeneous environments and random environments is developing prosperously in probability field. In this talk I first briefly review some basic facts on contact processes. Then I introduce some previous models on contact processes in inhomogeneous environments and random environments. Finally, I report some recent works on contact processes on some inhomogeneous graphs and random graphs. Some basic ideas of the proofs are given.

(Joint work with Xinxing Chen, Thomas Mountford, Jean-Christophe Mourrat and Daniel Valesin.)

**Title: **Models of gradient type with sub-quadratic actions

**Abstract: **We consider models of gradient type, which is the distribution of a collection of real-valued random variables $\phi :=\{\phi_x: x \in \Lambda\}$ given by $Z^{-1}\exp({-\sum\nolimits_{j \sim k}V(\phi_j-\phi_k)})$. We focus our study on the case that $V(\nabla\phi) = [1+(\nabla\phi)^2]^\alpha$ with $0 < \alpha < 1/2$, which is a non-convex potential. We introduce an auxiliary field $t_{jk}$ for each edge and represent the model as the marginal of a model with log-cancave density. Based on this method, we prove that finite moments of the fields $\left<[v \cdot \phi]^p \right>$ are bounded uniformly in the volume. However,$< e^{v \cdot \phi} >$does not exist for infinite volume measure, unless $\alpha > 1/2$.

**Title: **On time regularity of generalized Ornstein-Uhlenbeck processes with Lévy noises in Hilbert spaces

**Abstract: **In this paper, at first, we obtain a necessary condition of $H$-càdlàg modification and $H$-weakly càdlàg modification of generalized Ornstein-Uhlenbeck processes with Lévy noises in Hilbert spaces $H$. And then, we give a necessary and sufficient condition of $H$-càdlàg modification and $H$-weakly càdlàg modification of Ornstein-Uhlenbeck processes driven by cylindrical $\alpha$-semi-stable processes. Secondly, we investigate the properties of cylindrical càdlàg modification and $V$-cylindrical càdlàg modification. Applying the obtained results to the diagonal Ornstein-Uhlenbeck processes with $\alpha$-stable noises, we show a necessary and sufficient condition of cylindrical càdlàg modification and $V$-cylindrical càdlàg modification in symmetric case for $\alpha\in (0,1)$, and give a sufficient condition in general case for $\alpha\in (0,2)$. Some examples illustrate the relations among the concepts of various càdlàg modification.