Please note all the lectures will take place in **rooms G01 and G21** (next to each other) of the **Old Physics Building** which is marked as Building number 67 in this **map**. The lecture schedule is available **here**. Details of the plenary and invited talks are given below.

**PLENARY TALKS**

**Zdzislaw Brzezniak**(9 lectures/7 hours)

Title of the Mini-Course: *Introduction to Stochastic Partial Differential Equations*

**David Elworthy**(9 lectures/7 hours)

Title of the Mini-Course: *Introduction to Some Concepts in Stochastic Analysis*

Abstract and references of the mini-course can be found **here**.

**Istvan Gyongy**(6 lectures/6 hours)

Title of the Mini-Course: *An Introduction to Numerical Analysis of Stochastic Partial Differential Equations *

Abstract: Second order (possibly degenerate) parabolic stochastic PDEs will be considered. Applications in nonlinear filtering theory will be discussed to motivate the need for developing effective numerical methods to solve them.

Temporal and spatial approximation schemes with emphasis on their rate of convergence will be studied. In particular, Wong-Zakai uk essays writing approximations, Euler approximations, splitting-up methods, finite element and finite difference schemes will be studied. Extrapolation methods to accelerate numerical schemes will be investigated.

**Martin Hairer**(8 lectures/8 hours)

Title of the Mini-Course: *Stochastic PDEs and Renormalisation*

Abstract: Certain stochastic PDEs arising naturally in theoretical physics are classically considered to be severely ill-posed. As a matter of fact, besides the question of existence and uniqueness of solutions, it is not even clear in this case what the concept of a “solution” even means! The recently developed theory of regularity structures allows to answer this question in a way which is both mathematically unambiguous and physically meaningful. In this lecture series, we will give an overview of some of the main features of the theory with an emphasis on the notion of “renormalisation” in this context. Along the way, we will encounter a number of nice analytical objects, probabilistic bounds, and algebraic structures.

**Xue-Mei Li**(2 lectures/2 hours)

Title: *The Stochastic Interpolation Equation*

Abstract: I will discuss a stochastic model on a Lie group, which interpolate one parameter family subgroups with Brownian buy essay motions, and more generally hypoelliptic diffusions on a subgroup. We discuss the effective limit when we take one of the parameters to infinity, involving separation of scales, diffusion creation, and classification of limits on the orbit manifolds.

References:

- Stochastic Homogenisation on Homogeneous Spaces. (2015) Arxiv 1505.06772
- Limits of Random Differential Equations on Manifolds. (2015) Arxiv 1501.04793. To appear: Probability and Theory Related Fields. DOI 10.1007/s00440-015-0669-x

Also a related reference is: Random Perturbation to the Geodesic Equation. Arxiv 1402.5861. To appear: The Annals of Probability.

**Terry Lyons**(6 lectures/6 hours)

Title of the Mini-Course: *On Signatures*

**Alexander Yu Veretennikov**(3 lectures/3 hours)

Title of the Mini-Course: *On Direct Approach to Diffusion Filtering SPDEs and on Stability Problems*

**INVITED TALKS**

**David Applebaum**, University of Sheffield

Title: *Generalised Spherical Functions and Levy-Khintchine Formula on Groups and Symmetric Spaces*

Abstract: In 1964 Ramesh Gangolli published a Levy-Khintchine type formula which characterised $K$ bi-invariant infinitely divisible probability measures on a symmetric space $G/K$. His main tool was Harish-Chandra’s spherical functions which he used to construct a generalisation of the Fourier transform of a measure. In this talk I will introduce generalised spherical functions (or Eisenstein integrals), and extensions of these which are constructed using representation theory, to obtain such a characterisation for arbitrary infinitely divisible probability measures on a non-compact symmetric space.

*(Based on a joint work with Tony Dooley (Bath, Sydney))*

**Anup Biswas**, IISER Pune

Title: *Controlled Equilibrium Selection in Stochastically Perturbed Dynamics*

Abstract: We consider a dynamical system with finitely many equilibria and perturbed by small noise, in addition to being controlled by an `expensive’ control. It is shown that depending on the relative magnitudes of the noise variance and the `running cost’ for dynamics, one can identify three regimes in each of which the optimal control forces the process to concentrate near equilibria that can be characterised depending on the regime.

*(Based on a joint work with Ari Arapostathis and Vivek S. Borkar)*

**Imran Habib Biswas**, TIFR-CAM Bangalore

Title: *Conservation Laws Driven by Levy White Noise*

Abstract: We explore the entropy solution framework for scalar conservation laws that are perturbed by multiplicative Levy noise. The primary focus of this talk is to establish existence and uniqueness of entropy solutions of conservation laws with multiple spatial dimensions that are driven by jump processes. The entropy inequalities are formally obtained by Ito-Levy chain rule. The issue of multidimensionality requires generalized interpretation of entropy inequalities to accommodate Young measure valued solution. We first establish the existence of entropy solution in the generalised sense via vanishing viscosity approximation, and then establish the $L*^*1$-contraction principle which also requires vanishing viscosity regularisation. Finally, the $L*^*1$ contraction principle is used to argue that the generalised entropy solution is indeed the classical entropy solution.

*(Based on a joint work with K. H. Karlsen, U. Koley and A. K. Majee)*

**Vivek S Borkar**, IIT Bombay

Title: *Some Contributions to Risk-Sensitive Control*

Abstract: This talk will describe a new representation for risk-sensitive reward for a controlled Markov chain and point out its connection with the Donsker-Varadhan formula for principal eigenvalue. An analogous result for controlled diffusions will also be described.

(*Based on a joint work with V. Anantharam, A. Arapostathis and K. Suresh Kumar*)

**Mrinal K Ghosh**, IISc Bangalore

Title: *Nonzero-Sum Risk Sensitive Stochastic Games for Continuous Time Markov Chains*

Abstract: We study nonzero-sum stochastic games for continuous time Markov chains on a denumerable state space with risk sensitive ergodic cost criterion. Under a Lyapunov type stability assumption and a small cost condition, we show that the corresponding system of coupled HJB equations admits a solution which leads to the existence of Nash equilibrium in stationary strategies.

*(Based on a joint work with Chandan Pal)*

**Mathew Joseph**, University of Sheffield

Title: *Approximation of the Stochastic Heat Equation*

Abstract: We consider the stochastic heat equation and discuss how an analysis of the moments gives information about the solution. We will then indicate how one can approximate the equation by a system of interacting diffusions. From this, a few consequences follow, one of which is comparison inequalities for moments of the stochastic heat equation with different nonlinearities.

**Weijun Xu**, University of Warwick

Title: *When is $\Phi^4_3$ Universal?*

Abstract: The dynamical $\Phi^4_3$ equation is widely expected to be the universal model for phase coexistence in three dimensions. In this talk, we will see that although a large class of symmetric models near a pitchfork bifurcation are described $\Phi^4_3$ in large scales, this is in general not true when asymmetry is present. One interesting fact is that, asymmetry in the potential turns a pitchfork bifurcation into a saddle-node bifurcation. As a consequence, the large scale behaviour of asymmetric models near criticality are described by $\Phi^3_3$ instead.

*(Based on a joint work with Martin Hairer)*