# Workshop in stochastic analysis

26–30 November 2018

This workshop will bring together experts with diverse backgrounds in analysis, partial differential equations, and probability, working on stochastic differential equations. It will aim to facilitate exchanges of perspectives and techniques between different groups currently using substantially different approaches to solve a range of similar problems.

## Sessions

26 November 2018
Time Session
1:30pm
Schauder estimates and stochastic PDEs
Jiakun Liu, University of Wollongong, Australia
The talk has three parts: First in part 1, we introduce the standard Schauder estimates for elliptic PDEs, including Possion's equation and Monge-Ampere equations. Then in part 2, we move to stochastic settings where the Schauder estimate was an open problem raised by Krylov in 1999. Last in part 3, we state our recent work on the Schauder estimates for SPDEs and some applications, which is a joint work with Kai Du.
2:30pm
TBC
Laurence Field, The Australian National University, Australia
TBC
27 November 2018
Time Session
9:30am
Topics in Gaussian Harmonic Analysis
Wilfredo Urbina, Roosevelt University, USA
In the first part of the talk we briefly review the basic notions of Gaussian harmonic analysis: semigroups, maximal functions, Littlewood-Paley functions, spectral multipliers and singular integrals. In the second half we discuss new developments on continuity of those operators on variable exponent Lebesgue spaces.
10:30am
Morning tea
11am
Regularity and approximation of SPDEs
Petru A. Cioica-Licht, University of Otago, New Zealand
The speed of an approximation method is related to the regularity of the target function. As a rule of thumb, the convergence rate of classical uniform methods is governed by the Sobolev regularity, whereas adaptive methods correlate with the regularity in special scales of Besov spaces. In this talk, I am going to present some results concerning the spatial regularity of second order SPDEs in such scales. The main difficulty comes from the fact that the scales do not consist of Banach spaces but merely of quasi-Banach spaces. One way out is to extend the stochastic integration theory from UMD Banach spaces to proper classes of quasi-Banach spaces. Another strategy is to show how the singularities of the solution can be handled by appropriate weights and embed suitable weighted Sobolev spaces into the quasi-Banach Besov spaces of interest. I am going to elaborate on how far we can get (so far) in each direction. In particular, I will focus on what happens if the underlying domain is not smooth.
1:30pm
Stochastic heat equation and a central limit theorem
Jingyu Huang, University of Birmingham, UK
We study the stochastic heat equation on the real line \begin{equation*} \frac{\partial u}{\partial t} = \frac12 \frac{\partial^2 u}{\partial x^2} + u \dot{W} \end{equation*} (where $\dot{W}$ is a space time white noise). I will talk about the random field theory for stochastic heat equation. Later I will present a recent result: the spatial integral $\int_{-R}^R u(t,x) dx$ converges to a Gaussian distribution as $R \to \infty$, after renormalization. It is proved using Stein's method and Malliavin calculus, which will be introduced during the talk. This result is based on a joint work with Nualart and Viitasaari.
2:30pm
Sharp Strichartz estimates for Schrodinger equation
Zihua Guo, Monash University, Australia
We consider the $L_t^2L_x^r$ estimates for the solutions to the wave and Schrodinger equations in high dimensions. For the homogeneous estimates, we show $L_t^2L_x^\infty$ estimates fail at the critical regularity in high dimensions by using stable Levy process in $\mathbb{R}^d$. Moreover, we show that some spherically averaged $L_t^2L_x^\infty$ estimate holds at the critical regularity. As a by-product we obtain Strichartz estimates with angular smoothing effect. For the inhomogeneous estimates, we prove double $L_t^2$-type estimates. This is a joint work with Ji Li, Kenji Nakanishi and Lixin Yan.
6:00pm
Conference dinner
Mode 3, 22 Londsdale sT, braddon, ACT 2612
28 November 2018
Time Session
9:30am
TBC
Lutz Weis, Karlsruhe Institute of Technology, Germany
TBC
10.30am
Morning tea
11am
Maximal regularity for stochastic Volterra integral equations
Markus Antoni, Karlsruhe Institute of Technology, Germany
In this talk we discuss an approach to obtain maximal regularity estimates for solutions of stochastic Volterra integral equations driven by a multiplicative Gaussian noise. To achieve that, we mainly focus on suitable estimates for deterministic and stochastic convolution operators. Starting with the scalar-valued case, we use functional calculi results to lift the corresponding estimates to the operator-valued setting. Once maximal regularity estimates for convolutions are obtained, appropriate Lipschitz and linear growth assumptions on the nonlinearities will lead to unique mild solutions with H\"older continuous trajectories. This is joint work with Boris Baeumer and Petru Cioica-Licht.
1:30pm
PDEs with stochastic memory and reaction
Boris Baeumer, University of Otago, New Zealand
Based on a basic continuous time random walk model we carefully derive a general PDE with memory (aka Volterra integral equation). We show how a physical reaction term will impact the memory term and end up with a stochastic Volterra equation where the noise is part of the memory.
2:30pm
Strong solutions of stochastic models for viscoelastic flows of Oldroyd type
Debopriya Mukherjee, UNSW, Australia
In this talk, I will provide basic difference between Stratonovich integral and Ito integral for general stochastic partial differential equations. We then move to the study of stochastic Oldroyd type models for viscoelastic fluids in $\mathbb{R}^ d , d = 2, 3.$ In the current work, we have shown existence and uniqueness of strong local maximal solutions when the initial data are in $^ s$ for $s > d/2, d = 2, 3.$ Probabilistic estimate of the random time interval for the existence of a local solution is expressed in terms of expected values of the initial data.
29 November 2018
Time Session
9:30am
Stochastic maximal regularity for rough time-dependent problems
Pierre Portal, The Australian National University, Australia
To solve non-linear parabolic SPDE, one often uses a simple fixed point argument, based on a subtle regularity property of the linear part, called a stochastic maximal regularity estimate. Such estimates have a long history, including the 2012 milestone result of van Neerven-Veraar-Weis (NVW) taught in the mini-course. In this talk, I will present an extension of this result to situations where the coefficients are $L^{\infty}$ in time. It combines the NVW operator theoretic approach with Krylov's PDE approach, to deduce sharp results for coefficients that are continuous in the spatial variables. It also introduces a new method that combines the NVW approach with harmonic analysis to deduce stochastic maximal regularity estimates for coefficients that are mereley $L^{\infty}$ in BOTH space and time. This is joint work with Mark Veraar (Delft).
10.30am
Morning tea
11am
Stochastic PDE's and mixed PDE/Monte-Carlo methods for derivatives pricing
Gregoire Loeper, Monash University, Australia
In the spirit of [1] we propose a pricing method for derivatives when the underlying diffusion is given by a set of stochastic differential equations, with the objective of reducing the computing time. The numerical method is based on a joint use of Monte-Carlo simulations, PDE or analytical formulas. We show that this method has an natural interpretation in terms of stochastic pde's and from this observation propose a new way of implementing it. We also show how to implement the Least Square Monte-Carlo method proposed in [2] together with the mixed PDE/Monte-Carlo method. \\ [1] G. Loeper and O. Pironneau A Mixed PDE /Monte-Carlo Method for Stochastic Volatility Models CRASS 2009\\ [2] F.A. Longstaff and E.S. Schwartz Valuing American Options by simulations:A Simple Least-Squares Approach. Working Paper Anderson Graduate School of Management University of California 1998.
1:30pm
Burkholder–Davis–Gundy inequalities and stochastic integration in UMD Banach spaces
Ivan Yaroslavtsev, TU Delft, Netherlands
In this talk we will present Burkholder--Davis--Gundy inequalities for general UMD Banach space-valued martingales. Namely, we will show that for any UMD Banach space $X$, for any $X$-valued martingale $M$ with $M_0=0$, and for any $1\leq p<\infty$ $\mathbb E \sup_{0\leq s\leq t} \|M_s\|^p \eqsim_{p, X} \mathbb E \gamma([\![M]\!]_t)^p ,\;\;\; t\geq 0,$ where $[\![M]\!]_t$ is the covariation bilinear form of $M$ defined on $X^* \times X^*$ so that $[\![M]\!]_t(x^*, y^*) = [\langle M,x^* \rangle, \langle M, y^* \rangle]_t,\;\;\;x^*, y^*\in X^*,$ and $\gamma([\![M]\!]_t)$ is the $L^2$-norm of a Gaussian measure on $X$ having $[\![M]\!]_t$ as its covariance bilinear form. As a consequence we will extend the theory of vector-valued stochastic integration with respect to a cylindrical Brownian motion by van Neerven, Veraar, and Weis, to the full generality.
2:30pm
Quantifying uncertainty in Lagrangian coherent structures in unsteady flows
Sanjeeva Balasuriya, University of Adelaide, Australia
Over the past several decades, there has been considerable interest in tracking structures in unsteady flows which 'remain coherent,' as well as the boundaries of such structures. Such 'Lagrangian coherent structures' are fundamental in transport (of heat, salinity, humidity, pollutants, bio-organisms); examples are the Gulf stream, the Antarctic Circumpolar Vortex ('the ozone hole'), hurricanes, oceanic eddies within which there is high plankton concentration, the region on the west coast of Florida to which the Deepwater Horizon oil spill did not reach, and separated globules in a microfluidic device which resist mixing. Standard techniques for tracking these involve numerically advecting initial conditions by using the velocity field (usually available only from data), and then extracting from this flow map coherent structures based on a range of techniques. None of these explicitly address uncertainties (noise) in the velocity, which should be of concern because the data typically has very low resolution (oceanic measurements are on grids of size ~100 km). In this talk, I develop a stochastic differential equation model for analytically computing uncertainties of eventual trajectories, which allows for the velocity to be unsteady (the system is nonautonomous), and the noise to be not just time, but also location-dependent ('multiplicative noise'). Combining stochastic calculus techniques such as the Burkholder-Davis-Gundy inequality and Ito's isometry, with ordinary differential equations methods such as the derivation strategy of Melnikov's method, I derive the anisotropic variance of the eventual location explicitly. The 'stochastic sensitivity' field I additionally obtain enables the identification of evolving flow regions which are robust with respect to a user-specification of length-scale and noise-level in the data. It is expected that these will be invaluable new tools in ascribing uncertainties to Lagrangian coherent structures.
30 November 2018
Time Session
9:30am
On stochastic flows associated to stochastic equations in Hilbert spaces
Beniamin Goldys, University of Sydney, Australia
It is well known that finite dimensional stochastic differential equations with regular coefficients generate stochastic flows. In infinite dimensions this property of stochastic equations does not hold in general. Skorokhod constructed a very simple system of independent stochastic equations that does not generate a stochastic flow. Later a number of disconnected results have been obtained but our understanding of stochastic flows in infinite dimensions remains very limited. It turns out that the problem has some interesting connections with the theory of cylindrical processes on the algebras of Hilbert space operators and the theory of random matrices. We will present some new results in this area. This is a joint work with Szymon Peszat.
10.30am
Morning tea
11am
Ergodicity for stochastic dispersive equations
Leonardo Tolomeo, The University of Edinburgh, Scotland
In this talk, we study the long time behaviour of some stochastic partial differential equations (SPDEs). After introducing the notions of ergodicity, unique ergodicity and convergence to equilibrium, we will discuss how these have been proven for a very large class of parabolic SPDEs. We will then shift our attention to dispersive SPDEs, where the general strategy for the parabolic case fails. We will describe this failure for the wave equation on the 1-dimensional torus and present a recent result that settles unique ergodicity even in this case.

Registration is free and open to the public.

Seminar Room 1.33, Building #145, Science Road, The Australian National University

## About Canberra

Canberra is located in the Australian Capital Territory, on the ancient lands of the Ngunnawal people, who have lived here for over 20,000 years. Canberra’s name is thought to mean ‘meeting place’, derived from the Aboriginal word Kamberra. European settlers arrived in the 1830s, and the area won selection by ballot for the federal capital in 1908. Since then the ‘Bush Capital’ has grown to become the proud home of the Australian story, with a growing population of around 390,000.

Canberra hosts a wide range of tourist attractions, including various national museums, galleries and Parliament House, as well as beautiful parks and walking trails. Several attractions are within walking distance of the ANU campus, including the National Museum of Australia and the Australian National Botanic Gardens. Canberra is also a fantastic base from which to explore the many treasures of the surrounding region, including historic townships, beautiful coastlines and the famous Snowy Mountains. Learn more about what to do and see during your stay in Canberra here.

## Accommodation

Below are some accommodation options for your visit to Canberra.

http://www.anu.edu.au/study/accommodation/accommodation-alternatives

## Visas

International visitors to Australia require a visa or an electronic travel authority (ETA) prior to arrival. It is your responsibility to ensure documentation is correct and complete before you commence your journey. Information on obtaining visas and ETAs can be found here.

## Transportation

There are many ways to get around Canberra. Below is some useful information about Bus & Taxi transport around the ANU, the Airport and surrounding areas.

If you are catching a taxi or Uber to the ANU Mathematical Sciences Institute, ask to be taken to Building #145, Science Road, ANU. We are located close to the Ian Ross Building and the ANU gym. A Taxi will generally cost around $40 and will take roughly 15 minutes. Pricing and time may vary depending on traffic. Taxi bookings can be made through Canberra Elite Taxis - 13 22 27. ### Airport Shuttle the ACT government has implemented a public bus service from the CBD to the Canberra Airport via bus Route 11 and 11A, seven days a week. Services run approximately every half hour, and better during peak times (weekdays) and every hour (weekends). To travel just use your MyWay card or pay a cash fare to the driver when boarding. A single adult trip when paying cash will cost$4.80 with cheaper fares for students and children. Significant savings can be made when travelling with MyWay.

View MyWay and Fares information.

For more information about the buses to Canberra airport.

### Action Buses

Canberra buses are a cheap and easy way of getting around town once you're here.

For more information about bus services and fares.

Macarena Rojas
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