#### PROGRAM

- 10:50 - 11:35 M. Röckner (Bielefeld Universität): The Stochastic Porous Media Equations on
*ℝ*^{d} - 11:45 - 12:30 M. Neklyudov (Università di Pisa): New type of homogenisation problem for stochastic parabolic equations
- 12:30 - 14:20 Lunch break
- 14:20 - 15:05 C. Olivera (Universidade Estadual de Campinas): On a class of stochastic transport equations for
*L*vector fields^{2}_{loc} - 15:15 - 16:00 M. Leimbach (TU Berlin): Blow-up of stable stochastic differential equation

This talk is on existence and uniqueness of solutions to the stochastic porous
media equation

*dX - Δψ(X)dt = XdW*on*ℝ*. Here,^{d}*W*is a Wiener process and*ψ*is a maximal monotone graph in*ℝxℝ*such that*ψ(r)≤ C(|r|*, for every^{m}+ 1)*r∈ℝ*. In this general case the dimension is restricted to*d>3*. When*ψ*is Lipschitz, the well-posedness, however, holds for all dimensions on the classical Sobolev space*H*. If^{-1}(ℝ^{d})*ψ(r)r≥ρ|r|*and^{m+1}*m:= (d-2)/(d+2)*, we prove finite time extinction of the solutions with strictly positive probability.
We will show that the solution of 1D stochastic parabolic equation
with additive noise converges to a diffusion process independent
upon space variable when we rescale noise at the extremum points
of the process. We will discuss open problems and suggest future
directions of research. The talk is based on a joint work in
progress with Ben Goldys.

We study in this talk the existence and uniqueness of solutions to a class
of stochastic transport equations with irregular coefficients. Asking only
boundedness of the divergence of the coefficients (a classical condition
in both the deterministic and stochastic setting), we can lower the
integrability regularity required in known results on the coefficients
themselves and on the initial condition, and still prove uniqueness
of solutions.

We examine a 2-dimensional ODE which exhibits explosion in finite
time. Considered as an SDE with additive white noise, it is known to
be complete - in the sense that for each initial condition there is almost
surely no explosion. Furthermore, the associated Markov process even
admits an invariant probability measure. On the other hand, as we will
show, the corresponding local stochastic flow will almost surely not be
strongly complete, i.e. there exist (random) initial conditions for which
the solutions explode in finite time.

### Venue

#### Aula Magna

#### Dipartimento di Matematica

#### Università di Pisa

#### Largo Bruno Pontecorvo 5

#### 56127 Pisa

#### Italia

Supported by the project *PRIN 2011: Equazioni differenziali stocastiche*