177 research outputs found
Hitting properties of parabolic s.p.d.e.'s with reflection
We study the hitting properties of the solutions of a class of parabolic
stochastic partial differential equations with singular drifts that prevent
from becoming negative. The drifts can be a reflecting term or a nonlinearity
, with . We prove that almost surely, for all time , the
solution hits the level 0 only at a finite number of space points, which
depends explicitly on . In particular, this number of hits never exceeds 4
and if , then level 0 is not hit.Comment: Published at http://dx.doi.org/10.1214/009117905000000792 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Stochastic Cahn-Hilliard equation with double singular nonlinearities and two reflections
We consider a stochastic partial differential equation with two logarithmic
nonlinearities, with two reflections at 1 and -1 and with a constraint of
conservation of the space average. The equation, driven by the derivative in
space of a space-time white noise, contains a bi-Laplacian in the drift. The
lack of the maximum principle for the bi-Laplacian generates difficulties for
the classical penalization method, which uses a crucial monotonicity property.
Being inspired by the works of Debussche, Gouden\`ege and Zambotti, we obtain
existence and uniqueness of solution for initial conditions in the interval
. Finally, we prove that the unique invariant measure is ergodic, and
we give a result of exponential mixing
Macroscopic fluctuations theory of aerogel dynamics
We consider the thermodynamic potential describing the macroscopic
fluctuation of the current and local energy of a general class of Hamiltonian
models including aerogels. We argue that this potential is neither analytic nor
strictly convex, a property that should be expected in general but missing from
models studied in the literature. This opens the possibility of describing in
terms of a thermodynamic potential non-equilibrium phase transitions in a
concrete physical context. This special behaviour of the thermodynamic
potential is caused by the fact that the energy current is carried by particles
which may have arbitrary low speed with sufficiently large probability.Comment: final versio
Algebraic renormalisation of regularity structures
We give a systematic description of a canonical renormalisation procedure of stochastic PDEs containing nonlinearities involving generalised functions. This theory is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of their group of automorphisms. This subgroup is sufficiently large to be able to implement a version of the BPHZ renormalisation prescription in this context. This is in stark contrast to previous works where one considered regularity structures with a much smaller group of automorphisms, which lead to a much more indirect and convoluted construction of a renormalisation group acting on the corresponding space of admissible models by continuous transformations. Our construction is based on bialgebras of decorated coloured forests in cointeraction. More precisely, we have two Hopf algebras in cointeraction, coacting jointly on a vector space which represents the generalised functions of the theory. Two twisted antipodes play a fundamental role in the construction and provide a variant of the algebraic Birkhoff factorisation that arises naturally in perturbative quantum field theory
Bismut-Elworthy-Li formulae for Bessel processes
In this article we are interested in the differentiability property of the Markovian semi-group corresponding to the Bessel processes of nonnegative dimension. More precisely, for all δ ≥ 0 and T > 0, we compute the derivative of the function x↦PδTF(x), where (Pδt)t≥0 is the transition semi-group associated to the δ-dimensional Bessel process, and F is any bounded Borel function on R+. The obtained expression shows a nice interplay between the transition semi-groups of the δ—and the (δ + 2)-dimensional Bessel processes. As a consequence, we deduce that the Bessel processes satisfy the strong Feller property, with a continuity modulus which is independent of the dimension. Moreover, we provide a probabilistic interpretation of this expression as a Bismut-Elworthy-Li formula
Hot scatterers and tracers for the transfer of heat in collisional dynamics
We introduce stochastic models for the transport of heat in systems described
by local collisional dynamics. The dynamics consists of tracer particles moving
through an array of hot scatterers describing the effect of heat baths at fixed
temperatures. Those models have the structure of Markov renewal processes. We
study their ergodic properties in details and provide a useful formula for the
cumulant generating function of the time integrated energy current. We observe
that out of thermal equilibrium, the generating function is not analytic. When
the set of temperatures of the scatterers is fixed by the condition that in
average no energy is exchanged between the scatterers and the system, different
behaviours may arise. When the tracer particles are allowed to travel freely
through the whole array of scatterers, the temperature profile is linear. If
the particles are locked in between scatterers, the temperature profile becomes
nonlinear. In both cases, the thermal conductivity is interpreted as a
frequency of collision between tracers and scatterers
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