Page 572 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 572
STOCHASTIC EVOLUTION EQUATIONS (MS-68)
On a construction of martingale solutions of SPDEs
Martin Ondreját, ondrejat@utia.cas.cz
Institute of Information Theory and Automation, v.v.i.,
Czech Academy of Sciences, Czech Republic
We will discuss a simplified method of proving existence of martingale solutions to SPDEs
based on a compactness argument but not using the Skorokhod representation theorem. The
method works both for theoretical and numerical approximations of solutions.
Weighted Energy–Dissipation principle for nonlinear stochastic evolution
equations
Luca Scarpa, luca.scarpa@univie.ac.at
University of Vienna, Italy
Coauthor: Ulisse Stefanelli
We present the Weighted Energy–Dissipation (WED) principle for nonlinear stochastic evolu-
tion equations in variational form. The approach consists in minimizing suitable convex WED
functionals, defined on spaces of entire trajectories, and depending on an approximation pa-
rameter. The corresponding Euler–Lagrange equation is characterized as an elliptic-in-time
regularization of the original problem, which can be equivalently seen as a forward–backward
nonlinear stochastic evolution system. Finally, WED minimizers are shown to converge to the
solution of the original nonlinear evolution equation as the approximation parameter vanishes.
This study is based on a joint work with Ulisse Stefanelli (University of Vienna, Austria).
Local characteristics and tangent martingales in Banach spaces
Ivan Yaroslavtsev, yaroslavtsev.i.s@yandex.ru
Max Planck Institute for Mathematics in the Sciences, Germany
Let L be a real-valued Lévy martingale. Then we know that by the Lévy-Khinchin formula
EeiθLt = exp t −1σ2θ2 + eiθx − 1 − iθxν(dx) , t ≥ 0, θ ∈ R,
2
R
where σ ≥ 0 is responsible for the continuous part of L and the measure ν on R in responsible
for the discontinuous part of L, i.e. if we decompose L = W + N into a sum of a Brownian
motion W and a purely discontinuous Poisson martingale N , then the distributions of W and
N are uniquely determined by σ and ν respectively.
Any real-valued martingale M has a analogue of (σ, ν) which is called the local charac-
teristics of M and which is defined as the pair ([M c], νM ) of a quadratic variation [M c] of the
continuous part M c of M and the compensator νM of the jump measure of M . The local char-
acteristic have been defined and intensively studied in a number of works by Jacod, Kallenberg,
Kwapien´, Shiryaev, and Woyczyn´ski, and in particular it turned out that the local character-
istics uniquely determine the distribution of the corresponding martingale if and only if they
are constant. Therefore the notion of tangent martingales (i.e. martingales with the same local
characteristics) was introduced. In 2017 Kallenberg have shown sharp Lp bounds for tangent
570
On a construction of martingale solutions of SPDEs
Martin Ondreját, ondrejat@utia.cas.cz
Institute of Information Theory and Automation, v.v.i.,
Czech Academy of Sciences, Czech Republic
We will discuss a simplified method of proving existence of martingale solutions to SPDEs
based on a compactness argument but not using the Skorokhod representation theorem. The
method works both for theoretical and numerical approximations of solutions.
Weighted Energy–Dissipation principle for nonlinear stochastic evolution
equations
Luca Scarpa, luca.scarpa@univie.ac.at
University of Vienna, Italy
Coauthor: Ulisse Stefanelli
We present the Weighted Energy–Dissipation (WED) principle for nonlinear stochastic evolu-
tion equations in variational form. The approach consists in minimizing suitable convex WED
functionals, defined on spaces of entire trajectories, and depending on an approximation pa-
rameter. The corresponding Euler–Lagrange equation is characterized as an elliptic-in-time
regularization of the original problem, which can be equivalently seen as a forward–backward
nonlinear stochastic evolution system. Finally, WED minimizers are shown to converge to the
solution of the original nonlinear evolution equation as the approximation parameter vanishes.
This study is based on a joint work with Ulisse Stefanelli (University of Vienna, Austria).
Local characteristics and tangent martingales in Banach spaces
Ivan Yaroslavtsev, yaroslavtsev.i.s@yandex.ru
Max Planck Institute for Mathematics in the Sciences, Germany
Let L be a real-valued Lévy martingale. Then we know that by the Lévy-Khinchin formula
EeiθLt = exp t −1σ2θ2 + eiθx − 1 − iθxν(dx) , t ≥ 0, θ ∈ R,
2
R
where σ ≥ 0 is responsible for the continuous part of L and the measure ν on R in responsible
for the discontinuous part of L, i.e. if we decompose L = W + N into a sum of a Brownian
motion W and a purely discontinuous Poisson martingale N , then the distributions of W and
N are uniquely determined by σ and ν respectively.
Any real-valued martingale M has a analogue of (σ, ν) which is called the local charac-
teristics of M and which is defined as the pair ([M c], νM ) of a quadratic variation [M c] of the
continuous part M c of M and the compensator νM of the jump measure of M . The local char-
acteristic have been defined and intensively studied in a number of works by Jacod, Kallenberg,
Kwapien´, Shiryaev, and Woyczyn´ski, and in particular it turned out that the local character-
istics uniquely determine the distribution of the corresponding martingale if and only if they
are constant. Therefore the notion of tangent martingales (i.e. martingales with the same local
characteristics) was introduced. In 2017 Kallenberg have shown sharp Lp bounds for tangent
570
