Wong–Zakai approximation and support theorem for semilinear stochastic partial differential equations with finite dimensional noise in the whole space

2020 ◽  
Author(s):  
Timur Yastrzhembskiy
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Partial Differential ◽  
Support Theorem ◽  
Finite Dimensional ◽  
Whole Space

scholarly journals Variational principles for stochastic soliton dynamics

2016 ◽  
Vol 472 (2187) ◽  
pp. 20150827 ◽  
Author(s):  
Darryl D. Holm ◽  
Tomasz M. Tyranowski
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Variational Principles ◽  
Soliton Solutions ◽  
Stochastic Modelling ◽  
Partial Differential ◽  
Stochastic Perturbations ◽  
Finite Dimensional ◽  
Stochastically Perturbed

We develop a variational method of deriving stochastic partial differential equations whose solutions follow the flow of a stochastic vector field. As an example in one spatial dimension, we numerically simulate singular solutions (peakons) of the stochastically perturbed Camassa–Holm (CH) equation derived using this method. These numerical simulations show that peakon soliton solutions of the stochastically perturbed CH equation persist and provide an interesting laboratory for investigating the sensitivity and accuracy of adding stochasticity to finite dimensional solutions of stochastic partial differential equations. In particular, some choices of stochastic perturbations of the peakon dynamics by Wiener noise (canonical Hamiltonian stochastic deformations, CH-SD) allow peakons to interpenetrate and exchange order on the real line in overtaking collisions, although this behaviour does not occur for other choices of stochastic perturbations which preserve the Euler–Poincaré structure of the CH equation (parametric stochastic deformations, P-SD), and it also does not occur for peakon solutions of the unperturbed deterministic CH equation. The discussion raises issues about the science of stochastic deformations of finite-dimensional approximations of evolutionary partial differential equation and the sensitivity of the resulting solutions to the choices made in stochastic modelling.

THE MONOTONICITY INEQUALITY FOR LINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

2009 ◽  
Vol 12 (04) ◽  
pp. 575-591 ◽  
Author(s):  
L. GAWARECKI ◽  
V. MANDREKAR ◽  
B. RAJEEV
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Evolution Equations ◽  
Differential Operators ◽  
Probabilistic Representation ◽  
Partial Differential ◽  
Finite Dimensional ◽  
Monotonicity Inequality ◽  
Elliptic Differential Operators

We prove the monotonicity inequality for differential operators A and L that occur as coefficients in linear stochastic partial differential equations associated with finite-dimensional Itô processes. We characterize the solutions of such equations. A probabilistic representation is obtained for solutions to a class of evolution equations associated with time dependent, possibly degenerate, second-order elliptic differential operators.

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