Abstract
We consider an infinite system of coupled stochastic differential equations (SDE) describing
dynamics of the following infinite particle system. Each particle is characterised by its
position x ∈ Rd and internal parameter (spin) σx ∈ R. While the positions of particles form
a fixed ("quenched") locally-finite set (configuration) γ ⊂ Rd, the spins σx and σy interact
via a pair potential whenever |x−y| < ρ, where ρ > 0 is a fixed interaction radius. The
number nx of particles interacting with a particle in position x is finite but unbounded in x.
The growth of nx as |x|→∞ creates a major technical problem for solving our SDE system.
To overcome this problem, we use a finite volume approximation combined with a version
of the Ovsjannikov method, and prove the existence and uniqueness of the solution in a
scale of Banach spaces of weighted sequences. As an application example, we construct
stochastic dynamics associated with Gibbs states of our particle system.
dynamics of the following infinite particle system. Each particle is characterised by its
position x ∈ Rd and internal parameter (spin) σx ∈ R. While the positions of particles form
a fixed ("quenched") locally-finite set (configuration) γ ⊂ Rd, the spins σx and σy interact
via a pair potential whenever |x−y| < ρ, where ρ > 0 is a fixed interaction radius. The
number nx of particles interacting with a particle in position x is finite but unbounded in x.
The growth of nx as |x|→∞ creates a major technical problem for solving our SDE system.
To overcome this problem, we use a finite volume approximation combined with a version
of the Ovsjannikov method, and prove the existence and uniqueness of the solution in a
scale of Banach spaces of weighted sequences. As an application example, we construct
stochastic dynamics associated with Gibbs states of our particle system.
| Original language | English |
|---|---|
| Article number | 023508 |
| Number of pages | 25 |
| Journal | Journal of Mathematical Physics |
| Volume | 66 |
| Issue number | 2 |
| Early online date | 18 Feb 2025 |
| DOIs | |
| Publication status | E-pub ahead of print - 18 Feb 2025 |
Bibliographical note
© Author(s) 2025Keywords
- interacting particle systems
- infinite systems of stochastic equations
- scale of Banach spaces,
- Ovsjannikov’s method
- dissipativity.