#politics news is getting on top of me today. To calm down I watched some videos explaining how to implement #stochasticdifferentialequations in #julialang.

Thanks, #SciML -- I needed a break.

in many ways, #mathematical #physics is based on the #smoothing properties of the #integral

#integrableSystems #mathematicalPhysics #complexAnalysis #spectralTheory #harmonicAnalysis #functionalAnalysis #mathematicalAnalysis #differentialEquations #ODEs #PDEs #SDEs #DEs #equations #BrownianMotion #LangevinDynamics #dynamics #Langevin #StochasticDifferentialEquations #StochasticProcesses #WienerProcess #OrnsteinUhlenbeck #HarmonicOscillator #WaveEquation #Newton #Newtonian #Maxwell #Einstein

After attending a seminar by Pete Ashwin (Exeter) several weeks ago, quasipotentials (in the sense of Freidlin and Wentzell) have been on a mind.

Consider a gradient system, that is, a system of the form dx/dt = ∇V(x). Minima of the potential are attractors for this system. If we now add some noise, the equation becomes the stochastic differential equation dxₜ = ∇V(x) dt + σ dWₜ. The noise can push the system from one minimum to another. Large deviation theory gives a formula for the expected time that we have to wait for this to happen.

But what if we don't start with a gradient system? The equation is now dxₜ = f(x) dt + σ dWₜ for general f. The same formula holds if we replace the potential V with the so-called pseudopotential. The pseudopotential measures how much you have to go against the motion of the noise-free system dx/dt = f(t). Unfortunately, the pseudopotential is not so easy to compute and this is what I have been thinking about.

For a precise statement and references, maybe look at the Wikipedia page https://en.wikipedia.org/wiki/Freidlin%E2%80%93Wentzell_theorem (I don't know the topic well enough to understand the details).

#StochasticDifferentialEquations #LargeDeviationsTheory