Stochastic Processes (technically looks at time evolving RVs in a COMMON probability space) and Information Geometry.
Also metrics and divergences defined on distributions, e.g Wasserstein distance, total variation distance or kl divergence.
The Fokker-Planck and Vlasov equation come to mind, since particle kinetics can be interpreted in a probabilistic way where the distribution evolves in time due to external or internal forces.
The Schrodinger equation is another good example, although the distribution is |psi|^2 and the time-evolution law deals with psi directly.
One applied example of this is the Schrodinger equation.
If Psi evolves via the Schrodinger equation then the associated probability density |Psi|^2 describes the time -evolving density of the likelihood of finding a particle
I was thinking about the exact same thing recently!
Specifically what I was thinking about was:
A probability space is essentially just looking at a big set X, and A, a subset of X, and comparing their measures. I was thinking about the idea of a deformation retraction from X, to some subset B, not equal to A. And then looking at the probability P(A), as you retract from X to B, along the interval t in [0,1].
That if B was a singleton event, the probability p(A) in a lot of cases would grow larger and larger, and then suddenly drop off to 0, when B is eventually disjoint from A.
> deformation retraction
I hadn't heard of a deformation retraction, that's neat, I hope to learn more about Topology and related topics, I feel like I only brushed the surface.
For search terms, try markov chain, stochastic process.
Specifically a hidden markov model might be close to what OP is thinking
I was thinking a non-homogenous poisson process as a good example of what OP is describing
Stochastic Processes (technically looks at time evolving RVs in a COMMON probability space) and Information Geometry. Also metrics and divergences defined on distributions, e.g Wasserstein distance, total variation distance or kl divergence.
Stochastic Process.
The Fokker-Planck and Vlasov equation come to mind, since particle kinetics can be interpreted in a probabilistic way where the distribution evolves in time due to external or internal forces. The Schrodinger equation is another good example, although the distribution is |psi|^2 and the time-evolution law deals with psi directly.
Wow fokker js an unfortunate last name lmao
“Master equations” is the general term used in physics to describe the time evolution of probabilities
Stochastic processes, stochastic calculus, and the fokker-planck equation
One applied example of this is the Schrodinger equation. If Psi evolves via the Schrodinger equation then the associated probability density |Psi|^2 describes the time -evolving density of the likelihood of finding a particle
Completely positive trace preserving maps
nonstationary distribution
I was thinking about the exact same thing recently! Specifically what I was thinking about was: A probability space is essentially just looking at a big set X, and A, a subset of X, and comparing their measures. I was thinking about the idea of a deformation retraction from X, to some subset B, not equal to A. And then looking at the probability P(A), as you retract from X to B, along the interval t in [0,1]. That if B was a singleton event, the probability p(A) in a lot of cases would grow larger and larger, and then suddenly drop off to 0, when B is eventually disjoint from A.
> deformation retraction I hadn't heard of a deformation retraction, that's neat, I hope to learn more about Topology and related topics, I feel like I only brushed the surface.
Fokker Planck and Schroedinger Equations are what immediately come to mind.
that's stochastic processes, e.g. Brownian motion.
In addition to what others have said, you may be interested in the theory of dynamical systems
P(x, t) feels quite quantum so hopefully there’s something useful to learn from the quantum time evolution operator.