# How to compare pathwise convergence and convergence in probability

This question was asked quite sometime back in mathexchange and deleted, as it was downvoted, asked again but never got an answer. So I am asking here.

Motivation: It appears pathwise convergence can have a distinct advantage in the area of signal processing (I work on), specifically, we can improve filter operation https://en.wikipedia.org/wiki/Digital_filter. In fact I have written some matlab codes and applied it in real data, collected from sensor, and got some very interesting result while performing beamforming. But not sure how to prove the advantage of the algorithm mathematically. As I understand the pathwise convergence is possible through a graded vector spaces or direct sum of Bannach spaces. My algorithm is based on the functional on the element of individual Bannach spaces. I hope this makes sense.

EDIT: Motivation As I understand the difference between convergence in probability is more like global convergence and pathwise is like of local convergence. I am assuming that patwise convergence method gives some local infomation which is not there in the other methods which gives probability wise convergence. If this is true the local information must be embeded in the iterated integrals in case of rough path and in the model in regularity structure. I tried to extract these additional information (assuming it exists) from the model, which is also termed as signature in rough path literature.

Rough path theory which is a method to solve stochastic differential equation claims pathwise converse of the solution in contrast to Ito's integral and Stratonovich integral which claims convergence in probability. In addition regularity structure was introduced later to extend the method to solve stochastic partial differential equation. I understand that these methods works only under certain conditions. But my question is what are the differences we can obtain from a method which gives pathwise convergence compare to convergence in probability?

Edit: People are voting to close, without a comment, few comments would be appreciated.

Consider we are finding solution of the same problem by two different methods, A and B. Method A gives path-wise convergence and method B gives convergence in probability. I am trying to understand what can we infer from the solution of method A and solution of method B? Is there any different information we can infer from A compare to B? Can we somehow compare the solution (what we can learn) from A and B?

EDIT:The corresponidng equations showing probability wise convergence and pathwise convergence under the same setting would be highly informative. Would someone please help me.

Edit: The convergence referred specifically to PDE/SPDE. Really sorry if this question is annoying.

To elaborate the above an over-simplified example given below (but it might express the view). Consider we have stochastic process with $$n$$ points, each $$n$$ points is random variable. The "probability wise" convergence method would consider "characteristic" of each $$n$$ points. The "characteristic" is problem dependent and can be probability distribution of the variable at the points. On the other hand the path-wise convergence method would consider the characteristic of all $$n$$ points simultaneously. This is similar to the "model" in regularity structure and "signature" in the rough path theory.