This question on physics stackexchange https://physics.stackexchange.com/questions/12973/the-entropic-cost-of-tying-knots-in-polymers has a formulation which is perhaps more appropriate for this forum.

Given a Brownian motion for time t, link the ends to infinity by horizontal lines parallel to the x-axis, going in opposite directions. The walk will not intersect those lines generically, since a 2d random walk is marginally recurrent. You have then closed a loop on the one-point sphere compactification, and it makes sense to ask what knot you made.

There is a scaling problem, so that the knot you get might be very wild. But one can fix this by asking the right question in the limit. Approximate the Brownian motion with small randomly oriented straight line segments. Then, for long walks, the resulting knot will have a prime decomposition, and it is is very plausible to me that the number of prime knots of each type in the prime decomposition will converge to a fixed distribution in the limit of long walks.

Does this distribution exist?

Is there a more efficient method than simulation to get the distribution?