probability theory – Brownian motion first exit time from interval with deterministic structure

I’ve been working on the following problem from a survey paper. It says the following:

Let $(B_t: t geq 0)$ be a standard Brownian motion and $mu,$ a probability measure on $mathbb{R}$ such that $int_{mathbb{R}}xmu(dx) = 0.$ Then for any $lambda geq 0,$ define
$$-rho(lambda) = infleft{y in mathbb{R}: int_{mathbb{R}} {1}_{{x leq y} cup {x geq lambda}}x mu(dx) leq 0right}.$$
Let $R>0$ be a random variable independent of the Brownian motion. The distribution function of $R$ is $$mathbb{P}(R leq x) = int_{mathbb{R}} {1}_{y leq x}left(1+frac{y}{rho(y)}right)mu(dy).$$
If we set the stopping time $T = inf{t geq 0:B_t notin (-rho(R),R)},$ then the stopped process $B_T$ has also law $mu.$

Now, I know how to usually deal with this kind of exit times in the classical setting when for example $T = inf{t geq 0:B_t notin (-a,b)}.$ What I’ve tried so far is to write the following:

$$mathbb{P}(B_T geq z) = int_{z}^{infty} mathbb{P}(B_T geq z|R=x)mathbb{P}(R=x) = int_{z}^{infty} mathbb{P}(B_T geq z|R=x)left(1+ frac{x}{rho(x)}right)mu(dx).$$

I’m stuck on how to find $mathbb{P}(B_T geq z|R=x)$ as the end, after computing the integral, I should arrive at something like $int_{z}^{infty} mu(dx).$

Any help would be appreciated. Thank you!

unity – What is the standard practice for animating motion — move character or not move character?

I’ve downloaded a bunch of (free) 3d warriors with animations. I’ve noticed for about 25% of them, the ‘run’ animation physically moves the character forward in the z direction. For the other 75%, the animation just loops with the characters feet moving etc., but does so in place, without changing the character’s physical location.

I could fix this by:
1.) Manually updating the transform in code for this 75%, to physically move the character
2.) Alter the animation by re-recording it, with a positive z value at the end (when I did this it caused the character to shift really far away the rest of the units, probably something to do with local space vs world space I haven’t figured out yet).

But before I go too far down this rabbit hole, I wonder if there is any kind of standard? In the general case, are ‘run’ / ‘walk’ animations supposed to move the character themselves, or is it up to the coder to manually update the transform while the legs move and arms swing in place? Is one approach objectively better than the other, or maybe it depends on the use case? If so, what are the drawbacks of each? I know nothing about animation, so I don’t want to break convention (if there is one).

How to Make GameObject move in Circular Motion

I’ve been trying to work an enemy that moves in a circular motion for my RPG game, but for some reason, whenever I press play the GameObject instantly goes hundreds of units in the X and Y coordinates. It also move back to -1 on the Z axis. Here’s the script to my enemy’s movement:

using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class EnemyScript : MonoBehaviour

    private Rigidbody2D rb;

    float rotationRadius = 2f, angularSpeed = 2f;
    float posX, posY, angle = 0f;

    // Start is called before the first frame update
    void Start()
        // Gets the RigidBody2D component
        rb = GetComponent<Rigidbody2D>();

    // Update is called once per frame
    void Update()

    void Movement_1()
        posX = rb.position.x + Mathf.Cos(angle) + rotationRadius;
        posY = rb.position.y + Mathf.Sin(angle) + rotationRadius;
        transform.position = new Vector2(posX, posY);
        angle = angle + Time.deltaTime * angularSpeed;

        if (angle >= 360f)
            angle = 0f;

stochastic processes – Is this integral with respect a Brownian motion a Brownian motion?

Is there any theorem to see that $M= (M_t)_{tgeq 0}$ is a Brownian motion? Or should I check the sole definition? M is defined by
M_t = int_0^{e^t-1}frac{1}{sqrt{1+s}}d B_s

where $B = (B_t)_{tgeq 0}$ is a BM.

My attempt is to use Itô’s formula to $varphi(t,x) = frac{x}{sqrt{1+t}}$ to get that
frac{B_t}{sqrt{1+t}} = int_0^t frac{1}{sqrt{1+s}}dB_s -frac{1}{2} int_{0}^t B_s(1+s)^{-3/2}d s.

N_t = int_0^t frac{1}{sqrt{1+s}}dB_s

We see that
M_t = N_{e^t-1} = e^{-t/2}B_{e^t-1} + frac{1}{2}int_0^{e^t-1}B_s(1+s)^{-3/2}ds

Doing this, I’m struggling to see that $M_t-M_s sim M_{t-s}$. Am I in a correct way? Is there any other? Maybe Lévy’s theorem? Thanks

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python – Active Brownian Motion

I am attempting to write a Python code to simulate many particles in a confined box. These particles behave in such a way that they move in the box in straight lines with a slight angular noise (small changes in the direction of the particle path). They should interact by acknowledging the other particle and ‘shuffle/squeeze’ past each other and continue on their intended path, much like humans on a busy street. Eventually, the particles should cluster together when the density of particles (or packing fraction) reaches a certain value, but I haven’t got to this stage yet.

The code currently has particle interactions and I have attempted the angular noise but without success so far.

However, I have a feeling there are parts of the code that are inefficient or which could be either sped up or written more conveniently.

If anyone has any improvements for the code speed or ideas which may help with the interactions and/or angular noise that would be much appreciated. I will also leave an example of an animation which is my aim:

The above link shows the animation I am looking for, although I don’t need the sliders, just the box, and moving particles. The whole code is shown below:

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation

def set_initial_coordinates():
    x_co = (np.random.uniform(0, 2) for i in range(n_particles))
    y_co = (np.random.uniform(0, 2) for i in range(n_particles))
    return x_co, y_co

def set_initial_velocities():
    x_vel = np.array((np.random.uniform(-1, 1) for i in range(n_particles)))
    y_vel = np.array((np.random.uniform(-1, 1) for i in range(n_particles)))
    return x_vel, y_vel

def init():
    ax.set_xlim(-0.05, 2.05)
    ax.set_ylim(-0.07, 2.07)
    return ln,

def update(dt):
    xdata = initialx + vx * dt
    ydata = initialy + vy * dt
    fx = np.abs((xdata + 2) % 4 - 2)
    fy = np.abs((ydata + 2) % 4 - 2)

    for i in range(n_particles):
        for j in range(n_particles):
            if i == j:

            dx = fx(j) - fx(i)  # distance in x direction
            dy = fy(j) - fy(i)  # distance in y direction
            dr = np.sqrt((dx ** 2) + (dy ** 2))  # distance between x

            if dr <= r:
                force = k * ((2 * r) - dr)  # size of the force if distance is less than or equal to radius

                # Imagine a unit vector going from i to j
                x_comp = dx / dr  # x component of force
                y_comp = dy / dr  # y component of force

                fx(i) += -x_comp * force  # x force
                fy(i) += -y_comp * force  # y force
    ln.set_data(fx, fy)
    return ln,

# theta = np.random.uniform(0, 2) for i in range(n_particles)
n_particles = 10
initialx, initialy = set_initial_coordinates()
vx, vy = set_initial_velocities()
fig, ax = plt.subplots()
x_co, y_co = (), ()
ln, = plt.plot((), (), 'bo', markersize=15)  # radius 0.05
plt.xlim(0, 2)
plt.ylim(0, 2)

k = 1
r = 0.1
t = np.linspace(0, 10, 1000)

ani = FuncAnimation(fig, update, t, init_func=init, blit=True, repeat=False)

Instagram stories and local videos play in slow motion but sound is fine

A friend of mine started facing this problem yesterday night: when she browses through her Instagram stories, the video plays at a slower rate than normal, however the sound is totally fine. She went ahead and checked the gallery as well – same problem there. In TikTok though, there are no issues at all.

As she did not provide any more information (she is not tech savvy at all) here is the info that I know about her phone: it’s a Huawei Y5 2018, with Android 8.

I searched online about this, but I didn’t come up with anything related to Android. Any help is appreciated.

Motion camera app using video

I just started using Alfred camera which is a security camera app. I noticed in my Samsung note 9 that when I switch from photo to video the quality improves and location changes. Which tells me it is using another lenses since this one has several lenses. But I think Alfred camera app is only using the photo lense as the quality is bad and the placement seems to be coming from the camera lense.

Do apps like Alfred camera only access the camera lenses and not the video lense? Are there any apps that can access video lenses so quality is better? I need a mention sensor app that can tell me when there is motion. It does not have to do live view as it can just record motion and email or text me. Or do all these apps not access the better quality lenses?