Consider the special case where $a\equiv 0$ and $b\equiv 1\,.$ Then $x(t)$ is a standard Brownian motion. With reflection condition at zero this process has the transition density $$ p(t,x,y)=\frac{1}{\sqrt{2\pi t}}\left\{\exp\left(-\frac{(x-y)^2}{2t}\right)+\exp\left(-\frac{(x+y)^2}{2t}\right)\right\}\, $$ which solves the heat equation $$ \frac{\partial p}{\partial t}=-\frac{\partial^2 p}{\partial x^2}\,. $$ According to exercise 2.8.4 of [1], this is the transition density of the modulus of Brownian motion $|x(t)|\,.$

In other words: the trajectory of the reflected particle is $|x(t)|\,.$

The picture in [2] is also very enlightning.

[1] I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus. Springer Graduate Texts in Mathematics. 1991.

[2] Wikipedia Reflection principle (Wiener process).

Consider the special case where $a\equiv 0$ and $b\equiv 1\,.$ Then $x(t)$ is a standard Brownian motion. With reflection condition at zero this process has the transition density $$ p(t,x,y)=\frac{1}{\sqrt{2\pi t}}\left\{\exp\left(-\frac{(x-y)^2}{2t}\right)+\exp\left(-\frac{(x+y)^2}{2t}\right)\right\}\, $$ which solves the heat equation $$ \frac{\partial p}{\partial t}=\frac{1}{2}\frac{\partial^2 p}{\partial x^2}\,. $$ According to exercise 2.8.4 of [1], this is the transition density of the modulus of Brownian motion $|x(t)|\,.$

In other words: the trajectory of the reflected particle is $|x(t)|\,.$

The picture in [2] is also very enlightning.

[1] I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus. Springer Graduate Texts in Mathematics. 1991.

[2] Wikipedia Reflection principle (Wiener process).