We investigate the Hölder regularity of the local time of the fractional Ornstein-Uhlenbeck process <svg style="vertical-align:-3.3907pt;width:120.475px;" id="M1" height="20.237499" version="1.1" viewBox="0 0 120.475 20.237499" width="120.475" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,15.95)"><path id="x1D44B" d="M775 650l-6 -28q-60 -6 -81.5 -16t-61.5 -54l-175 -191l125 -243q30 -58 48.5 -71t82.5 -19l-5 -28h-275l7 28l35 4q31 4 37 12t-6 34l-108 216q-140 -165 -177 -219q-16 -22 -10.5 -30.5t41.5 -13.5l22 -3l-7 -28h-244l8 28q52 4 75 15.5t67 52.5q48 46 206 231
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q0 -37 -4 -106.5t-4 -98.5q0 -18 2 -30t10 -27t28 -23.5t51 -10.5v-29z" /></g> </svg>. As a related problem, we study the collision local time of two independent fractional Ornstein-Uhlenbeck <svg style="vertical-align:-3.59802pt;width:126.3625px;" id="M2" height="21.4" version="1.1" viewBox="0 0 126.3625 21.4" width="126.3625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,16.863)"><use xlink:href="#x1D44B"/></g> <g transform="matrix(.012,-0,0,-.012,13.525,8.7)"><use xlink:href="#x1D43B"/></g> <g transform="matrix(.008,-0,0,-.008,23.963,11.563)"><path id="x1D456" d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30
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