𝒫=ItoProcess[\[DifferentialD]x[t]==-x[t]\[DifferentialD]t+σ \[DifferentialD]w[t],x[t],{x,x0},t,w\[Distributed]WienerProcess[]]

Table[ListLinePlot[RandomFunction[𝒫/.{x0->5},{0,4,0.01},10],PlotLabel->Row[{"σ","==",N@σ}]],{σ,{1/4,1/2,1,2}}]

Mean[𝒫[t]]

x[t]/.DSolve[{x'[t]==-x[t],x[0]==x0},x[t],t]

Variance[𝒫[t]]

PDF[𝒫[t],x]