List process for current user

This is faster, one line, doesn't require admin.

Get-Process | ? {$_.SI -eq (Get-Process -PID $PID).SessionId}

You can do that through WMI. Here is an excerpt of an article you can find here.

$View = @(
 @{l='Handles';e={$_.HandleCount}},
 @{l='NPM(K)';e={ (Get-Process -Id $_.ProcessId).NonpagedSystemMemorySize/1KB -as [int]}},
 @{l='PM(K)';e={ $_.PrivatePageCount/1KB -as [int]}},
 @{l='WS(K)';e={ $_.WorkingSetSize/1KB -as [int]}},
 @{l='VM(M)';e={ $_.VirtualSize/1mB -as [int]}},
 @{l='CPU(s)';e={ (Get-Process -Id $_.ProcessId).CPU -as [int]}},
 @{l='Id';e={ $_.ProcessId}},
 'UserName'
 @{l='ProcessName';e={ $_.ProcessName}}
)
Get-WmiObject Win32_Process | % { $_ | 
    Add-Member -MemberType ScriptProperty -Name UserName -Value {
        '{0}\{1}' -f $this.GetOwner().Domain,$this.GetOwner().User
    } -Force -PassThru
} | ? UserName -match $env:USERNAME | ft $View -AutoSize

Get-Process alone will not give you this information, you'll need WMI:

$owners = @{}
gwmi win32_process |% {$owners[$_.handle] = $_.getowner().user}
$ps = get-process | select processname,Id,@{l="Owner";e={$owners[$_.id.tostring()]}}
foreach($p in $ps) {
    if($p.Owner -eq $env:USERNAME) {
        $p
    }
}